All Questions
Tagged with model-theory peano-axioms
128
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Order type of cuts satisfying $\mathsf I\Sigma_n$
When $M$ is a model of Peano arithmetic, a cut of $M$ is an initial segment $I$ of $M$ such that $I$ is closed under successor. There is some work on cuts that satisfy $\mathsf I\Sigma_n$, Peano ...
- 1,311
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Are the models of PA recursively enumerable?
Is there a Turing machine (TM from now on) which lists every model of PA (without the induction axiom schema, so just addition and multiplication)? More specifically, can such a TM list all infinite ...
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Can we conclude that Peano's axioms consistent from soundness?
One of the corollaries of soundness says that if $\Gamma$ is satisfiable, then $\Gamma$ is consistent. I am wondering whether we can conclude that Peano's axioms $\mathsf{PA}$ is consistent from the ...
- 589
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Model Theory in the Language of Peano Arithmetic
Most introductory textbooks on model theory establish the theory based on the ZF set theory (e.g. [1]). In particular, a structure is defined to be a 4-tuple of sets, and so on. In [2], I came to ...
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-3
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Why Löwenheim–Skolem theorem asserts the non-existence of such predicates in 1st order logic
Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it ...
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Is pointwise definability of a model of PA equivalent to it being the standard model? [duplicate]
The standard model of Peano Arithmetic is pointwise definable, because every finite natural number is parameter-free definable. What about the converse? That is, if a model $M$ of PA is pointwise ...
- 21.5k
3
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196
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What are the parameter-free definable elements of a model of Peano Arithemetic?
Let $M$ be a model of Peano Arithmetic. What are the parameter-free definable elements of $M$? I conjecture that they are precisely the standard natural numbers, meaning, no nonstandard infinite ...
- 21.5k
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On the consistency of satisfiable first order theories
Considering this question, we know that a first order theory that admits a model has to be consistent.
A model for a theory $T$ in a language $\mathcal L$ is an interpretation of $\mathcal L$ in which ...
- 987
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1
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115
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Are the natural numbers definable in ZFC-Inf
While reading up on the incompleteness theorems on this page, I came upon the statement that the incompletess theorems hold in ZFC-Inf. According to the page ZFC-Inf is 'proof-theoretically equivalent'...
- 149
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195
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If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?
Suppose Goldbach's conjecture $G$ is undecidable in first-order Peano arithmetic, $\sf{PA}$. That would mean there are models in which $G$ is true and other in which it is false. But intuitively, this ...
- 6,672
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Mistake in proof that $\textbf{P}^- +\textbf{I}\Delta^0_0$ proves infinitude of primes
I know that it is still an open question whether $\textbf{P}^- +\textbf{I}\Delta^0_0$ is able to prove that there are infinitely many primes. I know that $\textbf{P}^- +\textbf{I}Open$ cannot prove ...
- 1,926
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52
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Show there is a term t and formula $\theta$ such that $\mathbb{N} \models \forall x<t \theta(x)$ and $\mathbb{N} \models \forall x<t \lnot \theta(x)$
I've been reading Richard Kaye's book, Models of Peano Arithmetic, and I saw this question as an exercise.
Let $\mathbb{N}$ be the standard model of the natural numbers. Give a closed term $t$ and a ...
- 303
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74
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At what number of variables does provability in Peano Arithmetic differ from truth in the standard model?
Let our language be $\{+,\cdot,<,0,1\}$. Let $T_n$ be the set of of true statements in that language (true in the standard model $\mathbb{N}$) that have at most $n$ bound variables and no other ...
- 21.5k
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117
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Are there Nonstandard Models of Arithmetic that don't Add Additional Axioms?
Apologies if this is an elementary question that should have been obvious to me. I am learning about these topics very much from the perspective of an outside hobbyist, and am not a wizard of logic.
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288
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Can Goodstein's theorem be false in some model of Peano arithmetic? How?
I read that Goodstein's theorem is not provable in Peano arithmetic, but is provable in ZFC.
Now, using Godel's completeness theorem, we can say that, since Goodstein's theorem is not provable in ...
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