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I have a really hard time learning math when I can't find any justifications for the existence of infinity (in this case actual infinity). Are there any justifications for the Axiom of Infinity?

Thank you very much.

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    Is there justification for the existence of the number 3? Do numbers actually "exist?" Or are they merely abstractions which we find can be applied to things that exist?
    – Cort Ammon
    Feb 28 '18 at 18:45
  • Do you want to be able to talk about 'the Integers' as a group? If you never wish to be able to have a symbol for that collection or to talk about its structure, but only talk about processes that evolve through the enumeration of it, you can throw out the axiom of infinity. Hard core constructivists sometimes do so. But we all think of the integers as a valid collection of things to discuss. The question is whether discussions of them as a group are actual mathematics, or are just informal meta-mathematical stereotypes.
    – user9166
    Feb 28 '18 at 19:50
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    Possible duplicate of How does actual infinity (of numbers or space) work? This reviews Cantor's arguments for actual infinity. However, it makes no difference for mathematics whether actual infinity exists or not, the "justification" for adopting the axiom of infinity is that it makes for a handy formalism, not that it is "true".
    – Conifold
    Mar 1 '18 at 5:43
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    Cort Ammon, as you said, 3 can describe things in the real world. Can actual infinity (for example the set of all natural numbers) describe real things?
    – D. Hershko
    Mar 1 '18 at 8:01
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    I have no problem with potential infinities but see no example of a real one.
    – user20253
    Mar 1 '18 at 13:25
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The classical Intuitionist approach to this is to admit that infinities are just a figure of speech making it easier to discuss ongoing processes that can be repeated. There are an 'infinity' of points in a line because I can keep dividing it arbitrarily often.

From that point of view, 'completed' infinities simply don't exist. But then again, neither do sets. There is not some kind of cosmic baggie that springs into existence to separate the things you decide are in a set from the things that aren't.

Both are verbal shorthand for common ways of looking at things. So why not roll up the one verbal shorthand within the other? From this point of view, the axiom of infinity says, 'Yes we are going to include the human intuition of repetition in the notion of listing.' As long as you don't get carried away and forget these are abstractions of potentialities, rather than instantiated objects, no harm is done.

Sets of 'real' numbers of arbitrary precision are even reasonable, as you can picture a real number as an endless binary fraction, a procedure where we divided a length and chose either the left or the right half over and over. And we can imagine any number of these ongoing selection processes continuing in parallel.

But there are points where it is obvious that you have taken this convenience too far. For instance, the paradoxes like Tarski's, introduced by applying the Axiom of Choice to neighborhoods in the real plane should not sneak into your reasoning due to a mere convenience. They rely on tricks that defy our natural notion of infinity as incomplete progress.

The axiom of choice simply cannot apply in this case. There is no reasonable way to imagine a selection process that has something going on for each real number in a range. There is no such thing as each real number in a range. To collect them up and separate them all, the infinite sequences of digits would have to be completed, so we could sort through them. As long as they are mere potentialities modeled by processes ticking out boolean decisions sequentially you cannot tell apart the ones that are running at similar 'speeds' and converging to similar places. So strangely, we have a sort of linguistic gap, here we can talk about 'all' real numbers in a range, but not about 'each' one doing something.

So there is a reasonable defense for the Axiom of Infinity, as it captures the intuition of arbitrary repetition. But there is still a limit beyond which one must be careful about simply naming a set and pretending that the thing named is really something a human can actually find or imagine. A lot of cardinal and ordinal theory betrays the original understanding with linguistic tricks that create masses of references to indiscernible objects.

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The natural numbers are a mathematical topic that mathematicians study. Ergo, it must have an avatar in any reasonable formulation of mathematical foundations.

If you were only interested in studying finite sets (but still using ZFC-style material approach to the subject), you still need an avatar of the natural numbers somewhere in your mathematical foundation. Maybe it appears in the ambient first-order logic, in the form of the proposition "n is a natural number".

But in its usual role in foundations, set theory acts as a rich and full-featured incarnation of higher order logic. That proposition "n is a natural number" needs to be encoded as an actual object in the universe of sets.

The axiom of infinity in ZFC is a technical way to ensure this, for a specific model of how to encode the idea of natural numbers in terms of sets.

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It really hangs upon what you mean by existence.

In the formalist position in the philosophy of mathematics a mathematical object is granted existence when it's definition & construction is shown to be consistent.

Since the axiom of infinity is consistent with the other axioms of set theory it has been adopted. This is the smallest such axiom, there are other axioms of infinity with increasing larger cardinality.

Another position on existence is that an ideal object must refer. This is the correspondance theory of truth/existence.

The simplest infinite set is the set of all natural numbers. The set of all atoms in the universe is roughly 10^80, so we see that the set of all numbers can't refer. There is nothing physical that measures as infinity.

A similar point was made by Feynman, the American physicist when some mathematical friends of his were discussing infinity. In this context though it was infinite divisibility that was being discussed. Physicists after all aren't in mathematics for mathematics sake (though it can look like this sometimes in more advanced texts) but in order to describe physical reality so this is a very natural view to take.

I can't resist also pointing out that Aristotle thought that a completed infinity can't exist. This might seem a little odd in the modern theory of sets when we seem to have a completed infinity such as the set of all natural numbers. However as already pointed out there are other larger infinities, and there appears to be no end to them.

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