# Can we give a well-motivated distinction between finitary and non-finitary mathematics?

I'm reading up on Hilbert, and wondering if there's actually anything fundamental to his distinction between finitary and infinitary mathematics. His system seems to be an attempt to avoid too much ontological commitment (though from what I understand, he is still ontologically committed to finitary numbers as abstract objects, though he would protest otherwise - correct me if I'm wrong). However, given that his plans famously failed, might it be better for us to just have platonism with ZFC - for us to accept that we in some way intuit the truth of the relevant axioms (as we must at some point in our reasoning regardless) and deny any real epistemological distinction between finitary and infinitary mathematics - i.e. our claims about both are just as true.

I get the feeling I'm missing something epistemologically troubling about infinitary mathematics, so if I am please tell me!

• Except for a small number of hardcore platonists neither finitism nor ZFC are about "truth", and if they were what is "better" would be moot. What is "better" depends on one's purposes, among other things. There is more or less a consensus that the more finitist a theory is the "safer" it is, less likely to turn up paradoxes, unfeasible procedures, or behaviors sensitive to what is indistinguishable in practice. Classical mathematics of ZFC is the common compromise between safety and efficiency, grades of finitism sacrifice efficiency for more safety. May 26, 2018 at 4:28

The issue must be understood historically, in the context on the debate about de Foundation of mathematics (1900-1930s).

The development of Intuitionistic mathematics (introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966)) was a radical challenge for classical mathematics :

Besides the rejection of the principle of the excluded middle, intuitionism strongly deviates from classical mathematics in the conception of the continuum, which in the former setting has the property that all total functions on it are continuous. Thus, unlike several other theories of constructive mathematics, intuitionism is not a restriction of classical reasoning; it contradicts classical mathematics in a fundamental way.

David Hilbert, one of the most influential and universal mathematicians of the 19th and early 20th centuries, developed his Program as a way to overcame Brouwer's challenge :

to develop meta-mathematics as a rigorous theory about formal mathematical theories. The main goal was to formalize axiomatic theories and then prove their consistency.

If the Program would succeed in the process of proving the consistency of analysis and set theory (the theories involved with the infinite) with a limited amount of resources, i.e. resources accepted by Intuitionists (the so-called finitary mathematics), Brouwer's challenge will be defeated.

This is the root of the interest into finitary mathematics, like Primitive recursive arithmetic.

In 1931, Kurt Gödel - with his Incompleteness Theorems - showed that the "trick" of proving the consistency of arithmetic using as meta-theory a weaker theiry was impossible.

• There are some complete consistent arithmetics where Gödel's theorem can't be formulated. Like Presburger arithmetic (and any decidable arithmetic). They can prove their own consistency. May 25, 2018 at 19:17
• @rus9384 Another arithmetic which is a personal favorite of mine is that of Willard, which admit multiplication but it is not provably a total function. Jul 25, 2018 at 5:33
• @rus9384 Presburger arithmetic can't even state its own consistency in any meaningful sense: the language is too restrictive for any version of Godel numbering to be appropriately "internalizable" (indeed, we don't even have pairing operators). That said, self-verifying systems do exist - see the work of Dan Willard. Dec 26, 2019 at 21:58