a bizzare question

For the discussion of math. Duh.

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laserzt
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a bizzare question

Postby laserzt » Fri Jan 14, 2011 6:29 am UTC

on 7:30AM, after a sleepless night of beer, other alcoholic drinks which I'm familiar with their names in English, chatting on bdsm sites, reading posts and more,
a bizzare question came into my mind.
Let us assume that you can define a group of all mathematical objects in the world. (probably it's not well defined, but I guess that will be part of your answer).
What is its "size" (I know in hebrew the term is power, but I don't know the term in English)?
like the "size" of N is alef zero.
I really hope I would get some interesting answers for this question, because for all I know, it is really infinite, because it contains all groups with power 2^alef,
2^2^alef etc.

xepher
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Re: a bizzare question

Postby xepher » Fri Jan 14, 2011 1:54 pm UTC

I'd have to say uncountably infinite/undefined.

The real numbers are uncountably infinite. So a group of mathematical objects would be even larger.

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Re: a bizzare question

Postby Robert'); DROP TABLE *; » Fri Jan 14, 2011 2:18 pm UTC

If you're talking about this thing, the set of all sets, then the size is undefined, because the object itself is paradoxical.
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Re: a bizzare question

Postby jestingrabbit » Fri Jan 14, 2011 2:30 pm UTC

When you say group, the english word is set and when you say size the english word is cardinality. Also, it seems that when you're talking about the world, I would talk about the universe of mathematical discourse. With that said, and with a big caveat that I might have misunderstood you...

You seem to be talking about something like "the set of all sets" which isn't actually well defined as a set. Imagine if it was, call it S, then you could form [imath]P=\{ A\in S :\ A\not\in A\}.[/imath] If P were well defined, and it would be if S is well defined, then you could answer the question "is [imath]P\in P[/imath]?" Whichever way you answer that question, when you look at the definition of P, you'll realise that your answer is wrong. If it is logically impossible to decide whether a set contains a particular element, then its not well defined. So P isn't well defined and S isn't well defined. So, "the set of all sets" isn't something we can logically talk about.

You can talk about a thing called a proper class of all sets, and there is a set of numbers that is as large as it, called the surreal numbers. Where I said "as large as it" there, you can't really say much about how largeness works for proper classes.

So basically, when you get to really large colections of things, the idea of set and largeness breaks, at least as we currently understand them.
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Re: a bizzare question

Postby skeptical scientist » Fri Jan 14, 2011 5:53 pm UTC

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

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Re: a bizzare question

Postby Yakk » Fri Jan 14, 2011 7:12 pm UTC

Now, you could talk about the set of all mathematical objects for which we can specifically specify in a given framework.

That set is typically countable (of the cardinality of the natural numbers or less), as most frameworks only admit finite definitions for the things they specify.
jestingrabbit wrote:You can talk about a thing called a proper class of all sets, and there is a set of numbers that is as large as it, called the surreal numbers. Where I said "as large as it" there, you can't really say much about how largeness works for proper classes.

I wasn't aware there was anything fundamentally difficult about finding the size of classes. They aren't different from sets all that much, except they are of a 'higher kind' and hence are permitted to do things like quantify over all sets? So a theory of class cardinality would work much like set cardinality, with the standard issue that the class of all classes that do not contain themselves isn't properly defined (and the class of all classes is also undefined). Naturally doing a 'higher kind' of set-like structure keeps things intact.

Which leads to limit ordinal indexed kinds of sets, by which point you are doing something that is pretty silly.
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

laserzt
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Re: a bizzare question

Postby laserzt » Fri Jan 14, 2011 8:05 pm UTC

wow, you guys are really awesome!
thank you very much.
You brought peace to my ill mind!
Let x be me, and let A be the set of people in this forum.
then x wants to be in A.
geekily speaking :)


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