Questions tagged [ultrafinitism]
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22 questions
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How strong is exponentiation with only open induction? (Or: "how low can we go?")
Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
- 3,612
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votes
0
answers
281
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What is the meaning and proof of Harvey Friedman’s ultrafinite incompleteness sentence?
On page 7 of his paper “Adventures in Incompleteness”, Harvey Friedman states the following:
IN ANY LONG ENOUGH SEQUENCE $x_1,...,x_n$ FROM $\{1,2,3\}$, SOME $(x_i,...,x_{2i})$ IS A SUBSEQUENCE OF ...
- 4,587
8
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3
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Dedekind-Peano axioms, but numbers have at most one successor
One can consider a variant of the Dedekind-Peano axioms
in which one replaces the assumption that every number
has exactly one successor by the assumption that every
number has at most one successor, ...
- 19.7k
7
votes
1
answer
341
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Can this weakish system of arithmetic express multiplication for second-sort numbers?
Consider a 2-sorted first-order logic with equality (for first-sort entities). The first sort consists of numbers, the second sort (which will be capitalized) of unary functions. There is one constant,...
- 1,974
17
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3
answers
3k
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Did Edward Nelson accept the incompleteness theorems?
Edward Nelson advocated weak versions of arithmetic (called predicative arithmetic) that couldn't prove the totality of exponentiation. Since his theory extends Robinson arithmetic, the incompleteness ...
- 675
4
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0
answers
202
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The Return of Graham Arithmetics: adding induction up to $g_{64}$
In my previous question The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$, I introduced an extension of Robinson Arithmetics with the recursive definition of Tetraction, a small ...
- 7,853
12
votes
2
answers
865
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The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$
As you all know, Ronald Graham just passed away. He is famous for many fabulous contributions to finite combinatorics, and much much more, but perhaps none of them is as popular as the infamous ...
- 7,853
12
votes
1
answer
738
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When we count the same set, must the number always be the same?
Return to Frege's question, What justifies arithmetic? And consider the ur-proposition that counting a finite set always produces the same number, and ask whether this has a logical justification, ...
- 1,974
9
votes
3
answers
622
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Model-theoretic accounts of feasibility in bounded arithmetic and related systems
Various weak theories of arithmetic have been partially motivated by a concern with numbers (or functions/proofs) that are feasible. This concern is sometimes connected to an interest in strictly ...
- 91
5
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1
answer
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Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?
Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...
- 4,587
38
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4
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Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic
Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
- 4,587
12
votes
1
answer
1k
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Is an ultrafinitist Hilbert's program doomed?
Hilbert's program is popularly understood as an attempt to justify infinitary mathematics with a finitary consistency proof. Godel's Second Theorem is usually considered as showing this is not ...
- 1,974
20
votes
3
answers
2k
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Can FPA really prove its consistency?
I will ask the question first and then explain.
QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency?
FPA is a multi-sorted first-order theory,...
- 1,974
6
votes
3
answers
1k
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Provability in Second-Order Arithmetic without the Successor Axiom
Consider second-order Peano Arithmetic Z2, i.e. the two-sorted first-order theory with induction and comprehension. Remove the assumption about the totality of the successor relationship (the ...
- 1,974
9
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4
answers
1k
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Does there exist a non-trivial Ultrafinitist set theory?
Does there exist a set theory T-which has not yet been proved to be inconsistent-and in which
one can prove the existence of (1) the empty set (2) sets that are singletons and (3) sets which
have non-...
- 6,215
10
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3
answers
3k
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What is the status of irrational numbers within finitism/ultrafinitism?
According to constructivism, "it is necessary to find (or "construct") a mathematical object to prove that it exists". There are several formulas to calculate $\pi$, such as:
...
- 597
8
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1
answer
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Nelson natural number objects in a topos (say)
Nelson's predicative arithmetic (survey article) is a very weak system of arithmetic extending Robinson's $Q$ (Wikipedia).
We can have natural number objects in a topos, or even a merely finitely ...
- 35.4k
6
votes
1
answer
1k
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How are mathematical objects defined from an ultrafinitist perspective?
I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
- 4,341
0
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1
answer
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Is it possible to construct a finite mathematical universe? [duplicate]
Possible Duplicate:
Is there any formal foundation to ultrafinitism?
Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...
- 19
4
votes
2
answers
922
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Natural numbers of great kolmogorov complexity
Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, ...
- 7,853
89
votes
10
answers
17k
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Is there any formal foundation to ultrafinitism?
Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to Wikipedia, it has been ...
- 1,001
68
votes
4
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Nelson's program to show inconsistency of ZF
At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...
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