22
votes
Accepted
Did Edward Nelson accept the incompleteness theorems?
Gödel’s second incompleteness theorem requires neither exponentiation nor “impredicative concepts”. The systems Nelson works in are fragments of arithmetic interpretable on definable cuts in $Q$; one ...
- 44.7k
17
votes
Accepted
When we count the same set, must the number always be the same?
Such a model can be constructed using known independence results in bounded arithmetic as follows.
Let $I\Delta_0(f)$ be a theory in the usual language of arithmetic ($0,S,+,\cdot,<$) augmented by ...
- 44.7k
14
votes
Accepted
Dedekind-Peano axioms, but numbers have at most one successor
Victoria Gitman and I recently looked at the first-order version of this theory, which we called fPA, to contrast it with the theory FPA, which Woodin has looked into, which is a possibly stronger ...
- 225k
14
votes
Did Edward Nelson accept the incompleteness theorems?
(EDIT: I have substantially rewritten this answer in light of what I have learned from Emil Jeřábek and from reading some of the relevant references more carefully.)
As Emil Jeřábek has said, the ...
- 78.3k
11
votes
Accepted
The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$
At the request of the OP, I’m writing a lengthy nonanswer showing that there are short proofs of inconsistency of similar theories where the “big number” is given by a term in the usual language of ...
- 44.7k
10
votes
Dedekind-Peano axioms, but numbers have at most one successor
I've looked at the system you've mentioned, where you assume these as mathematical axioms:
(1) Uniqueness of successoring
(2) Uniqueness of predecessoring
(3) 0 doesn't have a predecessor
(4) ...
- 1,844
7
votes
The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$
The length of the least proof of contradiction in $\mathsf{Graham}+\forall n (n<g_{64})$ should be inbetween $(\log_2^*(g_{64}))^{1/N}$ and $(\ln^*(g_{64}))^{N}$, where $\ln^*(x)=\min\{n\mid \log_2^...
- 5,461
7
votes
Dedekind-Peano axioms, but numbers have at most one successor
As pointed out by Emil Jeřábek (in the comment to Joel Hamkins' answer), the system you are referring to is known as "Peano Arithmetic with a top". It has been studied for several decades. A ...
- 17.1k
5
votes
Did Edward Nelson accept the incompleteness theorems?
Your second question has been properly answered by Emil Jerabek, I would say. Reading some of the comments, I feel I should write the following about your first question:
From talking to Ed Nelson ...
- 3,969
5
votes
Can FPA really prove its consistency?
EDIT: I retract. See comments.
I claim the answer is "Yes," that is, FPA (and $IΔ_0+Ω_1$) can prove the consistency of $G$.
Consider the proof system $G$ as found in Logical Foundations of Proof ...
- 1,844
3
votes
Is there any formal foundation to ultrafinitism?
This is an old question, and I'm far from a specialist, but I've recently been reading Dmytro Taranovsky's 2016 preprint Arithmetic with Limited Exponentiation, which seems to me highly relevant to ...
- 3,512
2
votes
Is there any formal foundation to ultrafinitism?
There is a consistent logic which explicitly limits computational complexity of valid statements, and thus is ultrafinitist: https://arxiv.org/abs/2106.13309
The idea is that only computationally ...
- 129
1
vote
Can this weakish system of arithmetic express multiplication for second-sort numbers?
This is not an answer, but a claim that the question is hard. That is, I believe the answer is negative, and there is no formula defining second-sort multiplication. OTOH, if multiplication can be ...
- 1,844
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