Does infinity cause incompleteness in formal systems? Is a finite formal system complete?

Like most, I’m having a hard time understanding the consequences of Gödel’s Incompleteness Theorems. In particular, I’d like to understand their connection to the concept of infinite mathematical structures. In doing so, I hope to formulate a better opinion on the merits of constructivism and finitism in regards to Gödel’s theorems. Without being philosophical, I … Read more

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 would avoid these divergences) possibly help to justify renormalization? Is there a clear reason why this would or would not be plausible? Answer AttributionSource : Link … Read more

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 “homomorphism” $\phi:\mathbb{Q}\to\mathbb{Z}/p$ given by $\frac{a}{b}\mapsto a\cdot b^{-1}$, which satisfies all the usual requirements ($\phi(ab)=\phi(a)\phi(b)$, $\phi(a+b)=\phi(a)+\phi(b)$ etc.) except that it isn’t well-defined ($\phi(\frac{p\cdot a}{p\cdot … Read more

Is there a finitist semantics for transfinite mathematics? [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed last year. Improve this question I’m sympathetic to the Aristotelian view that potential infinity makes sense while actual (completed) infinity doesn’t. However, I … Read more

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

Closed. This question needs to be more focused. It is not currently accepting answers. Want to improve this question? Update the question so it focuses on one problem only by editing this post. Closed 3 years ago. Improve this question The axiom of infinity implies that there exist infinite sets. We can construct the natural … Read more

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-finite-math-need-the-axiom-of-infinity Math without infinity If all sets were finite, how could the real numbers be defined? Answer One issue here is the meaning of “finitistic system”. PRA is usually considered … Read more

Are the computable reals finitary?

In the comment thread of an answer, I said: The computable numbers are based on the intuitionistic continuum, and are not finitary. To which T.. replied: Computable numbers are not based on the intuitionistic continuum. This disagreement contains, I think, a good example of a philosophical question: are the computable reals within the scope of … Read more

Ultrafinitism and the denial of existence of ⌊eee79⌋\lfloor e^{e^{e^{79}}} \rfloor

I was reading about Ultrafinitism and the denial of existence of ⌊eee79⌋ by ultrafinitists. I am wondering if they were to deny the existence of ⌊eee79⌋ shouldn’t they actually deny the very existence of e in the first place, let alone forming eee79. Since e in itself is defined/obtained as a limit, if the ultrafinitists … Read more

Is an ultrafinitist way around Gödel incompleteness theorems?

I know that a similar question has been asked regarding finitism, but I’m interested in ultafinitism. That is, we define a set of numbers that has a specific upper limit. For argument’s sake – let’s say there are only 2 numbers: 0 and 1. So 1+1 is undefined because there is no number 2… Does … Read more

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 a certain sentence, and more generally that that theory does not prove any of a certain class of sentences. I … Read more