All Questions
12 questions
8
votes
3
answers
1k
views
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
17
votes
3
answers
3k
views
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
3
votes
0
answers
283
views
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 ...
- 4,597
7
votes
1
answer
341
views
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
68
votes
4
answers
12k
views
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 ...
- 25.5k
12
votes
2
answers
868
views
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
20
votes
3
answers
2k
views
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
4
votes
0
answers
203
views
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
6
votes
3
answers
1k
views
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
12
votes
1
answer
1k
views
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
4
votes
2
answers
924
views
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
8
votes
1
answer
897
views
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 ...
- 35.5k