4 ms·

> You can't create the set of natural numbers even with the axiom of infinity either, at least not in any first order logic. No first order logic is sufficient

> You can't create the set of natural numbers even with the axiom of infinity either, at least not in any first order logic. No first order logic is sufficient to strictly define the set of all natural numbers. Eh that's a fundamentally different problem though, although that is part of the reason why finitism and ultra-finitism are attractive. You can define a set that is the natural numbers for all intents and purposes from the viewpoint of your theory. If you were willing accept that something like PA is enough to do large parts of real analysis (and also believed in the usefulness of real analysis), then philosophically you should be totally fine with using theories that have models with nonstandard natural numbers, since "you can't observe any of the consequences from your theory" is the same bar in both cases (in the case of finitistic approaches to analysis it's even trickier because often you can only deal with encodings of objects rather than objects themselves and must rely on your metatheory to do the association of the encoding to the object).

Kranar 4y ago
I don't agree it's a different problem, as OP pointed out, ZF with the axiom of infinity replaced by its negation is equal to first order Peano Arithmetic, sometimes referred to as the theory of hereditarily finite sets. Said theory is also sufficient for all intents and purposes of doing real analysis.
- dwohnitmok 4y ago
  > I don't agree it's a different problem, as OP pointed out, ZF with the axiom of infinity replaced by its negation is equal to first order Peano Arithmetic, sometimes referred to as the theory of hereditarily finite sets. Being unable to talk about the set of natural numbers except through encodings (which is how you do infinitary math without infinity) is different than having a symbol for the set of natural numbers which may or may not have additional properties which cannot be discerned within the theory itself (e.g. Lowenheim-Skolem). > Said theory is also sufficient for all intents and purposes of doing real analysis. Large chunks of real analysis can indeed be carried out, but significant amounts cannot. Many theorems in analysis concerning perfect sets are not provable in that theory (usually stated as ACA0).

Kranar 4y ago

I don't agree it's a different problem, as OP pointed out, ZF with the axiom of infinity replaced by its negation is equal to first order Peano Arithmetic, sometimes referred to as the theory of hereditarily finite sets. Said theory is also sufficient for all intents and purposes of doing real analysis.

dwohnitmok 4y ago
> I don't agree it's a different problem, as OP pointed out, ZF with the axiom of infinity replaced by its negation is equal to first order Peano Arithmetic, sometimes referred to as the theory of hereditarily finite sets. Being unable to talk about the set of natural numbers except through encodings (which is how you do infinitary math without infinity) is different than having a symbol for the set of natural numbers which may or may not have additional properties which cannot be discerned within the theory itself (e.g. Lowenheim-Skolem). > Said theory is also sufficient for all intents and purposes of doing real analysis. Large chunks of real analysis can indeed be carried out, but significant amounts cannot. Many theorems in analysis concerning perfect sets are not provable in that theory (usually stated as ACA0).