Sizes of Infinity
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Sizes of Infinity
I have a question about the sizes of Infinity. I understand that the set of all decimals is larger than the set of all natural numbers. But I've heard two conflicting arguments about the size of all Integers vs the size of all Natural Numbers (or Positive Integers). Some say that there is a Bijection between all Integers and all Positive Integers (is there a difference between Positive Integers and Natural Numbers?), but just can't see how that is. It just seems natural to me that the set of all Integers would be larger than the set of all Positive Integers, thus being a larger infinity (as Georg Cantor might say) but I've been told otherwise by some. Am I missing something here, or are the those who've told me otherwise wrong?
I'm not the most mathematically knowledgable person, so I would not be surprised if I'm completely off here in some way. Let me know!
I'm not the most mathematically knowledgable person, so I would not be surprised if I'm completely off here in some way. Let me know!
Re: Sizes of Infinity
I guess the quickest thing I came up with offhand is this:
f(n) = 2n for n<0
f(n) = 2n+1 for n>0 or n=0
This maps the positive integers (and 0) to the odd positive integers and the negative integers to the even positive integers thus all integers are mapped to the set of positive integers and viceversa (a bijection).
The key thing here is to realize that there are different ways to compare the size of sets. As you are aware, one definition mathematicians take is that two sets have the same "size" if there is a bijection between those two sets. I have just provided a bijection between the positive integers and all of the integers so by that definition the two sets have the same "size".
However, another definition of "size" you might take is that one set A is "bigger" than set B if B is a subset of A and A contains elements that aren't in B. i.e. if A has everything that B has and then some we might say that A is "bigger" than B. By this definition you would say the set of all integers is "bigger" than the set of positive integers because the set of integers contains all positive integers AND nonpositive integers.
So you see that it all depends on what you take as your definition of "size" and furthermore that you get a different answer depending on which definition you choose! This idea of definitions is very important to mathematicians. It is very important in mathematics that you are clear what definition you are using for ANYTHING you say. When people are talking about sizes of infinity they are almost certainly talking about this 'bijection' notion of infinity. Have a look at cardinality if you are more interested.
edit: rewording
f(n) = 2n for n<0
f(n) = 2n+1 for n>0 or n=0
This maps the positive integers (and 0) to the odd positive integers and the negative integers to the even positive integers thus all integers are mapped to the set of positive integers and viceversa (a bijection).
The key thing here is to realize that there are different ways to compare the size of sets. As you are aware, one definition mathematicians take is that two sets have the same "size" if there is a bijection between those two sets. I have just provided a bijection between the positive integers and all of the integers so by that definition the two sets have the same "size".
However, another definition of "size" you might take is that one set A is "bigger" than set B if B is a subset of A and A contains elements that aren't in B. i.e. if A has everything that B has and then some we might say that A is "bigger" than B. By this definition you would say the set of all integers is "bigger" than the set of positive integers because the set of integers contains all positive integers AND nonpositive integers.
So you see that it all depends on what you take as your definition of "size" and furthermore that you get a different answer depending on which definition you choose! This idea of definitions is very important to mathematicians. It is very important in mathematics that you are clear what definition you are using for ANYTHING you say. When people are talking about sizes of infinity they are almost certainly talking about this 'bijection' notion of infinity. Have a look at cardinality if you are more interested.
edit: rewording
Re: Sizes of Infinity
Another way to think of the integers as being the same size as the naturals is to note that you can list the integers like: 0, 1, 1, 2, 2, 3, 3, and so on.
The fact that the integers have the same cardinality as the naturals despite the latter being a proper subset of the former is not unusual at all. In fact, every infinite set has a proper subset with the same cardinality, and this is sometimes taken to be the definition of an infinite set.
The fact that the integers have the same cardinality as the naturals despite the latter being a proper subset of the former is not unusual at all. In fact, every infinite set has a proper subset with the same cardinality, and this is sometimes taken to be the definition of an infinite set.
Re: Sizes of Infinity
Regarding "positive integers" and "natural numbers", the the latter isn't really universally defined as to whether or not it includes 0. Some will say it doesn't so there's no difference, and others will say it does include 0 so there's that small difference (although that difference doesn't change the cardinality of course).
It's often preferable to just say "positive integers" if you mean 1, 2, 3, ... and "nonnegative" integers if you mean 0, 1, 2, ... in order to limit potential ambiguity.
It's often preferable to just say "positive integers" if you mean 1, 2, 3, ... and "nonnegative" integers if you mean 0, 1, 2, ... in order to limit potential ambiguity.
Re: Sizes of Infinity
Dopefish wrote:Regarding "positive integers" and "natural numbers", the the latter isn't really universally defined as to whether or not it includes 0. Some will say it doesn't so there's no difference, and others will say it does include 0 so there's that small difference (although that difference doesn't change the cardinality of course).
It's often preferable to just say "positive integers" if you mean 1, 2, 3, ... and "nonnegative" integers if you mean 0, 1, 2, ... in order to limit potential ambiguity.
*shrug* I was taught in college that 0 is both positive and negative. We used the term "strictly positive integers" to emphasize that we didn't want to include 0.
Re: Sizes of Infinity
Tirian wrote:*shrug* I was taught in college that 0 is both positive and negative.
This is an... unconventional definition. Typically, 0 is taken to be neither positive nor negative. Then, the positive integers are those greater than 0, and the nonnegative integers are those greater than or equal to 0. You can use your definition if you want, but you should know that people will be confused by it.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
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 jestingrabbit
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Re: Sizes of Infinity
Cauchy wrote:Tirian wrote:*shrug* I was taught in college that 0 is both positive and negative.
This is an... unconventional definition. Typically, 0 is taken to be neither positive nor negative. Then, the positive integers are those greater than 0, and the nonnegative integers are those greater than or equal to 0. You can use your definition if you want, but you should know that people will be confused by it.
Strongly agree.
@OP Another set that is even more surprisingly countable is the set of rational numbers. The way to see this is to imagine placing the rational numbers at grid points ie associate p/q (in lowest terms) with the point (p, q). You can then imagine putting them in a list where the ones closest to the origin are listed first, and the ones further away are listed later. You can pretty easily prove that the number of grid points that have a fixed distance from the origin are finite, so the list makes sense. [You can even specify the order completely, by requiring that denominators be postive, and that p/q is before r/s iff p^2 + q^2 < r^2 + s^2 or p^2 + q^2 = r^2 + s^2 and arctan(p/q) < arctan(r/s)].
Other countable sets: countable unions of coutable sets, the finite subsets of a countable set, a product of n countable sets for any finite n.
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Re: Sizes of Infinity
jestingrabbit wrote:Other countable sets: countable unions of coutable sets
I'm nitpicking here, but that is not necessarily true, it is consistent with ZF that ℝ is a countable union of countable sets, you need (at least) AC_{ω} to prove that the countable union of countable sets is countable.
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Re: Sizes of Infinity
Twistar wrote:I guess the quickest thing I came up with offhand is this:
f(n) = 2n for n<0
f(n) = 2n+1 for n>0 or n=0
This maps the positive integers (and 0) to the odd positive integers and the negative integers to the even positive integers thus all integers are mapped to the set of positive integers and viceversa (a bijection).
The key thing here is to realize that there are different ways to compare the size of sets. As you are aware, one definition mathematicians take is that two sets have the same "size" if there is a bijection between those two sets. I have just provided a bijection between the positive integers and all of the integers so by that definition the two sets have the same "size".
Alright, it took a bit of rereading to wrap my head around this. But it makes sense (since, as a infinite set there will always be another item to which it can be matched) even if it does sort of feel like cheating to me. It could very well be my own lack of understanding, but this seems to presuppose that all countable infinities are the same size, which I have a hard time accepting. It seems that our limited ability to conceptualize infinity leads us to just say that all countable infinities are the same size. But as I said, that's probably a personal problem.
Twistar wrote:However, another definition of "size" you might take is that one set A is "bigger" than set B if B is a subset of A and A contains elements that aren't in B. i.e. if A has everything that B has and then some we might say that A is "bigger" than B. By this definition you would say the set of all integers is "bigger" than the set of positive integers because the set of integers contains all positive integers AND nonpositive integers.
This also makes sense, but also feels like cheating since: although I could say that ABCD is a subset of ABCD, I don't think anyone would be able to say that one is really bigger than the other.
Nyktos wrote:In fact, every infinite set has a proper subset with the same cardinality, and this is sometimes taken to be the definition of an infinite set.
That's an interesting definition for infinite sets, that I can't say I've ever heard. But I kind of like it.
jestingrabbit wrote:Another set that is even more surprisingly countable is the set of rational numbers.
This, I've actually had shown to me to my satisfaction (using a grid), before. (Thanks to Numberphile)
Thanks for all your responses everyone. I have to say that it still hasn't precisely clicked (though the arguments make sense) but there's still something in me that doesn't want to accept this. I have a very hard time believing that set ℤ, which contains all of the elements of set ℕ PLUS others which are not in set ℕ, can be the same size as set ℕ. It just doesn't seem to jive with my way of understanding the world. Though that probably just means I need to stretch my brain more to grasp it.
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Re: Sizes of Infinity
0x783czar wrote:It could very well be my own lack of understanding, but this seems to presuppose that all countable infinities are the same size, which I have a hard time accepting. It seems that our limited ability to conceptualize infinity leads us to just say that all countable infinities are the same size. But as I said, that's probably a personal problem.
All countable infinities are the same size, specifically the size of the natural numbers. That's what makes them countable in the first place  you can count them, i.e. correspond one element from them to each natural number.
Twistar wrote:However, another definition of "size" you might take is that one set A is "bigger" than set B if B is a subset of A and A contains elements that aren't in B. i.e. if A has everything that B has and then some we might say that A is "bigger" than B. By this definition you would say the set of all integers is "bigger" than the set of positive integers because the set of integers contains all positive integers AND nonpositive integers.
This also makes sense, but also feels like cheating since: although I could say that ABCD is a subset of ABCD, I don't think anyone would be able to say that one is really bigger than the other.
FWIW, I've highlighted the parts of Twistar's post you might've missed. ABCD is not greater than itself because it doesn't contain elements that are missing from itself.
Thanks for all your responses everyone. I have to say that it still hasn't precisely clicked (though the arguments make sense) but there's still something in me that doesn't want to accept this. I have a very hard time believing that set ℤ, which contains all of the elements of set ℕ PLUS others which are not in set ℕ, can be the same size as set ℕ. It just doesn't seem to jive with my way of understanding the world. Though that probably just means I need to stretch my brain more to grasp it.
This is actually a very old problem  it even tripped up Galileo, who posed the same question except with the squares rather than the negative numbers. If you can see why the set of perfect squares is the same size as the set of natural numbers, then you only have to do the "same" leap of reasoning again to see why the set of all integers is the same size as the naturals.
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 jestingrabbit
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Re: Sizes of Infinity
0x783czar wrote:Thanks for all your responses everyone. I have to say that it still hasn't precisely clicked (though the arguments make sense) but there's still something in me that doesn't want to accept this. I have a very hard time believing that set ℤ, which contains all of the elements of set ℕ PLUS others which are not in set ℕ, can be the same size as set ℕ. It just doesn't seem to jive with my way of understanding the world. Though that probably just means I need to stretch my brain more to grasp it.
This is actually a property of all infinite sets: a set is infinite iff it has a proper subset with the same cardinality.
It might also be worth mentioning that cardinality is only one concept of size. Another is a 'mean'. You calculate the the mean of A subset of Z, m(A), by setting
This then gives you results like "half the integers are even", "half the integers are positive", "a negligible amount of the integers are squares" etc which agree with the sort of intuition that you're bringing up.
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Re: Sizes of Infinity
0x783czar wrote:I have to say that it still hasn't precisely clicked (though the arguments make sense) but there's still something in me that doesn't want to accept this. I have a very hard time believing that set ℤ, which contains all of the elements of set ℕ PLUS others which are not in set ℕ, can be the same size as set ℕ. It just doesn't seem to jive with my way of understanding the world. Though that probably just means I need to stretch my brain more to grasp it.
Consider the nonnegative integers, ℕ={0, 1, 2, 3, ...}, and the even nonnegative integers, which I'll call 𝕊={0, 2, 4, 6, ...}. 𝕊 is clearly a proper subset of ℕ, in that only even elements of ℕ are in 𝕊. However, if we take all the elements of ℕ and double them, we get 𝕊, so they must be of the same size.
Taking only the even elements of ℕ, we get something like this:
ℕ, 𝕊
0, 0
1
2, 2
3
4, 4
5
6, 6
...
𝕊 is clearly a proper subset of ℕ, and so intuitively feels smaller (as you already know).
Doubling all the elements of ℕ, we get something like this:
ℕ, 𝕊
0, 0
1, 2
2, 4
3, 6
4, 8
5, 10
6, 12
...
In other words, 𝕊={2n: n∊ℕ}, which just means it's all the elements of ℕ doubled, and so intuitively feels the same size, doesn't it?
And now take only those elements of 𝕊 that are multiples of four, 𝕋={0, 4, 8, 12, 16, ...}, so that you've clearly got a proper subset of 𝕊 that intuitively feels smaller than 𝕊. Divide the elements of 𝕋 by four, to get ℕ. ℕ should intuitively feel the same size as 𝕋, since it's all the same elements divided by four. So ℕ should now intuitively feel smaller than 𝕊 itself. Yeah?
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 Forest Goose
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Re: Sizes of Infinity
Another problem with the notion of using subsets, is that it falls apart in "obvious" cases where you aren't dealing with exact subsets, and that comparisons that fit intuition aren't usually allowed:
1.) Are the positive numbers larger than the evens and {1} together?
2.) Are the negative numbers larger than the positive evens?
In the first case, the obvious answer seems to be that adding one more element cannot make up for the infinity of missing positives, but there is no reasonable proof of this unless you just state it, which seems ad hoc, or add some form of putting them on equal footing...like bijections, which undermines the whole point of using something else.
In the second case, intuition wants to say positives and negatives are the same size, so yes, since positives > positive evens; but, again, you are stuck, and cannot easily justify that step.
There's a lot of other weird edge cases that can come up, especially once you move to bigger sets than the naturals (in normal terms of cardinality)  you either end up with a really really ad hoc system of rules, have to admit that you can't cover most cases, or have to specialize to sets of a certain size (which is hard to specify since you can't just say "countable", as that is the whole point, to not use that definition).
1.) Are the positive numbers larger than the evens and {1} together?
2.) Are the negative numbers larger than the positive evens?
In the first case, the obvious answer seems to be that adding one more element cannot make up for the infinity of missing positives, but there is no reasonable proof of this unless you just state it, which seems ad hoc, or add some form of putting them on equal footing...like bijections, which undermines the whole point of using something else.
In the second case, intuition wants to say positives and negatives are the same size, so yes, since positives > positive evens; but, again, you are stuck, and cannot easily justify that step.
There's a lot of other weird edge cases that can come up, especially once you move to bigger sets than the naturals (in normal terms of cardinality)  you either end up with a really really ad hoc system of rules, have to admit that you can't cover most cases, or have to specialize to sets of a certain size (which is hard to specify since you can't just say "countable", as that is the whole point, to not use that definition).
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Re: Sizes of Infinity
alessandro95 wrote:jestingrabbit wrote:Other countable sets: countable unions of coutable sets
I'm nitpicking here, but that is not necessarily true, it is consistent with ZF that ℝ is a countable union of countable sets, you need (at least) AC_{ω} to prove that the countable union of countable sets is countable.
I think this deserves a bit more discussion. Notably, I am pretty sure that we only need the ability to find a choice function on a countable collection of sets whose cardinality is ω^{ω} (though I’m not quite sure what it takes to prove that ω^{ω}=2^{ω}).
Given a countable collection of sets, there exists a bijection from the positive integers to that collection (by definition of being countable). So the set of such bijections—call it J—is nonempty. We can pick an element j of J directly without any form of choice because we only need one element from one set. So j provides an index of the countable sets we are unioning. For notation, let A_{k}=j(k) be those sets.
For each A_{k}, the set of bijections from the positive integers to that set is nonempty, and we can call it J_{k}. We need to pick one representative j_{k} from each J_{k}, which requires the axiom of countable choice for sets of cardinality q=J_{k}. But q is just the number of permutations of the positive integers, which has cardinality ω^{ω}—or at least its cardinality is no larger than that, because ω^{ω} is by definition the number of functions from the positive integers to themselves, and we are only counting bijective functions therebetween.
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Re: Sizes of Infinity
Qaanol wrote:I think this deserves a bit more discussion.
I am completely selftaught when it comes to set theory (and maths in general for that matter) so I'm still halfway through understanding your post and I'll update this post afterward, but my "at least" was meant to emphasize the fact that we don't need AC, I'm actually not sure whether results weaker than AC_{ω} are enough.
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 Forest Goose
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Re: Sizes of Infinity
I return from an early Christmas party a bit past my Ballmer Peak, apologies on any stupidity.
Countable unions of countables is countable does not imply countable choice:
countable choice implies (dedekind finite iff finite) in ZF. In the basic Fraenkel Model, the result on unions is valid, the result about dfin iff fin is not (more about the latter in comb, and the former in cons). Using Pincus's transfer theorem, there is a model of ZF with the same. Thus, if the result on unions implies cc in ZF, then we couldn't have a model in which result on unions but not on dfin were true. In short, the implication does not reverse in ZF.
In another direction:
choice for countable collections of countable sets does not imply the result about unions; see cccnu for proof.
For interesting results linking countable unions up with delta systems, see dsys.
(This, in general, is interesting when discussing choice principles: ams)
@Qaanol: Are you doing ordinal exponents, or using exponent notation for the set of functions? If you are using ordinal exponents 2^omega is omega, so I'm guessing not. Your proof works if you assume the axiom of choice for the specific set you are choosing over, obviously; but when you are saying of cardinality up to something, that makes me leery, in general, in ZF since it seems to assume cardinals can be linearly ordered, but that is equivalent to AC, using hartog. I'm hazy on what exactly you are using: "Choice over this specific set", which is true, but not a specifically interesting principle or "Choice over sets up to size x", which is interesting, but something that requires being cautious with the "up to". Apologies if that sounds bitchy, it isn't meant to be, I always sound a bit cranky, even when completely unintended.
Countable unions of countables is countable does not imply countable choice:
countable choice implies (dedekind finite iff finite) in ZF. In the basic Fraenkel Model, the result on unions is valid, the result about dfin iff fin is not (more about the latter in comb, and the former in cons). Using Pincus's transfer theorem, there is a model of ZF with the same. Thus, if the result on unions implies cc in ZF, then we couldn't have a model in which result on unions but not on dfin were true. In short, the implication does not reverse in ZF.
In another direction:
choice for countable collections of countable sets does not imply the result about unions; see cccnu for proof.
For interesting results linking countable unions up with delta systems, see dsys.
(This, in general, is interesting when discussing choice principles: ams)
@Qaanol: Are you doing ordinal exponents, or using exponent notation for the set of functions? If you are using ordinal exponents 2^omega is omega, so I'm guessing not. Your proof works if you assume the axiom of choice for the specific set you are choosing over, obviously; but when you are saying of cardinality up to something, that makes me leery, in general, in ZF since it seems to assume cardinals can be linearly ordered, but that is equivalent to AC, using hartog. I'm hazy on what exactly you are using: "Choice over this specific set", which is true, but not a specifically interesting principle or "Choice over sets up to size x", which is interesting, but something that requires being cautious with the "up to". Apologies if that sounds bitchy, it isn't meant to be, I always sound a bit cranky, even when completely unintended.
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 MartianInvader
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Re: Sizes of Infinity
0x783czar, the main issue here is that we need to nail down what we mean by "size". Size is really an abstract concept, and if we want to speak mathematically then we need to pin down this concept with a mathematical definition. The usual definition people use for the "size" of sets is that of cardinality: two sets have the same cardinality if they can be put in onetoone correspondence. Under this definition, the natural numbers are the same size as the integers, which are the same size as the rationals, and the interval [0,1] is the same size as the interval [0,2], which is the same size as the entire real line (or plane, for that matter).
Now your brain doesn't like this, because look! There are obviously half as many natural numbers as integers! And that dislike is fine  you're just thinking of a slightly different notion of "size" than what cardinality represents, more along the lines of "how much space does it take up?" as opposed to "how many elements does it have?" And it turns out we can make mathematical definitions of these other notions of size as well. There are mathematical ways to define the measure of a subset of the integers, and under such definitions the natural numbers are half the size of the integers. There are also ways to measure subsets of the real numbers, and under these definitions the interval [0,2] is twice the size of the interval [0,1], and the real line is infinitely larger than either of these. I won't get into the details of how to do this because it's a little complicated, but these definitions have their drawbacks  they typically only apply to subsets of a given set, and can't compare arbitrary sets the way cardinality can.
The important thing to remember is that concepts don't always map perfectly onto mathematical definitions, but finding where the definition behaves the way the concept predicts it should, and where the two diverge, is one of the most fun parts of mathematics.
Now your brain doesn't like this, because look! There are obviously half as many natural numbers as integers! And that dislike is fine  you're just thinking of a slightly different notion of "size" than what cardinality represents, more along the lines of "how much space does it take up?" as opposed to "how many elements does it have?" And it turns out we can make mathematical definitions of these other notions of size as well. There are mathematical ways to define the measure of a subset of the integers, and under such definitions the natural numbers are half the size of the integers. There are also ways to measure subsets of the real numbers, and under these definitions the interval [0,2] is twice the size of the interval [0,1], and the real line is infinitely larger than either of these. I won't get into the details of how to do this because it's a little complicated, but these definitions have their drawbacks  they typically only apply to subsets of a given set, and can't compare arbitrary sets the way cardinality can.
The important thing to remember is that concepts don't always map perfectly onto mathematical definitions, but finding where the definition behaves the way the concept predicts it should, and where the two diverge, is one of the most fun parts of mathematics.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
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Re: Sizes of Infinity
The problem with the "if A is a proper subset of B then A is smaller than B" definition for infinite sets is that by simply renaming the elements of B, I can easily have A = B or A > B. That's why for infinite sets, this is not usually considered a meaningful measure of relative size.
 Forest Goose
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Re: Sizes of Infinity
Eebster the Great wrote:The problem with the "if A is a proper subset of B then A is smaller than B" definition for infinite sets is that by simply renaming the elements of B, I can easily have A = B or A > B. That's why for infinite sets, this is not usually considered a meaningful measure of relative size.
Yes, but that that "renaming" works the way you think relies on the intuition, and acceptance, that the cardinal notion of size is the correct one to use. Someone in the other camp can just as easily argue that the function is "compressing" B to A and that you should be comparing the outputs acting on B and on A  and there is no bijection that maps B onto A and A onto A. There is nothing inherently illogical about the concept of "subset" = "smaller", it is a partial order, but the key thing is that it is not a useful replacement for cardinality as size and that cardinality as size is useful too.
Actually, that comes to another point: we see cardinality and bijections, and etc., as measuring size, but that's not really relevant. We can just as easily say that cardinals measure "klubzgrug" and say your notion is the definitive notion of size. What matters is that "klubzgrug" is useful and leads to a rich theory that informs us about other relations (as does the subset relation too). Mathematics is not philosophy, even if cardinality turned out to be an exceptionally poor measure of what we would call size, it still tells us a whole damn lot of other interesting things  and it is interesting to us as well.
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Re: Sizes of Infinity
Put another way: the "renaming" you're talking about is a bijection, and while bijections preserve cardinality, they don't preserve this "proper subset" notion of size.
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 Eebster the Great
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Re: Sizes of Infinity
somehow wrote:Put another way: the "renaming" you're talking about is a bijection, and while bijections preserve cardinality, they don't preserve this "proper subset" notion of size.
That was pretty much my entire point.
Re: Sizes of Infinity
Okay. Sorry if I misinterpreted you or came off as condescending. I guess it just seemed a little odd to me to describe that as a "problem" with the partial order given by "A is a proper subset of B". Bijections work nicely with cardinality because cardinality is defined in terms of them; the fact they don't work nicely with this other notion of size doesn't seem in and of itself like an argument against the subset order.
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Re: Sizes of Infinity
somehow wrote:I guess it just seemed a little odd to me to describe that as a "problem" with the partial order given by "A is a proper subset of B". Bijections work nicely with cardinality because cardinality is defined in terms of them; the fact they don't work nicely with this other notion of size doesn't seem in and of itself like an argument against the subset order.
Can the proper subset notion of 'size' really be anything more than the notion of proper subsets?
Are {a, b, c} and {1, 2, 3} of the same 'size'? Neither is a proper subset of the other. Are {a, b, c} and {a, b, 1, 2} of the same 'size'?
Is the set of positive integers, ℕ^{+}, the same 'size' as {1/n: n∊ℕ^{+}}?
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 Eebster the Great
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Re: Sizes of Infinity
That's the thing. Not only does cardinality impose a (nonstrict) total order, but the proper sets partial order only works on nested sets, i.e. one of the sets has no unique elements. It doesn't seem to me to represent "size" so much as mere containment, which is after all what it's called.
Cardinality isn't called "size" arbitrarily. For finite sets, it is clear that cardinality gives an intuitive notion of size, in that it literally "counts" the elements (associating with each element a natural number). Pretty much the same procedure is extended to countably infinite sets, which have the same cardinality as a proper subset of themselves. Any notion of size which attempts to get around that seems to me to give an incorrect result. The way we usually mean "size," either in math or in conversation, the set of natural numbers is just as "big" as the set of rational numbers. There exists a partial order a < b iff a ⊃ b, but that order is not size.
Cardinality isn't called "size" arbitrarily. For finite sets, it is clear that cardinality gives an intuitive notion of size, in that it literally "counts" the elements (associating with each element a natural number). Pretty much the same procedure is extended to countably infinite sets, which have the same cardinality as a proper subset of themselves. Any notion of size which attempts to get around that seems to me to give an incorrect result. The way we usually mean "size," either in math or in conversation, the set of natural numbers is just as "big" as the set of rational numbers. There exists a partial order a < b iff a ⊃ b, but that order is not size.
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Re: Sizes of Infinity
Eebster the Great wrote:That's the thing. Not only does cardinality impose a (nonstrict) total order
In ZFC, yes; in ZF, you have a choice (lol), either not every set has a cardinality or some are incomparable.
Eebster the Great wrote:Pretty much the same procedure is extended to countably infinite sets, which have the same cardinality as a proper subset of themselves. Any notion of size which attempts to get around that seems to me to give an incorrect result.
I don't follow  that seems to be less an intuition about size and more an assumption that size is measured by cardinality. What about the notion of size indicates that the above should hold? It's usually something that is considered unintuitive  indeed, that is exactly how this thread came to exist.
Eebster the Great wrote:The way we usually mean "size," either in math or in conversation, the set of natural numbers is just as "big" as the set of rational numbers.
I disagree. Conversationally, no one is using "size" in a way that is anything remotely like comparing infinite sets  there are plenty of concepts that fall apart in weird ways when infinity becomes involved, just because finite counting can be accurately captured by bijections does not necessarily conceptually justify the extension to infinite systems. But, moreover, there are plenty of ways that "size" is used, mathematically, in which that result surely doesn't hold, as in discussing density, and other such.
Somehow wrote:Put another way: the "renaming" you're talking about is a bijection, and while bijections preserve cardinality, they don't preserve this "proper subset" notion of size.
That depends on what is meant. If f:X > Y is a bijection and X >= Z, then Y >= Z; but that that should be the case requires already assuming that cardinality is the right notion and that bijections "rename" in an absolute sense. It also assumes that all bijections should preserve size, even the most pathological and weird ones.
For another sense of preserve, all bijections are order isomorphisms of the subset order.

It should also be pointed out that there are some "weird" aspects of cardinality. Like countability not being absolute; so either some models just "get it wrong" or our intuition about "size" doesn't translate as well as we think. Hell,even "finite" isn't absolute.

I'm not arguing that subsets measure size, nor that we should stop thinking that cardinals do  and pathological weirdness doesn't bother me either. However, I don't think cardinals, in the infinite especially, behave like we intuitively expect "size" to behave  and it doesn't appear that it does except to those people that already have an intuition and knowledge of the subject; and there are still tons of mysteries and confusions about cardinals, tons and tons and tons, which seems to imply that cardinality is a lot more mysterious than it first seems (that ZFC can barely scratch the surface of "size", but can do most of modern mathematics seems to indicate that there is a lot beneath the covers).
The cardinal numbers are just as, probably more, mysterious to someone who is used to dealing with normal notions of size as Quantum Mechanics is to someone only familiar with everyday scales  and just like the latter, it eventually seems intuitive and makes clear sense, but that's because of education, not because it is the obvious candidate.
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 Eebster the Great
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Re: Sizes of Infinity
Forest Goose wrote:Eebster the Great wrote:Pretty much the same procedure is extended to countably infinite sets, which have the same cardinality as a proper subset of themselves. Any notion of size which attempts to get around that seems to me to give an incorrect result.
I don't follow  that seems to be less an intuition about size and more an assumption that size is measured by cardinality. What about the notion of size indicates that the above should hold? It's usually something that is considered unintuitive  indeed, that is exactly how this thread came to exist.
The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
Forest Goose wrote:Eebster the Great wrote:The way we usually mean "size," either in math or in conversation, the set of natural numbers is just as "big" as the set of rational numbers.
I disagree. Conversationally, no one is using "size" in a way that is anything remotely like comparing infinite sets  there are plenty of concepts that fall apart in weird ways when infinity becomes involved, just because finite counting can be accurately captured by bijections does not necessarily conceptually justify the extension to infinite systems. But, moreover, there are plenty of ways that "size" is used, mathematically, in which that result surely doesn't hold, as in discussing density, and other such.
Except you can count both sets using the same numbers. That is, you can map them both onetoone into the set of natural numbers. It is a direct generalization of counting finite sets. Your method is not very useful for most finite sets.
I think most people would agree that the set {1,2,3} is bigger than the set {a,b}.
Forest Goose wrote:Somehow wrote:Put another way: the "renaming" you're talking about is a bijection, and while bijections preserve cardinality, they don't preserve this "proper subset" notion of size.
That depends on what is meant. If f:X > Y is a bijection and X >= Z, then Y >= Z; but that that should be the case requires already assuming that cardinality is the right notion and that bijections "rename" in an absolute sense. It also assumes that all bijections should preserve size, even the most pathological and weird ones.
For another sense of preserve, all bijections are order isomorphisms of the subset order.
I don't know how many senses of "rename" exist, but bijections do it pretty well. You are associating every object x with exactly one name y, and {y=f(x)} is precisely the range of f(x) (i.e. you use every name).
Forest Goose wrote:It should also be pointed out that there are some "weird" aspects of cardinality. Like countability not being absolute; so either some models just "get it wrong" or our intuition about "size" doesn't translate as well as we think. Hell,even "finite" isn't absolute.
That sets will have different sizes in different models of ZFC is inevitable and unsurprising.
Forest Goose wrote:I'm not arguing that subsets measure size, nor that we should stop thinking that cardinals do  and pathological weirdness doesn't bother me either. However, I don't think cardinals, in the infinite especially, behave like we intuitively expect "size" to behave  and it doesn't appear that it does except to those people that already have an intuition and knowledge of the subject; and there are still tons of mysteries and confusions about cardinals, tons and tons and tons, which seems to imply that cardinality is a lot more mysterious than it first seems (that ZFC can barely scratch the surface of "size", but can do most of modern mathematics seems to indicate that there is a lot beneath the covers).
The cardinal numbers are just as, probably more, mysterious to someone who is used to dealing with normal notions of size as Quantum Mechanics is to someone only familiar with everyday scales  and just like the latter, it eventually seems intuitive and makes clear sense, but that's because of education, not because it is the obvious candidate.
I would argue most people have a pretty firm grasp on the finite cardinals.
Anyway, using containment to order sets is just a special case of using injections, where f(x)=x.
Re: Sizes of Infinity
I'm just rephrasing what Eebster's been saying here, but maybe this is a helpful reframing: there is a strongly intuitive notion of size for finite sets, and while cardinality extends that notion, this proper subsetrelated thing does not.
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Re: Sizes of Infinity
Eebster the Great wrote:The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
Except that that is a rephrasing, not an intuition. "Has the same number of elements" is justified by "Same size", both of those root in the same notion.
Eebster the Great wrote:Except you can count both sets using the same numbers. That is, you can map them both onetoone into the set of natural numbers. It is a direct generalization of counting finite sets. Your method is not very useful for most finite sets.
Lot's of things that are direct generalizations don't work out, or are extremely counterinuitive, or turn out to be acceptable, yet useless. While most people aren't thinking about density for finite sets, a lot of people do have that intuition for the rationals  and a lot of people find it curious that the rationals as as big as the naturals.
And, again, the idea that size is a total order and applies to all sets does require choice (it's equivalent to choice, actually). Most people do not find that to be nearly as intuitive.
Eebster the Great wrote:I think most people would agree that the set {1,2,3} is bigger than the set {a,b}.
Yes, they would, so what? Every finite set is recursive too. And any surjection, or injection, between finite sets of the same size is a bijection. In fact, there's a lot of things that hold, and make sense, for finite objects that do not for the infinite, and vice versa, so that it works in the finite realm does not mean it is the "right" extension in the infinite. Again, that infinite sets have equipollent proper subsets is something that does not hold for finite sets and seems wonky when first encountered.
Eebster the Great wrote:I don't know how many senses of "rename" exist, but bijections do it pretty well. You are associating every object x with exactly one name y, and {y=f(x)} is precisely the range of f(x) (i.e. you use every name).
That this holds for the extremely pathological is the point. I have no objection to it, but that weird unusual objects should be thought of a "renaming" is something that isn't immediately obvious  one might, for example, insist on some form of computational property holding, or may intuit that there ought to be such a principle.
Eebster the Great wrote:That sets will have different sizes in different models of ZFC is inevitable and unsurprising.
Which is odd, because lots of people find it surprising and a bit unsettling. It is unsurprising if you have a background and education in the matter, it is not something people "intuit" from casual discussions about size and a basic appreciation with naive set theory. Which is exactly my point. Your response is like saying "The Banach Tarski paradox is inevitable and unsurprising", it's true, for exactly those people who have a firm grasp of the concepts and formalism, already  it is quite surprising for people who are thinking purely intuitively.
Eebster the Great wrote:I would argue most people have a pretty firm grasp on the finite cardinals.
I was arguing about cardinals, in general, and most people, even a lot of people working with them, do not have a strong intuition, nor firm grasp, for what cardinals will do once you leave the finite case; in fact, they seem to do lots of powerful things and are quite impossible to pin down with any axioms that are not overly strict. Which, again, is not what you expect of "size" from an intuitive and finite understanding.
Eebster the Great wrote:Anyway, using containment to order sets is just a special case of using injections, where f(x)=x.
You don't say. You'll notice I'm not arguing that we should use subsets, and you'll also notice I'm not saying we shouldn't use cardinality. I'm arguing that infinite cardinality is not as straight forward and obvious a matter as you seem to think, that it does exhibit some weird behaviour one wouldn't anticipate, and that there is no notion of "size" that is philosophically justified. That cardinality measures size is a neat way to look at it, it's wear it started, but it's not intrinsically linked to it and oughtn't be seen as the "right" way to do it, but a useful one. All I'm saying is that an appeal to intuition isn't going to justify accepting cardinals, especially when the whole problem is that they behave uninituitively  most people do find the rationals being the size of the naturals to be unintuitive, so saying it is the intuitive notion of "size" doesn't really help.
there is a strongly intuitive notion of size for finite sets, and while cardinality extends that notion, this proper subsetrelated thing does not.
I can't tell if you're replying to me or not, I'm assuming yes, though. Yes, I agree that it is intuitive for finite sets and that cardinality does extend it  however, I am not arguing that the subset relation extends it, nor that it should be called "size". I'm arguing that cardinality, in general, is not straightforward and that most people do not find it, initially, intuitive in the infinite case (hence why questions, like this very thread, arise). In short, an appeal to intuition doesn't counter "I don't find this intuitive".
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 Eebster the Great
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Re: Sizes of Infinity
Forest Goose wrote:Eebster the Great wrote:The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
Except that that is a rephrasing, not an intuition. "Has the same number of elements" is justified by "Same size", both of those root in the same notion.
Then that makes my point for me. Two sets have the same number of elements if you can pair them up such that every element in each set is paired with exactly one element in the other set. That is exactly how counting works, and is literally the meaning of "number of elements."
Imagine a young child counting pebbles. She lines the pebbles up, then points at them one at a time. For each pebble she points at, she says the next natural number. "One ... two ... three ... ." She is associating with each element of the set of pebbles a unique natural number.
Forest Goose wrote:Eebster the Great wrote:Except you can count both sets using the same numbers. That is, you can map them both onetoone into the set of natural numbers. It is a direct generalization of counting finite sets. Your method is not very useful for most finite sets.
Lot's of things that are direct generalizations don't work out, or are extremely counterinuitive, or turn out to be acceptable, yet useless. While most people aren't thinking about density for finite sets, a lot of people do have that intuition for the rationals  and a lot of people find it curious that the rationals as as big as the naturals.
Some generalizations don't turn out how you expected, but any general notion of size ought to work for all sets, both finite and infinite. As long as "size" has any consistent meaning, there must exist some such generalization. If there does not, then there is no single term "size" that can accurately describe both finite and infinite sets, and that can describe sets with disjoint elements.
Forest Goose wrote:And, again, the idea that size is a total order and applies to all sets does require choice (it's equivalent to choice, actually). Most people do not find that to be nearly as intuitive.
I'm not sure that's such a big deal. Most sets can be easily assigned a cardinality without choice. Only pathological sets that you could never construct require choice. It's kind of analogous in this context (though not quite identical) to saying that there exist some things that can never be measured, but postulating that measurements exist for them anyway.
Forest Goose wrote:Eebster the Great wrote:I think most people would agree that the set {1,2,3} is bigger than the set {a,b}.
Yes, they would, so what? Every finite set is recursive too. And any surjection, or injection, between finite sets of the same size is a bijection. In fact, there's a lot of things that hold, and make sense, for finite objects that do not for the infinite, and vice versa, so that it works in the finite realm does not mean it is the "right" extension in the infinite. Again, that infinite sets have equipollent proper subsets is something that does not hold for finite sets and seems wonky when first encountered.
Are you conceding that cardinality properly describes size for finite sets and that containment does not? Because then my argument about generalizations above seems to hold a lot of weight. It doesn't make sense to have two incompatible notions of size anymore than it would make sense for a summation method used for divergent series to give the wrong sum for convergent or finite series.
Forest Goose wrote:Eebster the Great wrote:I don't know how many senses of "rename" exist, but bijections do it pretty well. You are associating every object x with exactly one name y, and {y=f(x)} is precisely the range of f(x) (i.e. you use every name).
That this holds for the extremely pathological is the point. I have no objection to it, but that weird unusual objects should be thought of a "renaming" is something that isn't immediately obvious  one might, for example, insist on some form of computational property holding, or may intuit that there ought to be such a principle.
I don't understand. It is a renaming by simple definition. If it looks "weird unusual" from another perspective, that doesn't change this fact. Sometimes simple procedures, even renaming, can produce complex results.
Forest Goose wrote:Eebster the Great wrote:That sets will have different sizes in different models of ZFC is inevitable and unsurprising.
Which is odd, because lots of people find it surprising and a bit unsettling. It is unsurprising if you have a background and education in the matter, it is not something people "intuit" from casual discussions about size and a basic appreciation with naive set theory. Which is exactly my point. Your response is like saying "The Banach Tarski paradox is inevitable and unsurprising", it's true, for exactly those people who have a firm grasp of the concepts and formalism, already  it is quite surprising for people who are thinking purely intuitively.
It is not like the BanachTarski paradox. The same set has different constructions in different models of ZF, and is "counted" in a completely different way. i.e. the definition of the set may not be changed, but the definition of "cardinality" is changed. Similarly, if I measured a racetrack by laying out meter sticks, I would find it to have finite length, but if I measured it by laying out infinitesimal points, I would find it to have infinite length, but this fact would not surprise anybody.
Forest Goose wrote:Eebster the Great wrote:I would argue most people have a pretty firm grasp on the finite cardinals.
I was arguing about cardinals, in general, and most people, even a lot of people working with them, do not have a strong intuition, nor firm grasp, for what cardinals will do once you leave the finite case; in fact, they seem to do lots of powerful things and are quite impossible to pin down with any axioms that are not overly strict. Which, again, is not what you expect of "size" from an intuitive and finite understanding.Eebster the Great wrote:Anyway, using containment to order sets is just a special case of using injections, where f(x)=x.
You don't say. You'll notice I'm not arguing that we should use subsets, and you'll also notice I'm not saying we shouldn't use cardinality. I'm arguing that infinite cardinality is not as straight forward and obvious a matter as you seem to think, that it does exhibit some weird behaviour one wouldn't anticipate, and that there is no notion of "size" that is philosophically justified. That cardinality measures size is a neat way to look at it, it's wear it started, but it's not intrinsically linked to it and oughtn't be seen as the "right" way to do it, but a useful one. All I'm saying is that an appeal to intuition isn't going to justify accepting cardinals, especially when the whole problem is that they behave uninituitively  most people do find the rationals being the size of the naturals to be unintuitive, so saying it is the intuitive notion of "size" doesn't really help.
I don't really understand what you are arguing. You are apparently not arguing for the alternative notion you first proposed, nor are you arguing against mine. You are just arguing that "size" is the wrong word, and I cannot understand why. Because it's weird? Because you only want to use monosyllabic words for very simple ideas?
Look, the idea of cardinality, the definition, is very simple, and corresponds exactly with what we mean by size. The fact that actually measuring it can get difficult doesn't make it any less a measurement of size. The fact that the rationals are the same size as the integers may be surprising, but it is nevertheless true, in any meaningful sense.
Forest Goose wrote:there is a strongly intuitive notion of size for finite sets, and while cardinality extends that notion, this proper subsetrelated thing does not.
I can't tell if you're replying to me or not, I'm assuming yes, though. Yes, I agree that it is intuitive for finite sets and that cardinality does extend it  however, I am not arguing that the subset relation extends it, nor that it should be called "size". I'm arguing that cardinality, in general, is not straightforward and that most people do not find it, initially, intuitive in the infinite case (hence why questions, like this very thread, arise). In short, an appeal to intuition doesn't counter "I don't find this intuitive".
There are many facts people do not initially find intuitive in mathematics. Consider 0.(9) = 1. You might argue that while equality is intuitive for nonrepeating and terminating decimals, for repeating decimals like this one it can lead to unintuitive results. You can argue that equality doesn't really mean these have the same "size" in the intuitive sense, it's just a math word with math meaning. You might even introduce another notion of equality, a = b iff a and b represent the same irrational number. That avoids this problem entirely! Too bad it doesn't work for rationals.
Re: Sizes of Infinity
Eebster the Great wrote:The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
…which is why the interval from 0 to 1 is the same size as the interval from 0 to 3: they both have the same number of elements, and there is even a nice obvious bijection between them. We may as well just replace all yardsticks with 1foot rulers, after all the same bijection works so they must be the equal sizes.
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Re: Sizes of Infinity
Qaanol wrote:Eebster the Great wrote:The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
…which is why the interval from 0 to 1 is the same size as the interval from 0 to 3: they both have the same number of elements, and there is even a nice obvious bijection between them. We may as well just replace all yardsticks with 1foot rulers, after all the same bijection works so they must be the equal sizes.
Assuming that a yardstick is equivalent to a "set" of all real numbers between 0 and 3 seems weird to me. It is more like the set of all integers between 0 and 3 (excluding the 0, including the 3). You can measure the interval [0,3] and find that its measure is three times the measure of [0,1], but that's not the same thing. That is, there are just as many points in the yardstick and the ruler, even if one is longer than the other.
What I mean is that the size of a set should not depend on its elements. The set {1,2,3} is the same size as the set {10,20,30} or the set {a,b,c} or the set {0,1,i}. The set [0,1] is the same size as the set [0,3]. But that doesn't mean a segment of length 1 is the same length as a segment of length 3. You are asking very different questions.
The idea of length relies on simply using multiples of some given ruler (metric) in a given topology over a given set. It doesn't even really matter what the underlying set itself is, but rather what structure you build on top of it. "length 3" is in no sense an intrinsic property of the set of real numbers between 0 and 3.
Re: Sizes of Infinity
…which is why the interval from 0 to 1 is the same size as the interval from 0 to 3: they both have the same number of elements, and there is even a nice obvious bijection between them. We may as well just replace all yardsticks with 1foot rulers, after all the same bijection works so they must be the equal sizes.
In some sense you can replace yardsticks with rulers. Except your new "inch" is three times your old "inch".
Because an "inch" is a measurable amount, 3 inches is longer than 1 inch. When it comes to infinite sets of points, each point has a "length" that approaches zero. Three times zero is still zero...
The fundamental concept behind these bijections is that three times the "length" of a single point is still the "length" of a single point. A single point is equal to a single point, regardless of relative "lengths". I also dislike this concept, but it is what it is. Everything that approach zero uncertainty have the same uncertainty, even if you define one to always be X times greater than the other.
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Re: Sizes of Infinity
*whoo—wait both of you really? Okay I’ll just explain.
The claim was about what “seems pretty intuitive” regarding when “two sets are the same size”.
The real interval [0, 1] and the real interval [0, 3] are intuitively different sizes, despite the fact that they are sets of equal cardinality.
Quod erat stopincorrectlyappealingtointuition.
The claim was about what “seems pretty intuitive” regarding when “two sets are the same size”.
The real interval [0, 1] and the real interval [0, 3] are intuitively different sizes, despite the fact that they are sets of equal cardinality.
Quod erat stopincorrectlyappealingtointuition.
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 Eebster the Great
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Re: Sizes of Infinity
The real intervals [0,1] and [0,3] are indeed sets, and the only meaningful way we could say [0,3] > [0,1] is given above, because [0,3] ⊃ [0,1]. However, the metric sense you are giving does not apply. The set [0,1] ∩ R should be larger than the set [0,3] ∩ N, but by your logic, the latter is three times as large. The latter only seems larger because of the quality of the elements, not because of the quantity. In the same way that the set {3} is not larger than the set {1}, the set [0,3] is not larger than the set [0,1]. You are confusing the issue.
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Re: Sizes of Infinity
Eebster the Great wrote:Then that makes my point for me. Two sets have the same number of elements if you can pair them up such that every element in each set is paired with exactly one element in the other set. That is exactly how counting works, and is literally the meaning of "number of elements."
I don't believe you understood what I said. "Same size" and "Same number of elements" already assume the context that cardinality is the "true" notion of "size". You are justifying "Same size" as intuitive by appealing to "Same number of elements", all of which presumes the concept at hand as being "right". In other words, your appeal to intuition assumes the person already agrees with, and understands, what you are trying to show them...which doesn't work.
Eebster the Great wrote:Imagine a young child counting pebbles. She lines the pebbles up, then points at them one at a time. For each pebble she points at, she says the next natural number. "One ... two ... three ... ." She is associating with each element of the set of pebbles a unique natural number.
I understand the concepts involved, I research them daily, they aren't something I'm confused about. However, I don't think you understand what my argument is. Allow me to clarify, it has three points:
1.) This isn't philosophy, this isn't physics, it's mathematics  terms are convenient and inspired by concepts, but frequently end up no longer related to those concepts. Moreover, there is no "right" definition that goes with a word. Cardinality is not the "right" notion of size, "size" is the word that best fits where cardinality came from and provides a little background to help the intuition. Cardinality is not studies because set theorists care about "size", it is studied because it leads to a rich and interesting, and powerful, theory  if it did not, it would be abandoned and we would be studying something else.
2.) Cardinals are not intuitive, in the infinite, to people who aren't already used to them, so appeals to developed intuition about cardinals will not make "The rationals are the same size as the naturals" seem any more sensible, on those grounds alone.
3.) We should be really really cautious about treating things as intuitive and the "right" generalization, that's how loads of bad thinking comes to exist. The cardinals do all sorts of weird things, in the infinite, that our intuition about the finite would not lead us to expect  indeed, look at logic and set theory in the past, a big hurdle was breaking away from intuitions about the finite and infinite and how things "should" behave, rather than how they do.
Eebster the Great wrote:Some generalizations don't turn out how you expected, but any general notion of size ought to work for all sets, both finite and infinite. As long as "size" has any consistent meaning, there must exist some such generalization. If there does not, then there is no single term "size" that can accurately describe both finite and infinite sets, and that can describe sets with disjoint elements.
This is the exact problem thinking I'm talking about. First, right below, you dismiss the sets that wouldn't have a cardinality when this wouldn't work as "pathological" (despite that they break your "for all sets" mentioned here); and you also don't seem to notice that nearly every set cannot be constructed, in any reasonable sense, unless you're working in V = L, or something else.
Second: "size" isn't accurate, there is no notion of "accurate". Is it "accurate" to the word "measure" to allow measures to take values in a Banach Space? The answer is: "who cares?" or "that question doesn't make sense". It's not the meaning of "size" that matters, it's the theory of cardinality that matters  if we never thought of it as "size" and called it "krimbgah" instead, it would be just as important because of the results. You are getting hung up on if the name fits with some intuitive concept, but that's philosophy, not mathematics  and we can discuss things like that, all day, if we want, and it's neat, but it has more to do with metaphysics and ontology than it does mathematics.
Eebster the Great wrote:I'm not sure that's such a big deal. Most sets can be easily assigned a cardinality without choice. Only pathological sets that you could never construct require choice. It's kind of analogous in this context (though not quite identical) to saying that there exist some things that can never be measured, but postulating that measurements exist for them anyway.
That seems a lot of hurdles to jump through. You seem concerned that "size" applies to all sets, then you seem to be okay as long as it applies to sets that aren't "pathological"  in other words, "cardinality is intuitive and the right meaning of "size" because it works exactly like I expect exactly where I can intuit." Which is kind of my point: your intuition only helps exactly where you started, it is a hindrance.
Moreover, most sets are not easily assigned a cardinality without choice  most sets are not easily assigned one with choice. What is the cardinality of the reals? And, let's not even get started with large cardinal numbers  when they even exist is highly model dependent, and doesn't always workout how you might think.
It's like saying that particles intuitively behave like billiard balls, except where all that pathological quantum stuff happens. All that pathological set theory stuff is, honestly, pretty important when you start talking about cardinals (and trying to intuit set theory has gotten just as many people in trouble as trying to intuit particle physics).
Eebster the Great wrote:Are you conceding that cardinality properly describes size for finite sets and that containment does not? Because then my argument about generalizations above seems to hold a lot of weight. It doesn't make sense to have two incompatible notions of size anymore than it would make sense for a summation method used for divergent series to give the wrong sum for convergent or finite series.
There is not correct notion of "size", it's just a word. Yes, cardinality captures what we do with finite sets when we count, but that doesn't really mean that much  we use that generalization because it worked out to a rich theory, not because we care about accurately describing the "size" of sets in the right way. If I came up with a really useful and rich theoretical definition and called it "size", for no reason other than liking the word, people would still care  and if I came up with a perfect notion of "size" that lead nowhere, no one would care.
Eebster the Great wrote:I don't understand. It is a renaming by simple definition. If it looks "weird unusual" from another perspective, that doesn't change this fact. Sometimes simple procedures, even renaming, can produce complex results.
My problem is, again, you are relying on "renaming" as the right and true description of what bijections do, not merely as a convenient explanation. A lot of philosophical debates come up from things like this: some might argue that highly weird bijections are not "renaming", only "good" one's do, or something like that. You can have that debate all day, but it doesn't actually matter to the mathematics, the theorems still work the same.
Moreover, that bijections "rename" things isn't an intuition that someone who feels uneasy about bijections meaning equal "size" is going to have  so appealing to one intuition fails to justify the other.
Eebster the Great wrote:It is not like the BanachTarski paradox. The same set has different constructions in different models of ZF, and is "counted" in a completely different way. i.e. the definition of the set may not be changed, but the definition of "cardinality" is changed. Similarly, if I measured a racetrack by laying out meter sticks, I would find it to have finite length, but if I measured it by laying out infinitesimal points, I would find it to have infinite length, but this fact would not surprise anybody.
Actually, the BanachTarski paradox is different in different models of ZF. (I can't tell where your second sentence applies).
The definition of cardinality is not changed between models, that doesn't even make sense  unless you, again, are talking from intuition; and if so, it doesn't serve well here as it is inaccurate to say in a rigorous sense. Different models have different witnesses, that is true, but I wouldn't say that that is "counting" differently  you can say that; but that we disagree on the intuitive meaning of things is just one more reason we should, maybe, not rely on intuition so strongly here (because you can't really dispute that I'm wrong about my intuition). I also really really don't like your racetrack analogy, that's not a great way to explain how certain cardinal terms are not absolute  it would be useful if you were talking calculus, but, really, it's much more a matter of logic and what witnesses for what, and it does not function like you suggest (there is no measuring stick that is being applied in the sense of the analogy).
Eebster the Great wrote:I don't really understand what you are arguing. You are apparently not arguing for the alternative notion you first proposed, nor are you arguing against mine. You are just arguing that "size" is the wrong word, and I cannot understand why. Because it's weird? Because you only want to use monosyllabic words for very simple ideas?
My problem is that you are treating "size as cardinals" as the intuitively right definition, which is more philosophy than mathematics  and leads to bad practice when it comes to actually studying cardinals in any real fashion. For example: "Should a Reinhardt Cardinal be consistent with ZFC? With ZF?" that's the type of things one might wonder in the theory of cardinals  is there some intuition about "size" that is of value here? Nope, not at all. Intuition is great, it is, but it also causes lots of bad thinking and assumptions about how things work (and about what should be studied about them), that can be very very bad. Indeed, such thinking was what made a lot of early work surprising  people expected terms to behave like the words used, instead of how the concepts named actually did.
I'm not sure what, at any point, indicated I was arguing for either definition to be right  I've trying very hard to make it clear that the whole problem is that one of them is "right" and that this kind of thinking is not so good, especially for someone who already lacks that intuition (such as the op).
Eebster the Great wrote:Look, the idea of cardinality, the definition, is very simple, and corresponds exactly with what we mean by size. The fact that actually measuring it can get difficult doesn't make it any less a measurement of size. The fact that the rationals are the same size as the integers may be surprising, but it is nevertheless true, in any meaningful sense.
First, you can very rarely measure it, even conceptually (it's not just difficult)  it's undecidable, usually, even in basic cases (the reals). Second, a major genesis of my point is that you were saying that it is intuitive that the rationals were equipollent the naturals...despite that that is surprising (as you say). In short, an appeal to intuition doesn't help when the people you are talking to don't have that intuition (the one you have from having studied the concept).
In short: it's not that the result is intuitive; it's that you are comfortable with that result and it have gotten it to make sense for you.
Eebster the Great wrote:There are many facts people do not initially find intuitive in mathematics. Consider 0.(9) = 1. You might argue that while equality is intuitive for nonrepeating and terminating decimals, for repeating decimals like this one it can lead to unintuitive results. You can argue that equality doesn't really mean these have the same "size" in the intuitive sense, it's just a math word with math meaning. You might even introduce another notion of equality, a = b iff a and b represent the same irrational number. That avoids this problem entirely! Too bad it doesn't work for rationals.
Yeah, that's my whole argument...except you keep talking about "size", and else, as if they were more than just terms. As if there was a "right" notion of "size" that was captured by the cardinals, and that they behave intuitively. My point isn't: "Cardinals are the wrong notion", but "They don't behave intuitively, in general, and "size" is just a term".
Eebster the Great wrote:The real intervals [0,1] and [0,3] are indeed sets, and the only meaningful way we could say [0,3] > [0,1] is given above, because [0,3] ⊃ [0,1]. However, the metric sense you are giving does not apply. The set [0,1] ∩ R should be larger than the set [0,3] ∩ N, but by your logic, the latter is three times as large. The latter only seems larger because of the quality of the elements, not because of the quantity. In the same way that the set {3} is not larger than the set {1}, the set [0,3] is not larger than the set [0,1]. You are confusing the issue.
This is exactly the thing I'm talking about: you are assuming that cardinality is the right and true meaning of size and quantity and etc. You are philosophizing your mathematics...you can argue all day that, for you, [0, 1] and [0, 3] are the same size, but you can't argue (in any mathematical sense) that they are, truly, the same size, for everyone, independent of a specific definition of size. Intuition does not direct the show, it's the background music that keeps our attention and something helpful to summarize what you are working on when nonmath people ask (yep, I'm being glib). But, seriously, there is no right definition, one can just as validly call anything they want "size" and be perfectly fine in doing so.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
 Eebster the Great
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Re: Sizes of Infinity
Forest Goose wrote:Eebster the Great wrote:Then that makes my point for me. Two sets have the same number of elements if you can pair them up such that every element in each set is paired with exactly one element in the other set. That is exactly how counting works, and is literally the meaning of "number of elements."
I don't believe you understood what I said. "Same size" and "Same number of elements" already assume the context that cardinality is the "true" notion of "size". You are justifying "Same size" as intuitive by appealing to "Same number of elements", all of which presumes the concept at hand as being "right". In other words, your appeal to intuition assumes the person already agrees with, and understands, what you are trying to show them...which doesn't work.
Well the argument is about a definition in the first place. You so far have tried to keep your position so vague as to be impossible to argue against. I.e., I am presenting what I consider to be incontrovertibly the closest approximation to "size" set theory has to offer, and I was hoping you would present anything resembling a counterexample. But you seem to be rejecting that without offering any useful alternative. You say that size is a good way to describe cardinality. Fine. I say that cardinality is a good way to define size. I don't understand what the difference is, or even what you're arguing anymore.
Forest Goose wrote:1.) This isn't philosophy, this isn't physics, it's mathematics  terms are convenient and inspired by concepts, but frequently end up no longer related to those concepts. Moreover, there is no "right" definition that goes with a word. Cardinality is not the "right" notion of size, "size" is the word that best fits where cardinality came from and provides a little background to help the intuition. Cardinality is not studies because set theorists care about "size", it is studied because it leads to a rich and interesting, and powerful, theory  if it did not, it would be abandoned and we would be studying something else.
2.) Cardinals are not intuitive, in the infinite, to people who aren't already used to them, so appeals to developed intuition about cardinals will not make "The rationals are the same size as the naturals" seem any more sensible, on those grounds alone.
3.) We should be really really cautious about treating things as intuitive and the "right" generalization, that's how loads of bad thinking comes to exist. The cardinals do all sorts of weird things, in the infinite, that our intuition about the finite would not lead us to expect  indeed, look at logic and set theory in the past, a big hurdle was breaking away from intuitions about the finite and infinite and how things "should" behave, rather than how they do.
1. Fine, there is no "right" definition of anything in any context, but there are clearly some definitions that are better and some that are worse at expressing yourself. If you want to get an idea of how "big" a set is, cardinality is pretty much your only option.
2. This is the equivalent of throwing up your hands in surrender. The fact that you can count the naturals and the rationals using the same set of numbers seems to me a pretty good reason to say they are the same size. You can insist that I am being unintuitive, but I don't really know how to reply to that. How do you argue intuition?
3. I don't recall writing anywhere about the way things should be.
Forest Goose wrote:Eebster the Great wrote:Some generalizations don't turn out how you expected, but any general notion of size ought to work for all sets, both finite and infinite. As long as "size" has any consistent meaning, there must exist some such generalization. If there does not, then there is no single term "size" that can accurately describe both finite and infinite sets, and that can describe sets with disjoint elements.
This is the exact problem thinking I'm talking about. First, right below, you dismiss the sets that wouldn't have a cardinality when this wouldn't work as "pathological" (despite that they break your "for all sets" mentioned here); and you also don't seem to notice that nearly every set cannot be constructed, in any reasonable sense, unless you're working in V = L, or something else.
The question of whether a given set has a cardinality that is impossible to determine or has no cardinality at all seems academic to me, at least in this context. The point wasn't that you could necessarily generalize all notions to "all sets", but that if a notion can be applied to two different types of sets, it should be applied consistently.
Forest Goose wrote:Second: "size" isn't accurate, there is no notion of "accurate". Is it "accurate" to the word "measure" to allow measures to take values in a Banach Space? The answer is: "who cares?" or "that question doesn't make sense". It's not the meaning of "size" that matters, it's the theory of cardinality that matters  if we never thought of it as "size" and called it "krimbgah" instead, it would be just as important because of the results. You are getting hung up on if the name fits with some intuitive concept, but that's philosophy, not mathematics  and we can discuss things like that, all day, if we want, and it's neat, but it has more to do with metaphysics and ontology than it does mathematics.
You are confusing things even further by assuming my argument is somehow fundamentally mathematical. There is no mathematical convention on "size." It obviously is a question of "philosophy, not mathematics," or perhaps more accurately "philosophy of mathematics." More specifically, it's an attempt to find a formal, mathematical term most closely resembling an informal, natural language term. There is, as far as I can tell, absolutely no other way one could look at this thread, and I can't figure out what you think I've been saying.
Forest Goose wrote:Eebster the Great wrote:I'm not sure that's such a big deal. Most sets can be easily assigned a cardinality without choice. Only pathological sets that you could never construct require choice. It's kind of analogous in this context (though not quite identical) to saying that there exist some things that can never be measured, but postulating that measurements exist for them anyway.
That seems a lot of hurdles to jump through. You seem concerned that "size" applies to all sets, then you seem to be okay as long as it applies to sets that aren't "pathological"  in other words, "cardinality is intuitive and the right meaning of "size" because it works exactly like I expect exactly where I can intuit." Which is kind of my point: your intuition only helps exactly where you started, it is a hindrance.
Would you be this nitpicky about everything I say that has no formal meaning? Would you argue that a predecessor is not the "previous" ordinal, because not every ordinal number has a predecessor? For such sets that have no cardinality in ZF, we might say that they have no size, i.e. that there is no way to apply any notion of size to them and to other disjoint sets consistently. In ZFC, the axiom of choice demands that, by our previous, formal definition, some such cardinal number must exist, but there is no way to actually find it. Either way, it is not much of a problem. There is nothing "intuitive" or whatever about "size" that demands it be applicable to everything. However, it clearly should be applicable to finite sets. So if the issue is how to best generalize this to infinite sets, again, I reach the question of what better approach you think there is.
Forest Goose wrote:Moreover, most sets are not easily assigned a cardinality without choice  most sets are not easily assigned one with choice. What is the cardinality of the reals? And, let's not even get started with large cardinal numbers  when they even exist is highly model dependent, and doesn't always workout how you might think.
It's like saying that particles intuitively behave like billiard balls, except where all that pathological quantum stuff happens. All that pathological set theory stuff is, honestly, pretty important when you start talking about cardinals (and trying to intuit set theory has gotten just as many people in trouble as trying to intuit particle physics).
I recognize this, but that hasn't stopped people from trying to apply natural language to quantum physics. Particles might not literally be the same "particles" we normally think of intuitively, but they are still the closest things to actual particles quantum physics has to offer. And the question of whether these objects are "not really particles" or whether "particles are different than we thought" again becomes academic.
Forest Goose wrote:Eebster the Great wrote:Are you conceding that cardinality properly describes size for finite sets and that containment does not? Because then my argument about generalizations above seems to hold a lot of weight. It doesn't make sense to have two incompatible notions of size anymore than it would make sense for a summation method used for divergent series to give the wrong sum for convergent or finite series.
There is not correct notion of "size", it's just a word. Yes, cardinality captures what we do with finite sets when we count, but that doesn't really mean that much  we use that generalization because it worked out to a rich theory, not because we care about accurately describing the "size" of sets in the right way. If I came up with a really useful and rich theoretical definition and called it "size", for no reason other than liking the word, people would still care  and if I came up with a perfect notion of "size" that lead nowhere, no one would care.
Fine, but I didn't ask how the theory was developed or for a lesson on foundational set theory. The question was just a factual one, if for finite sets the notion of "size" corresponds to the notion of "number of elements," which is merely an English language description of cardinality. And clearly it does. This might not make it "right" or "wrong" in some sort of ethical or formal sense, but informally I think few would argue that someone claiming "the size of the set {a,b,c} is three" was not "right" or that the same person claiming its size was four was "wrong." And if I asked Matlab for size({1,2,3},1) and it returned anything other than 3, I would complain. Clearly this is the expected meaning for finite sets.
Forest Goose wrote:Eebster the Great wrote:I don't understand. It is a renaming by simple definition. If it looks "weird unusual" from another perspective, that doesn't change this fact. Sometimes simple procedures, even renaming, can produce complex results.
My problem is, again, you are relying on "renaming" as the right and true description of what bijections do, not merely as a convenient explanation. A lot of philosophical debates come up from things like this: some might argue that highly weird bijections are not "renaming", only "good" one's do, or something like that. You can have that debate all day, but it doesn't actually matter to the mathematics, the theorems still work the same.
Moreover, that bijections "rename" things isn't an intuition that someone who feels uneasy about bijections meaning equal "size" is going to have  so appealing to one intuition fails to justify the other.
I still don't understand the argument. Rather than saying "some people might argue" or "you assume the consequent" or whatever, why not give an actual counterexample? A bijection that one might not consider to be "renaming" and why. After all, any function is merely a set of ordered pairs. If you want to call the first element in the pair an "element" of the set and the second element in the pair a "name," then the function by definition renames every element in its domain, even from a formal perspective.
Forest Goose wrote:Eebster the Great wrote:It is not like the BanachTarski paradox. The same set has different constructions in different models of ZF, and is "counted" in a completely different way. i.e. the definition of the set may not be changed, but the definition of "cardinality" is changed. Similarly, if I measured a racetrack by laying out meter sticks, I would find it to have finite length, but if I measured it by laying out infinitesimal points, I would find it to have infinite length, but this fact would not surprise anybody.
Actually, the BanachTarski paradox is different in different models of ZF. (I can't tell where your second sentence applies).
The definition of cardinality is not changed between models, that doesn't even make sense  unless you, again, are talking from intuition; and if so, it doesn't serve well here as it is inaccurate to say in a rigorous sense. Different models have different witnesses, that is true, but I wouldn't say that that is "counting" differently  you can say that; but that we disagree on the intuitive meaning of things is just one more reason we should, maybe, not rely on intuition so strongly here (because you can't really dispute that I'm wrong about my intuition). I also really really don't like your racetrack analogy, that's not a great way to explain how certain cardinal terms are not absolute  it would be useful if you were talking calculus, but, really, it's much more a matter of logic and what witnesses for what, and it does not function like you suggest (there is no measuring stick that is being applied in the sense of the analogy).
Well I wasn't the one to bring up the comparison between rulers and yardsticks. Intuition breaks down here because we are trying to use incompatible (informal) notions of size, one being spatial extent, and the other being numerical quantity. So I agree my analogy was not the best. The best analogy I could give would be to topology, which much of a leap from set theory.
Forest Goose wrote:Eebster the Great wrote:I don't really understand what you are arguing. You are apparently not arguing for the alternative notion you first proposed, nor are you arguing against mine. You are just arguing that "size" is the wrong word, and I cannot understand why. Because it's weird? Because you only want to use monosyllabic words for very simple ideas?
My problem is that you are treating "size as cardinals" as the intuitively right definition, which is more philosophy than mathematics  and leads to bad practice when it comes to actually studying cardinals in any real fashion. For example: "Should a Reinhardt Cardinal be consistent with ZFC? With ZF?" that's the type of things one might wonder in the theory of cardinals  is there some intuition about "size" that is of value here? Nope, not at all. Intuition is great, it is, but it also causes lots of bad thinking and assumptions about how things work (and about what should be studied about them), that can be very very bad. Indeed, such thinking was what made a lot of early work surprising  people expected terms to behave like the words used, instead of how the concepts named actually did.
I'm not sure what, at any point, indicated I was arguing for either definition to be right  I've trying very hard to make it clear that the whole problem is that one of them is "right" and that this kind of thinking is not so good, especially for someone who already lacks that intuition (such as the op).
As soon as I used the informal word "size," and especially when I put it in quotes, I thought I was being pretty clear I wasn't trying to prove a formal statement about mathematics. That doesn't mean there is no argument to be had. Similarly, a "set" in the intuitive sense in the word is not identical to a "set" in a formal set theory, but that doesn't stop us from calling it a set.
Forest Goose wrote:Eebster the Great wrote:Look, the idea of cardinality, the definition, is very simple, and corresponds exactly with what we mean by size. The fact that actually measuring it can get difficult doesn't make it any less a measurement of size. The fact that the rationals are the same size as the integers may be surprising, but it is nevertheless true, in any meaningful sense.
First, you can very rarely measure it, even conceptually (it's not just difficult)  it's undecidable, usually, even in basic cases (the reals). Second, a major genesis of my point is that you were saying that it is intuitive that the rationals were equipollent the naturals...despite that that is surprising (as you say). In short, an appeal to intuition doesn't help when the people you are talking to don't have that intuition (the one you have from having studied the concept).
In short: it's not that the result is intuitive; it's that you are comfortable with that result and it have gotten it to make sense for you.
You don't have to resort to the results to find a definition intuitive. I find it intuitive that "north" is in the direction towards the north pole along the surface of the Earth. I might not find it intuitive that at the south pole, every direction is north. But I would not consider this to be a flaw in my definition, just an unintuitive result. Alternatively, I might find Simpson's Paradox unintuitive, but I wouldn't therefore decide that probability no longer described the likelihood of an event. Rather, I would decide that determining the likelihood of certain events from certain data can be unintuitive.
It should not surprise you that my decision to label cardinality as "size" is based on its definition, not on every result in mathematics. We are arguing definitions, after all.
Forest Goose wrote:Eebster the Great wrote:There are many facts people do not initially find intuitive in mathematics. Consider 0.(9) = 1. You might argue that while equality is intuitive for nonrepeating and terminating decimals, for repeating decimals like this one it can lead to unintuitive results. You can argue that equality doesn't really mean these have the same "size" in the intuitive sense, it's just a math word with math meaning. You might even introduce another notion of equality, a = b iff a and b represent the same irrational number. That avoids this problem entirely! Too bad it doesn't work for rationals.
Yeah, that's my whole argument...except you keep talking about "size", and else, as if they were more than just terms. As if there was a "right" notion of "size" that was captured by the cardinals, and that they behave intuitively. My point isn't: "Cardinals are the wrong notion", but "They don't behave intuitively, in general, and "size" is just a term".
So you are arguing that because there are unintuitive equalities, we cannot have an intuitive notion of equality at all?
Forest Goose wrote:Eebster the Great wrote:The real intervals [0,1] and [0,3] are indeed sets, and the only meaningful way we could say [0,3] > [0,1] is given above, because [0,3] ⊃ [0,1]. However, the metric sense you are giving does not apply. The set [0,1] ∩ R should be larger than the set [0,3] ∩ N, but by your logic, the latter is three times as large. The latter only seems larger because of the quality of the elements, not because of the quantity. In the same way that the set {3} is not larger than the set {1}, the set [0,3] is not larger than the set [0,1]. You are confusing the issue.
This is exactly the thing I'm talking about: you are assuming that cardinality is the right and true meaning of size and quantity and etc. You are philosophizing your mathematics...you can argue all day that, for you, [0, 1] and [0, 3] are the same size, but you can't argue (in any mathematical sense) that they are, truly, the same size, for everyone, independent of a specific definition of size. Intuition does not direct the show, it's the background music that keeps our attention and something helpful to summarize what you are working on when nonmath people ask (yep, I'm being glib). But, seriously, there is no right definition, one can just as validly call anything they want "size" and be perfectly fine in doing so.
Yes, one could call anything they want "size," but I would not say they were equally "valid." That claim would amount to saying English words have no meaning. Of course the English language is ambiguous, incomplete, and imprecise from a formal standpoint, but it is blatantly not useless. We would probably both agree that the definition "the size of a set is equal to the last number I thought of before considering that set" would not be a very good definition. Or from a descriptive standpoint, that does not closely match the meaning most Englishspeakers associate with "size." The word "size" has a real (if informal) meaning in English, and some things are more "sizelike" than others.
 Forest Goose
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 Joined: Sat May 18, 2013 9:27 am UTC
Re: Sizes of Infinity
Eebster the Great wrote:Well the argument is about a definition in the first place. You so far have tried to keep your position so vague as to be impossible to argue against. I.e., I am presenting what I consider to be incontrovertibly the closest approximation to "size" set theory has to offer, and I was hoping you would present anything resembling a counterexample. But you seem to be rejecting that without offering any useful alternative. You say that size is a good way to describe cardinality. Fine. I say that cardinality is a good way to define size. I don't understand what the difference is, or even what you're arguing anymore.
...I'm not rejecting anything, I'm saying that once you start talking cardinals, in full generality, your intuitions about "size" don't do anything. And that's true. Seriously, give me some examples of intuition from finite sets that actually translates over to full generality. I can't think of any.
I can think of a bunch that don't extend, of course.
Hence my point: "size", at that point, is just a term. Sure, it works in the finite (let's just say that it does), but so will tons and tons of stuff  even if they aren't as pretty. The point is that there aren't any intuitive arguments that start just from thinking about how "size" should work once you leave the realm of the finite. Thus, how do you know that it is the intuitive meaning? It's not from the reasoning. So from what? The senses you attach to the words involved "sound right" to you? Something else?
And, originally, it was a running response to:
Eebster the Great wrote:The idea that two sets are the same size if they have the same number of elements seems pretty intuitive to me.
Both concepts are the same concept  they come from the same source. The meaning of the things in that statement are what they are because of your intuition about what they should mean  it seems circular.
Eebster the Great wrote:1. Fine, there is no "right" definition of anything in any context, but there are clearly some definitions that are better and some that are worse at expressing yourself. If you want to get an idea of how "big" a set is, cardinality is pretty much your only option.
2. This is the equivalent of throwing up your hands in surrender. The fact that you can count the naturals and the rationals using the same set of numbers seems to me a pretty good reason to say they are the same size. You can insist that I am being unintuitive, but I don't really know how to reply to that. How do you argue intuition?
3. I don't recall writing anywhere about the way things should be.
1.) Who wants an idea of how big sets are? And I don't mean in the sense of what their cardinality is, but who want to know how big a set is and, thus, works that out with the cardinals, because they fit that definition?
I think, more likely, people are interested in set theory and the cardinals are a great tool and integral part of that.
2.) You keep talking about counting rationals and naturals like that is the major bulk of what people do with cardinals. Not to be a dick, but you do realize that we can almost never figure out what cardinality a set is, in general, and, in many cases, even say what cardinals exist. That seems really damn unintuitive; and no finite conception of size leads one to assume that that will be the case. Once you move past comparing the most basic sets, everything goes to hell  there are countable ordinals, the existence of which, are independent of ZFC...that's still in the realm of the smallest level, already things start going weird.
And, again, I challenge you: show me a result about finite cardinals, something nontrivial, that generalizes to cardinality in general  do this on the basis of what you intuit about how "size" should behave. How would your intuition lead you to approach any of the results that make up the theory of cardinals? I don't even know how to answer that...does Easton's Theorem fit your intuition about the size of the reals based on finite experience?
3.) Explain the following quotes, by you, because they seem that you are:
I am presenting what I consider to be incontrovertibly the closest approximation to "size" set theory has to offer,
the idea of cardinality, the definition, is very simple, and corresponds exactly with what we mean by size
It doesn't make sense to have two incompatible notions of size
Longer statements seem to indicate the same reasoning too, but that's a lot harder to quote and keep this manageable.
Eebster the Great wrote:The question of whether a given set has a cardinality that is impossible to determine or has no cardinality at all seems academic to me, at least in this context. The point wasn't that you could necessarily generalize all notions to "all sets", but that if a notion can be applied to two different types of sets, it should be applied consistently.
Such questions are the entire bread and butter of cardinality, look at a lot of the results, they are all "academic", most of them way way more so. I'm not trying to be a jerk, but to what extent have you studied cardinality and set theory? Everything you've said about it seems to come from how it works with the naturals, rationals, and things of that nature  once you get beyond this very very limited realm, things don't behave nicely at all, you have to be incredibly careful in what you say, how you say it, etc. It's not a straightforward area of mathematics and thinking about "size" doesn't shed any light on things.
Eebster the Great wrote:You are confusing things even further by assuming my argument is somehow fundamentally mathematical. There is no mathematical convention on "size." It obviously is a question of "philosophy, not mathematics," or perhaps more accurately "philosophy of mathematics." More specifically, it's an attempt to find a formal, mathematical term most closely resembling an informal, natural language term. There is, as far as I can tell, absolutely no other way one could look at this thread, and I can't figure out what you think I've been saying.
Then, I guess, have fun with philosophy. Just be aware that all of the reasoning you know about "size" from normal experience, essentially, fails once you leave that realm. So, I'm not sure what the point is unless it's to say what definition sounds "best".
Eebster the Great wrote:Would you be this nitpicky about everything I say that has no formal meaning? Would you argue that a predecessor is not the "previous" ordinal, because not every ordinal number has a predecessor? For such sets that have no cardinality in ZF, we might say that they have no size, i.e. that there is no way to apply any notion of size to them and to other disjoint sets consistently. In ZFC, the axiom of choice demands that, by our previous, formal definition, some such cardinal number must exist, but there is no way to actually find it. Either way, it is not much of a problem. There is nothing "intuitive" or whatever about "size" that demands it be applicable to everything. However, it clearly should be applicable to finite sets. So if the issue is how to best generalize this to infinite sets, again, I reach the question of what better approach you think there is.
There is no "better" approach...there are approaches. I'm not seeing how this intuition does anything besides say you happen to think it is fitting  it does not seem to lead to any results, nor to inspire any arguments.
Eebster the Great wrote:I recognize this, but that hasn't stopped people from trying to apply natural language to quantum physics. Particles might not literally be the same "particles" we normally think of intuitively, but they are still the closest things to actual particles quantum physics has to offer. And the question of whether these objects are "not really particles" or whether "particles are different than we thought" again becomes academic.
This kind of thinking leads to all sorts of misunderstanding and things that need to be unlearned. Your "academic" seems to include anything that is actually doing math, save for a few nice cases where things work out well. Again, I challenge you: show me that intuition at work, explain some of the actual results relating to cardinals via intuition.
Eebster the Great wrote:Fine, but I didn't ask how the theory was developed or for a lesson on foundational set theory. The question was just a factual one, if for finite sets the notion of "size" corresponds to the notion of "number of elements," which is merely an English language description of cardinality. And clearly it does. This might not make it "right" or "wrong" in some sort of ethical or formal sense, but informally I think few would argue that someone claiming "the size of the set {a,b,c} is three" was not "right" or that the same person claiming its size was four was "wrong." And if I asked Matlab for size({1,2,3},1) and it returned anything other than 3, I would complain. Clearly this is the expected meaning for finite sets.
I keep stressing those words "infinite" "in general"...you want to talk about finite, and only finite, fine, I'll just concede at that point. You have yet to show that your intuition works anywhere outside of that realm  to me, it seems like it would be mislead you instead...
And, for the record, the infinite cardinals are, pretty much, impossible to discuss without a bunch of foundational set theory. It's like starting out with basic curves, saying that Algebraic Geometry is intuitive, as a result, then wondering why someone is going about all those academic cohomology theories.
Eebster the Great wrote:I still don't understand the argument. Rather than saying "some people might argue" or "you assume the consequent" or whatever, why not give an actual counterexample? A bijection that one might not consider to be "renaming" and why. After all, any function is merely a set of ordered pairs. If you want to call the first element in the pair an "element" of the set and the second element in the pair a "name," then the function by definition renames every element in its domain, even from a formal perspective.
People object to all sorts of such things, a lot of people aren't okay with arbitrary things implied to exist being considered as on the same footing. I don't think that way, but some do, and others may. My point was that you are using concepts that already assume "rationals equipollent naturals" is intuitive, the op won't find these intuitive  if they did, they wouldn't have a problem to begin with. That's my point 2 above: you are appealing to intuition based on a thing to explain the thing, intuitively...that don't help.
Eebster the Great wrote:As soon as I used the informal word "size," and especially when I put it in quotes, I thought I was being pretty clear I wasn't trying to prove a formal statement about mathematics. That doesn't mean there is no argument to be had. Similarly, a "set" in the intuitive sense in the word is not identical to a "set" in a formal set theory, but that doesn't stop us from calling it a set.
Again, call it the intuitive meaning of size all you want, but, please, show me how this does anything anywhere  and, again, in the general theory. Where of use does such intuition lead you? How does it develop proofs? Does it actually do anything besides restate that that all seems sensible to you?
Eebster the Great wrote:So you are arguing that because there are unintuitive equalities, we cannot have an intuitive notion of equality at all?
Equality is a tricky thing, like most basic things. I would be very cautious about talking about equality from my gut intuition, especially once we start talking actual theory  logic ain't simple after all:) Which is my whole point, intuition is neat and all, but so are diagrams, and other such. Neat tools, but they don't carry any weight, they don't justify anything. And at this depth, they tend to not help as much as most would think  or they hurt, that happens too; there's a lot of confused students and mathematicians when it comes to sets.
Eebster the Great wrote:Yes, one could call anything they want "size," but I would not say they were equally "valid." That claim would amount to saying English words have no meaning. Of course the English language is ambiguous, incomplete, and imprecise from a formal standpoint, but it is blatantly not useless. We would probably both agree that the definition "the size of a set is equal to the last number I thought of before considering that set" would not be a very good definition. Or from a descriptive standpoint, that does not closely match the meaning most Englishspeakers associate with "size." The word "size" has a real (if informal) meaning in English, and some things are more "sizelike" than others.
Sure, why not. Does this actually help anyone? Does this lead us to new results, in this case? Not that I've seen.
I have seen this use of intuition hurt thinking, though. I have seen it lead to all sorts of confusion. I have seen people get hung up on the "philosophy" of terms. And etc. It kind of sucks to see people spend inordinate amounts of time realizing what a subject is actually about and actually saying  and set theory has reams of that, more than most places.
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Re: Sizes of Infinity
I feel like we're talking past each other, and it's making me frustrated and my behaviour pissy. So, let's try something different.
Which of the following do you agree with (it can be none, one, some, or all):
1.) The definition of cardinality is how I think about size, it appeals to my intuition of the finite.
2.) Cardinality is the natural definition of size for sets.
3.) The concept of size is useful guide for how we should approach cardinals and things related to then.
4.) The cardinal numbers behave as I expect size to behave.
5.) Intuition about size is useful in reasoning about cardinals and makes sense of results.
The first seems quite reasonable, but I don't see that it does anything more than say "this definition seems nice". I say that because 2 through 5 will lead to issues with doing actual set theory and thinking about cardinals; which is bad. To me, I feel you are saying all of them, at various points  and I don't see that the first is helpful, really, even if accurate, since it leads a lot of people to the other four.
If the interest is merely to learn a few basic facts and definitions, it doesn't really matter, but if you want to reason about the topics, the latter points aren't going to help, probably hurt instead. Hence, we should be very cautious of our intuition, because trusting it too much, in this case, will bite us on the ass as much as aid us.
Does that make more sense? Does that clarify where I'm coming from? I apologize if I came off as rude, walls of text get frustrating, and both of use were writing them
Which of the following do you agree with (it can be none, one, some, or all):
1.) The definition of cardinality is how I think about size, it appeals to my intuition of the finite.
2.) Cardinality is the natural definition of size for sets.
3.) The concept of size is useful guide for how we should approach cardinals and things related to then.
4.) The cardinal numbers behave as I expect size to behave.
5.) Intuition about size is useful in reasoning about cardinals and makes sense of results.
The first seems quite reasonable, but I don't see that it does anything more than say "this definition seems nice". I say that because 2 through 5 will lead to issues with doing actual set theory and thinking about cardinals; which is bad. To me, I feel you are saying all of them, at various points  and I don't see that the first is helpful, really, even if accurate, since it leads a lot of people to the other four.
If the interest is merely to learn a few basic facts and definitions, it doesn't really matter, but if you want to reason about the topics, the latter points aren't going to help, probably hurt instead. Hence, we should be very cautious of our intuition, because trusting it too much, in this case, will bite us on the ass as much as aid us.
Does that make more sense? Does that clarify where I'm coming from? I apologize if I came off as rude, walls of text get frustrating, and both of use were writing them
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Re: Sizes of Infinity
Intuitively, you just can't run out of an endless supply of something. So what on earth was Georg Cantor doing that meant he ran out of naturals when considering lists of reals?
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Re: Sizes of Infinity
FancyHat wrote:Intuitively, you just can't run out of an endless supply of something. So what on earth was Georg Cantor doing that meant he ran out of naturals when considering lists of reals?
The thing is the naturals and the reals are "endless" in different ways, and the reals are "more endless" than the naturals. Really, the best way to understand what that exactly means is just looking up the diagonalization proof: it's more straightforward than any analogy.
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