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

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 there is a solid mathematical theory that posits the…
user865086
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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…
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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 would this look like? When working in ZFC, it is…
Mario Carneiro
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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, BST...) $$\Gamma \triangleleft \Lambda \quad\text{…
Popov Florino
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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 always thought of math as being equivalent to…
Carla_
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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" in the sense they can be discussed as a whole,…
user361424
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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, I’ve had uncertainty about the precise role and…
Alex
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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
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