# Questions tagged [finitism]

This tag concerns topics in finitist philosophy, its implications in mathematical logic, and the practical consequences to other areas of mathematics. Use (finitism) for classical finitism and strict finitism, and (ultrafinitism) for ultrafinitism.

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### What part of arithmetic can be founded on recursive functions and without unbounded quantification?

Reading Skolem's 1923 Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlicher mit unendlichem Ausdehnungsbereich (Foundation of elementary ...
325 views

### A model-theoretic question re: Nelson and exponentiation

EDIT: I am not asking about the validity of exponentiation, or PA. My question is about a specific technical claim which Nelson makes in this article (pp. 9-12): that a certain theory does not prove ...
521 views

### Why do finitists reject the axiom of infinity? [closed]

The axiom of infinity implies that there exist infinite sets. We can construct the natural numbers without this axiom, but we cannot put them together in a set, as this would violate this axiom. The ...
126 views

### Precise definition of relative consistency in Kunen's “Set Theory”

I'm reading Kunen's "Set Theory" (Revised edition 2013). On page 108 he defines for axiomatic set theories $\Lambda, \Gamma$ which are strong enough to formalize finitistic arguments (e.g. ZFC, Z, ...
52 views

### Regularity in finitist models?

I understand there are various definitions of "finitist," so I'll be clear: by "finitist," I mean that any collection not finite is treated as a proper class. That is to say, such collections "exist" ...
59 views

### Can there be a number which is provably larger than any number, yet is provably not infinite [closed]

Suppose a natural number N. Is it possible for this number to have the following properties: The number is finite. The number is greater than any other natural number.
148 views

### How do we distinguish between characteristic 0 and characteristic p for very large p?

This is a somewhat soft question, apologies if it turns out to be trivial/nonsensical. Background: I was half-asleep one morning, not quite through my first cup of coffee, and thought about the "...
627 views

### Calculus in finitistic systems

I was just curious if there were some approaches to prove major theorems of calculus in finitistic systems like PRA? Some related questions are, e.g., https://mathoverflow.net/questions/551/does-...
150 views

### Can finitism justify renormalization?

If ultraviolet divergences in Feynman diagrams involve arbitrarily short time periods, approaching infinity, then can a finitist approach to time (assuming, perhaps, a limit to the time lengths that ...
8k views

### How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
671 views