gmalivuk wrote:Fine, but that only gets you out of that one particular paradox. What about the property, "is even (x)or odd"? 1 is even xor odd, 2 is even xor odd, ... n is even xor odd, but their limit is neither. In fact, I can think of very few typical properties of integers that could possibly be applicable to infinity. "Is 0 or 1 or strictly less than its square", "is 0 or has a unique prime factorization (modulo sign)", "is congruent to K modulo N" for any integer choice of K and any N with |N|>1, and on and on and on.
No contradiction arises from claiming that numbers remain "even or odd" when the digits are infinite. An infinite-digited integer, if it could be meaningfully written at all, would have to be written from right to left with either a repeating pattern indicated or a simple truncation. Either way, defining it as even or odd based on the value of its ones digit should be fine as always. And if the number cannot be meaningfully written, then whether it is specifically even or specifically odd cannot be determined, but there's no contradiction in saying that it satisfies the criteria "is even or odd".
And if you come at it from the angle of simply incrementing 1 and taking the limit of the "is even" property or "is odd" property, then it becomes obvious that that property simply doesn't converge. Sometimes that happens when taking a limit as something approaches infinity.
Similarly, there's no contradiction in claiming that the number always has a unique prime factorization. Want to actually compute that factorization? You can't, because it's infinite. Any property that only retains relevance when the number is computable becomes meaningless for the same reason that the "is finite" property becomes meaningless. The whole point of defining infinity's properties in terms of such limits is precisely because it defies normal computational methods. So properties that, like finiteness, are directly related to computability, become meaningless as an inherent consequence of the problem being addressed in the first place. Likewise with your other examples.
SunAvatar wrote:In addition to the above objections, even if you create some self-consistent system of arbiteroftruth-integers, some of which have infinitely many digits, I can still say "consider the set of all arbiteroftruth-integers with finitely many digits" and once again have an infinite set, all of whose elements are finite. Unless you want to posit that some finite numbers are also infinite (what would this even mean?), or alternately declare that certain sets are inadmissible. The latter is actually viable--it's more or less how the hyperintegers work--but it's up to you to give a precise rule for when a set is or isn't admissible. So this extension doesn't actually solve the problem (not that there is any problem to be solved in the first place).
This is the most relevant objection I've seen here. My system doesn't escape the counterintuitive issue, because the counterintuitive issue doesn't arise from the way we define integers, but from the entire concept of being "finite".
In any event, I have figured out a way to intuitively understand the situation. Normally, I think of finite numbers as being the more 'fundamental' concept, and infinity being a hypothetical extension of that(hence the entire basis of my hypothetical system). But things work much better if I think of infinite numbers as the more fundamental type of object, and define finite numbers as a subset. That is, I can think of a standard number as having an infinite number of digits both on the right and on the left of the decimal point. The finite numbers can then be defined simply as the subset of numbers that meet the criteria "has an infinite string of zeroes on the left side of the decimal". Likewise, rational numbers would be defined as having two infinite strings of zeroes, one on each side of the decimal, and integers would be defined as rational numbers where the right-side string of zeroes begins immediately after the decimal. In this way, finite numbers can be thought of as just a special case of infinite numbers, so the fact that they still retain some properties of infinite numbers (like lack of an upper bound) makes intuitive sense.
fishfry wrote:So you are saying, as a general principle, that a set of things should have the same property as the things?
Not at all. I'm saying that, when a set of things is designed specifically
to share certain properties with the things, then intuitively one would expect the set to actually share those properties with the things. Sets in generally obviously can be completely unrelated to the things they contain, but I'm talking about a specific case in which that is explicitly avoided.
After playing around for a while with the idea of extending standard arithmetic into infinite digits to the left, the results are interesting. I doubt I'm the first to do it, so this may already be well known, but the results end up having some similarities to the p-adics, though there are differences. In general, any infinite-digited integer (I'll call them 'infintegers') written as a repeating pattern to the left is equal to the negative equivalent repeating decimal. So ...999 = -0.999 = -1, ...111 = -0.111 = -1/9, and so on. Because such values can only be easily defined with repeating patterns, terminating decimals have no equivalent representation as infintegers, so the infintegers don't quite equal the rational numbers, as they miss any fractions with denominators that include any of the same factors as your base. But a terminating decimal can be approximated arbitrarily closely by using an arbitrarily long pattern. For example, ...5000050000 = -0.50000500005... to approximate -0.5 . Because it is only the repeating pattern that equates to a negative decimal, positive decimals can be achieved by having digits that don't fit the pattern, as these digits still function as normal integers. For example, since ...888 = -8/9, ...8889 = 1/9, as you've simply added 1(or alternately, multiplied by 10 then added 9, either way it works out). Addition, subtraction, and multiplication all work exactly as usual. Division works for any divisor that isn't a factor of your base, otherwise you're forced to either use a standard decimal place or use remainder notation.
An infinteger that isn't a repeating pattern, but can still be defined by some algorithm for computing the digits(such as simply writing the digits of pi in the opposite direction), might or might not be reasonably considered a number, but it couldn't be converted into any equivalent standard real number. Given the general rule that an infinteger equates to the equivalent negative decimal with the same pattern, a non-repeating infinteger would equate to a non-repeating negative decimal, except that the right-most digits of an infinteger pattern equate to the right-most digits of the equivalent decimal pattern, so a non-repeating infinteger written from right to left and truncated after n digits would tell us the "last" n digits of the non-repeating decimal. Perhaps that could be given some degree of meaning along the lines of surreal numbers, but it obviously can't be interpreted meaningfully as a real number.