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.
17
questions
1
vote
1answer
74 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 ...
8
votes
1answer
309 views
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 ...
3
votes
1answer
66 views
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 ...
3
votes
0answers
76 views
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 ...
19
votes
1answer
365 views
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 ...
5
votes
3answers
674 views
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 ...
2
votes
1answer
158 views
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, ...
1
vote
0answers
57 views
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" ...
-1
votes
1answer
64 views
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.
3
votes
2answers
166 views
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 "...
6
votes
1answer
661 views
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-...
3
votes
0answers
158 views
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 ...
155
votes
10answers
8k views
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 ...
10
votes
1answer
761 views
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}}} \...
38
votes
1answer
4k views
$e^{e^{e^{79}}}$ 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. ...
7
votes
4answers
429 views
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 ...
38
votes
4answers
3k views
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 ...
