I have in mind something like the following:




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Start with some suitable version of "finite" mathematics. Some possibilities might be maybe ZFC with a suitable anti-infinity axiom, the topos $\mathbf{FinSet}$, Peano arithmetic, Turing machines... something whose objects are suitably "finite".


Then, posit the existence of both a standard and a non-standard model.


Now, in this setting, where we have access both to a standard model and a non-standard extension, use the non-standard objects as proxies for infinite objects (e.g. maybe some sort of set theory that has a set of natural numbers), and develop ordinary mathematics this way.




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Has anybody worked on such a thing? Does anyone know of references of it being done? Or suggestions that it can't work out?


(P.S. I wasn't sure how to tag this....)




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Edit: After more thought and reviewing the answers thus far, I think I can state an example of the sort of thing i was imagining. Define a first-order theory with two types $T\_1$ and $T\_2$, two binary relation symbols $\in\_1, \in\_2$ (one for each sort), and a map $\tau : T\_1 \to T\_2$ satisfying:


* $(T\_1, \in\_1)$ satisfies the axioms of finite set theory
* $(T\_2, \in\_2)$ satisfies the axioms of finite set theory
* $\tau$ is injective
* $\tau$ is not surjective
* $\tau$ satisfies an axiom schema that says it's an elementary embedding


and the question is to what extent we can develop infinite set theory in this theory.



Infinite mathematics as non-standard finite mathematics?
The standard system of a first-order model of Peano Arithmetic works in this way.


The standard model $\mathbb{N}$ is an initial segment of every nonstandard model $\mathcal{M}$. Pick a nonstandard element $w$ of $\mathcal{M}$. The binary expansion of numbers in $\mathcal{M}$ below $2^w$ define nonstandard binary strings of length $w$. For each such $x$, there corresponds a subset $X$ of $\mathbb{N}$ where a standard number $n$ belongs to $X$ if and only if the $n$-th bit of $x$ is $1$. The collection of all these sets $X$ is called the standard system $\mathrm{SSy}(\mathcal{M})$ of $\mathcal{M}$. This standard system is always a Scott set, so it can be used to form a standard model of second-order arithmetic where the weak König lemma is true. The structure of the first-order model $\mathcal{M}$ dictates a lot of the properties of this second-order model.


Similar constructions can be done in different contexts, but this is surely the most common one.


