Questions tagged [ultrafinitism]

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

Finitism concerns the philosophy of mathematics that rejects the existence of uncountably infinite sets. Some finitists also reject countably infinite sets.

A rough taxonomy of finitists is as follows.

  • classical finitists, who reject only uncountably infinite sets.

  • strict finitists, who reject all infinite sets, including $\mathbb{N}$.

  • ultrafinitists, who reject all infinite sets, and additionally finite structures which are for any conceivable practical purpose.

Many finitists also identify as finitists in practice, meaning that although they acknowledge the existence of various infinite sets, they do not use them when doing mathematics.

11 questions
40
votes
1 answer

$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. I was wondering if that's the case because of…
10
votes
1 answer

Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$

I was reading about Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$ by ultrafinitists. I am wondering if they were to deny the existence of $\lfloor e^{e^{e^{79}}} \rfloor$ shouldn't they actually deny the very…
user17762
10
votes
1 answer

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…
Greg Viers
  • 219
  • 1
  • 5
7
votes
1 answer

What do ultrafinitists think about Graham's number?

I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time). So I wonder about something like the famous "Graham's number" . It's used as (non-binding) upper…
7
votes
1 answer

Zeilberger's potential proof of Fermat's last theorem.

Doron Zeilberger suggested the following potential proof for Fermat's last theorem: Let's define: $$W(n,a,b,c) \equiv (a^n + b^n - c^n)^2$$ I am almost sure that there exists a polynomial, discoverable by computer, with positive coefficients…
nbubis
  • 31,733
  • 7
  • 76
  • 133
4
votes
1 answer

How is ultrafinitism imprecise?

This is similar to "Why isn't finitism nonsense?" but instead of asking about the practical nature (applications), or instead of asking about the definition, I'm trying to understand how ultrafinitism is imprecise (foundations/philosophy). Consider…
Burnsba
  • 877
  • 6
  • 14
4
votes
0 answers

Book on ultrafinitism?

After reading What is "ultrafinitism" and why do people believe it?, I became interested in ultrafinitism and am looking for books or references that I can read to learn more about it. Any recommendations? I am particularly intrigued by the…
3
votes
2 answers

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…
2
votes
0 answers

How much arithmetic can Predicative Second-Order EFA do?

As discussed in this MathOverflow question, I'm trying to find what the result would be of applying a Feferman-Scutte-like analysis to the predicativism of Edward Nelson and Charles Parsons, who believe that mathematical induction is impredicative. …
1
vote
1 answer

Did some ultra-finitists suggest which number should be the largest?

I came across the ultra-finitism, the idea that there is a "largest number". Even most ultra-finitists admit that the "largest number" cannot be exactly defined. Therefore my question : Did some ultra-finist make any suggestion of at least an…
Peter
  • 78,494
  • 15
  • 63
  • 194
-1
votes
1 answer

Can there be a number which is provably larger than any number, yet is provably not infinite

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.
Anon21
  • 2,003
  • 11
  • 20