In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the possibility of extra natural numbers beyond the familiar ones. It doesn't accomplish that goal; there ...

What does “standard” in internal set theory really mean? Is it secretly a way of reconciling conventional mathematics with (ultra)finitism? Until recently I thought “standard” was just a way of talking about an elementary submodel but after reading Nelson's later writings it seems to me there is ...

Is there a theory T such that: T includes all the axioms of CZF. T includes the Idealization, Standardization, and Transfer schemas from IST. Every axiom of T is a theorem of IST. T has Church's rule. Explicitly, for every formula $\phi$ in IST's language, if $T \vdash \forall^{st} x \in \omega....

I have in mind something like the following: 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". Th...

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's lemma in constructive mathematics has been extensively studied as noted here. What is the status ...

This question is motivated by the discussion in the comments to this post. The question concerns a comparison of model-theoretic (extension) approaches to nonstandard analysis, and axiomatic (syntactic) approaches such as IST, BST, HST, and others. Consider the following two examples. An intern...