All Questions
Tagged with theories-of-arithmetic foundations
14 questions
3
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0
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283
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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 ...
- 4,597
1
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0
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Can this type theory interpret second order arithmetic?
Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
- 11.3k
8
votes
1
answer
574
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Iterated Gentzen: or, a Sith objection to the proof of consistency of PA
$\DeclareMathOperator\PRA{PRA}\DeclareMathOperator\WF{WF}\DeclareMathOperator\Con{Con}\DeclareMathOperator\PA{PA}$Preamble: In the year … in a galaxy far far away, a nasty Sith named Darth Dubious (...
- 7,853
10
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1
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An internalized version of Tennenbaum's Theorem
Tennenbaum's celebrated 1959 theorem (see here for a reference) is certainly one of the key theorems in mathematical logic. Not so much for its proof, but because it helps "isolating" $N$ ...
- 7,853
1
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1
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397
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Complete and consistent first-order theories that contain interesting phenomena
Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete.
I think there is some sentimental value in working with a theory ...
user158636
12
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2
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868
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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
16
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2
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2k
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Could Kronecker accept a proof of Goodstein's theorem?
A famous result of Goodstein asserts that the Goodstein sequence of integers terminates.
For a precise statement and a short proof, see https://en.wikipedia.org/wiki/Goodstein%27s_theorem.
A well ...
- 28k
3
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What does second order set theory give us that is new?
There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...
- 18.7k
6
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2
answers
1k
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Are omega-consistent extensions of PA always consistent with each other?
The question is as in the title. In the edit history you can find my attempt to formalise the question, but that was a failure, for reasons stated clearly in the comments. Thus, my question is just:
...
- 1,344
2
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1
answer
459
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Is the statement "All numbers are counting numbers" independent of $PA$?
In his paper, "Completed versus Incomplete Infinity in Arithmetic" (which can be found here), the late Edward Nelson defines the notion of 'counting number' as follows:
0 is a counting ...
- 6,132
9
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1
answer
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Does Nelson try to prove PA inconsistent directly?
Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...
- 28.2k
8
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3
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Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?
A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
- 4,597
3
votes
1
answer
405
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Godel 's Ladder: Undecidable PI_N sentences for N =2, 3, ......
After Godel's groundbreaking results, a plethora of $\Pi_1^0$ undecidable arithmetical sentences have been found by many authors.
But what about $\Pi_n^0$ for $n=2,3,.....$ ?
There are, to my ...
- 7,853
32
votes
11
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11k
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Is PA consistent? do we know it?
1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs (...