@JadeVanadium Yup I'm aware that physicists believe the universe is as flat as can be measured, and hence would be spatially infinite. However, we are incapable of accessing all of it. Besides that, we would need at least one discrete operation with 100% fidelity, but we currently don't know whether such a thing exists. If for whatever interesting or strange reason the universe does not have such an operation, then it would be very unlikely that we can have anything like the natural numbers.

@JadeVanadium I don't buy this modal argument at all, because I don't think you can assert "it is possible for an infinite space to exist" without using a pre-existing assumption of existence of an infinite collection.

However, you're right that finiteness of the real world doesn't imply there is no infinite structure. But it would simply remove our ability to justify assuming existence of infinity.

@user21820 I'm not sure what this means exactly.

Can you elaborate?

I mean that I don't get how you can make any modal statement that mentions the concept "infinite".

Unless you already have assumed some sufficiently strong concept of "infinite".

@JadeVanadium I have thought about a related mathematical theory. Let FPA (for Finitististic PA) be PA but with all quantifiers restricted to "<M", where M is a new constant-symbol. Add axioms 1 < M and ∀x,y<M ( x+y < M ). Then we can get ∀x,y<M ( x·y < M ). And any model of FPA has an initial segment that models PA. And any model of PA can be used to construct (by compactness) a model of FPA. And PA and FPA have essentially the same theorems (modulo the quantifier restriction).

But whenever we want to apply any theorem of FPA to the real world, we must get a lower bound for M, but we can only get 2^O(t) < M after t time steps...

@user21820 If your objection is that we can't define "(in)finite" without reference to the naturals, I'd say sure, but we can still precisely define a relatively weak proposition which entails that a collection is infinite, despite not having a proposition which correctly labels finite sets as finite. Would this address your concern?

For example, having a function which is injective but not surjective intuitively entails that a collection is not finite, although it's not equivalent to saying it's infinite. We can ask questions about whether such a function could exist despite not having an infinite domain. In that way, it makes sense to talk about whether infinity is possible, by appealing to some inexact proxy which is not substantially stronger.

@JadeVanadium Sure. And I don't think you can justify (without prior belief in ℕ) that it's possible for there to be such a function, right?

We can prove lots of things about such a function if it exists, but I don't think we can get any modal possibility out of it.

If you're asking me if I personally feel that it's possible, I'd say I don't really use modal semantics personally hahah

@JadeVanadium Oh then I guess you mean something different from what I interpreted.

My point was more that, if we can say that it's possible, then the question of whether it's actual becomes irrelevant.

I thought you were saying that we can justify existence of infinity by first justifying the possibility of it and then going from that to actual existence.

But it seems like you're just saying that it's equivalent.

Yes, that's approximately what I'm saying. I think this would be classified as a form of modal collapse.

And if I were to formalize what exactly that means, I'd probably just recite the model existence theorem.

LOL.. which uses WKL...

Tsk tsk.. you are using way more than a finitist would..

:p

Honestly, the model existence theorem feels more fundamental to me than any other mathematical principle. That we can prove it using other theories is very cool, but to me it might as well be an axiom.

I guess that's just how I think.

Why is it natural to believe that consistency implies model existence?

In fact, if I take finitism seriously, I would say that PA is consistent in the sense of not having a real-world proof of ⊥ but has no real-world model.

And nobody can refute this...

XD

In the sense that consistency=possibility, to say that an abstract structure is possible is just to say that it's actual. That's just what it feels like to me, it seems like a defining feature of what an abstract structure even is.

But consistency is not possibility... It's finite-subset possibility.

Every finite subset of ℕ has a maximum, but their union does not.

What makes consistency different?

I think it just depends on how you define possibility. That's why I said "if I were to formalize [it], I'd ... recite the model existence theorem".

So you are assuming that theorem to be true. But how do you refute my "seriously finitistic" idealogy?

Mere semantic disagreement? :3

Haha..

How about the fact that we can sort of verify FPA for M such that we can compute M·M in the real world?

Which tells us empirically that PA is consistent, but doesn't require us to believe that there is a model of PA?

When a physicalist (or someone similar) says something like "xyz does (not) exist", I just interpret them as saying "xyz does (not) physically exist". Then if they say "nonphysical objects don't exist", that's just to say "nonphysical objects don't physically exist", which is tautological. The a similar interpretation applies to anyone who wants to restrict existence by any other adjective; "real" or "current (temporal)" or whatever else.

If they don't want me thinking that way, then they'll have to find a way to rule out conceptual objects on their own terms... but to say that a concept doesn't conceptually exist is just to say that it's impossible, i.e. contradictory.

(To be clear, I meant above that M 'keeps increasing'; it can't be constant after all.. It's supposed to reflect the maximum natural we have computed so far. From the perspective of ACA0, it's non-standard.)

@JadeVanadium Sure, but my whole point of talking about real-world models is that it seems to provide justification for existence of a model of PA. Without the real world having anything like ℕ, we wouldn't even come up with PA, let alone believe it is consistent.

If we don't believe that a real-world model of PA exists, why should we believe a conceptual one exists?

I don't know, honestly. I have trouble imagining a world where something like PA isn't useful.

Same here.

But don't forget that the Henkin construction uses finite strings (modulo provable equality of associated objects), and for us to believe that this makes sense we need to believe that there is a collection of all finite strings (or at least potentially).

We believe that because so far whenever we have two strings in physical medium we can concatenate them or examine their symbols...

XD

But because PA is so useful, therefore I come up with FPA to try to explain why perhaps no model of PA exists but it is still consistent (as far as we can ever tell).

The catch is that from a set-theoretic model of PA we can construct a model of Z2 (full 2nd-order arithmetic), but from FPA we can't get much at all...

Hmm I need to go now so.. see you next time! =)

Okie see ya later :)