All Questions
Tagged with theories-of-arithmetic mathematical-philosophy
11 questions
68
votes
4
answers
12k
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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 ...
- 25.5k
32
votes
11
answers
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 (...
19
votes
3
answers
2k
views
Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
- 261
15
votes
5
answers
2k
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In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
- 961
13
votes
2
answers
1k
views
nonstandard models and mathematical theorems
Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...
7
votes
1
answer
489
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Is $ACA_0$ + 'True Arithmetic exists' interpretable in $ACA$?
Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a $\Pi_1^1$...
- 71
6
votes
1
answer
209
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Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory?
As is well known, the following theory is equiconsistent with $PA$:
$ZFC$ with the axiom of infinity replaced by its negation.
Since this theory is equiconsistent with $PA$, it would seem ...
- 6,132
5
votes
1
answer
485
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Extensions of the Ackermann interpretation to nonstandard theories of arithmetic
In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set ...
- 6,132
2
votes
1
answer
459
views
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
1
vote
1
answer
629
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In what sense is the "descending chain principle" for ordinals less than $\epsilon_0$ 'infinitary?
In the introduction to his paper "Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type", W.A. Howard writes:
Gentzen...showed that the consistency of first order (...
- 6,132
0
votes
3
answers
1k
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Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:
The axioms of arithmetic are obviously correct, and the ...
- 6,132