The hereditarily finite sets(a subclass of the Von Neumann universe) are an axiomatic model that corresponds to the usual axioms of set theory but with the axiom of infinity replaced by its negation(showing its independency from the other axioms of set theory).(adsbygoogle = window.adsbygoogle || []).push({});

Some mathematicians (a minority) that are called strict finitists, reject the notion of infinite sets and have this consistent axiom system of hereditarily finite sets model to work with, with wich apparently one can do most if not all of what one can do with the usual axiom system at least in the field of applied mathematics in as much as one can model infinity with finite sets approximately(like computers performing analytic tasks that formally require infinite sets with floating points...) or doesn't have to deal with actual infinite sets in applied science.

There is an important foundational issue that would concern both the finitists and non-finitists that has to do with the very distinction between finite and infinite that is vital for set theory(one can read about it here: https://en.wikipedia.org/wiki/Finite_set#Foundational_issues) but I won't dwell on it now unless necessary (it's thorny and can take us too far from my point).

My questions: is this consistent finitist model an example of the flexibility of the Formalist way to see mathematics and axioms. I mean, is the central idea that one can choose different sets of axioms depending on what one intends to do with them as long as they are consistent, and that there is no specific set of axioms that is "truer" or more fundamental and therefore that should be considered "the foundation of mathematics"?

It apears to me that this is the idea behind Formalism, i.e. that there is no single set of axioms that can be considered the foundation of all math(I know that the particular flavor of Hilbert formalism attempted this in the past but it is my understanding that Godel's theorems of incompleteness prevent it). If so, the reason that most mathematicians consider set theory itself and particularly ZF(C) axiomatic set theory as fundamental is that they are not actually formalists, but rather platonists or something else?

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# A Formal axiom systems and the finite/infinite sets

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