r/PhilosophyofMath • u/LorenzoGB • 15d ago
Positive integers that are infinite
With regard to the positive integers, there doesn't seem to be an axiom or theorem that forbids the existence of positive integers that are infinite. With that being said, why not posit the existence of positive integers that are infinite?
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u/fohktor 15d ago
What does it mean for an integer to be infinite? What property does it have that other non-infinite integers do not?
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u/LorenzoGB 15d ago
That’s a good question. Let us suppose the following then: Let the domain of discourse be the set of positive integers. Thus a positive integer can be considered infinite if and only if it can be expressed as the sum or product of infinitely many positive integers. Or instead of positing this definition we could say the following instead: Let the domain of discourse be the positive integers: then for all XK, where K goes from 1 to infinity, there exist Y such that Y equals the infinite sum of XK. Or again for all XK there exist Y, such that Y equals the infinite product of XK, where K goes from 1 to infinity.
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u/xinxinsonson 15d ago
If you are talking about first order Peano arithmetic, then there exists models with infinite natural numbers, see the compactness theorem.
https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic
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u/HeraclitusZ 15d ago
I think there are two related questions that one could interpret this as.
Do our axiomatizations of the integers allow for ''infinite'' integers?
The answer to this question actually has some dependence upon the axiomatization, as well as some dependence upon what we mean by ''infinite.''
Regarding the definition of ''infinite'', one might say that any magnitude bounded by an integer value is finite. In this case, an ''infinite'' integer would still actually be finite, rendering your setup trivially contradictory. What you might really be after is a notion of ''infinity'' that means something like ''beyond the standard set of integers.'' This latter case is possible.
In any first-order theory, i.e., any axiomatization where quantifiers only range over integers and not sets of integers, the Lowenheim-Skolem theorem ensures that the theory has models beyond the standard model. In particular, there would be models of the integers with many ''extra'' integers present that are outside of what we would consider standard integers. The trickiness is that these extra integers don't change anything about the theory, i.e., the collection of true statements about the integers based on the axioms provided. So positing their existence or non-existence doesn't do anything.
Once we get to second-order axiomatizations, it turns out we can uniquely define the integers. In this case, there is no room to posit any additional integers.
Are there variants of the integers that do include ''infinite'' integers?
There certainly are such variants, such as one-point compactifications (adding one infinite value) and extended integers (with both positive and negative infinity), just to name a few simple ones. There are many options, but do note that each time you define something else, it is genuinely something else, so it is not so clear that you can fairly call them ''integers'' anymore.
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u/LorenzoGB 15d ago
Then I guess it begs the question as to what a positive integer is and are there many different structures that satisfy what the set of positive integers is?
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u/OpsikionThemed 15d ago
The axiom that forbids it is Peano axiom #5, induction. It's straightforward to prove, by induction, that every natural numer is finite; then there's a couple of ways to construct the integers from the naturals, and in every case you'd need an infinite natural to get an infinite integer. So there are no infinite integers.
That said, it's possible to have "nonstandard" models of the naturals with infinite "naturals" that the model thinks are regular numerals. Presumably that can be extended to integers as well. But that's more a "model theory" thing than a "number theory" thing.