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.

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.

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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 him? The algebraist argues that the real numbers…
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If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural…
AgCl
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$e^{e^{e^{79}}}$ 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…
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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…
Noah Schweber
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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
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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
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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…
Rubi Shnol
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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…
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Why do finitists reject the axiom of infinity?

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 only reason I can think off why you would reject…
user370967
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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 arithmetic through the recurrent method of thought…
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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…
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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…
cartomancer
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Can one do calculus in finitist mathematics

Can one "do calculus" in weak formal systems like primitive recursive arithmetic or elementary function arithmetic? Can one at least explain the physics of classical mechanics in an elementary formal mathematical system like the ones described…
Amr
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Do Gödel's incompleteness results make it futile to use a finitist metatheory?

Most books on introductory logic seem to work on a metatheory where infinite sets are allowed to exist. This seems unnecessary: everything humans do is finite, so seems like it should be enough to assume existence of only finite things. The more we…
Carla_
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Is there a finitist semantics for transfinite mathematics?

I'm sympathetic to the Aristotelian view that potential infinity makes sense while actual (completed) infinity doesn't. However, I also find transfinite set theory to be fascinating, and I'm under the impression that this domain of mathematics is…
Matt D
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