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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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 ...
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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 ...
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Introduction to mathematical logic from a finitist (and preferably also formalist) perspective

I'm soon going to be a 3rd year undergrad in pure math, and next semester i will have a class about logic. I already had a logic class before so have some familiarity with first order logic. I've ...
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Establishing First Order Logic and basic results with PRA

I am just beginning my study of mathematical logic (I’ve worked through the first 7 chapters of Kleene’s Introduction to Metamathematics) and like many others who are studying FOL for the first time, ...
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Is there a finitist semantics for transfinite mathematics? [closed]

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 ...
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Is there a mathematical theory postulating something like the existence of the maximum (natural) number? [closed]

My math got pretty rusty since college, so please forgive the naivete and imprecise formulation of my question. I vaguely recall a mathematician telling me at a party something to the effect that ...
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2 votes
1 answer
201 views

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 ...
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10 votes
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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 ...
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Developing model theory in the language of PA

Is it possible to develop model theory for models of $PA$, inside $PA$ itself (augmented with consistency raising assumptions such as $Con(ZFC)$ if necessary, but still in the language of $PA$)? What ...
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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 ...
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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 ...
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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 ...
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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, ...
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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" ...
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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.
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3 votes
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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 "...
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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-...
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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 ...
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167 votes
10 answers
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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 ...
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10 votes
1 answer
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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 $\lfloor e^{e^{e^{79}}} \...
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40 votes
1 answer
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$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. ...
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7 votes
4 answers
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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 ...
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41 votes
4 answers
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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 ...
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