# Questions tagged [nonstandard-models]

For questions specifically concerning models of arithmetic (which could be Peano Arithmetic, the first-order theory of the natural numbers, or some other system) which differs from the standard model by the existence of nonstandard numbers.

85 questions
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### Internal Set Theory: Start with nonstandard finite E, then the standard subset of E is infinite?

I'm working on understanding Edward Nelson's Internal Set Theory. I wonder if folks can check my reasoning. I need to reference a result which says that if E is a set then the following two ...
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### Show that there is a nonstandard model of the set of all sentences that are true under natural numbers

Let $L$ be a language for arithmetic on the natural numbers $\mathbb{N}(=\{0,1,2,...\})$ including $0,1,+,\cdot$ and $>$. Let $Th(\mathbb{N})$ be the set of all sentences of $L$ true in $\mathbb{N}$...
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### Intuitive Demonstration of Tennenbaum Theorem?

Does the following provide an "intuitive demonstration" of Tennenbaum's theorem ? A countable Non Standard Model of PA has its domain of the form ($\mathbb{N}$, Z-Chain). As the domain is countable ...
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### Are we in a nonstandard model of arithmetic?

My question is: is $\mathbb{N}$ a nonstandard model of arithmetic for someone else? Let $\mathbb{N}^*$ be a nonstandard model of Peano Arithmetic. Then that consists of a copy of $\mathbb{N}$ ...
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### Nonstandard Model of PA where the carrier set is N

The link Not Skolem's Paradox - Part 3 includes the following words in a comment : "By carefully tracing the details of the usual compactness argument and the proof of the completeness theorem, ...
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### Doing math with nonstandard numbers

Assume we have a computable, nonstandard model of Robinson arithmetic consisting of the set of integer-coefficient polynomials with positive leading coefficient, plus the zero polynomial, with their ...
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### example of nonstandard model of PA that is not recursively saturated

I know that every nonstandard model of PA realizes any recursive type of given quantifier complexity (say $\Sigma_n$, for some $n$). I suppose there must be recursive types that are not always ...
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### What is an example of a non standard model of Peano Arithmetic?

According to here, there is the "standard" model of Peano Arithmetic. This is defined as $0,1,2,...$ in the usual sense. What would be an example of a nonstandard model of Peano Arithmetic? What would ...
1answer
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### Overspill in computable nonstandard models

Tennenbaums' theorem does not apply to Robinson Arithmetic ($Q$). There is a computable, nonstandard model of $Q$ "consisting of integer-coefficient polynomials with positive leading coefficient, plus ...
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### How does induction fail in computable nonstandard models?

Tennenbaum's theorem does not apply to Robinson Arithmetic ($Q$). There is a computable, nonstandard model of $Q$ "consisting of integer-coefficient polynomials with positive leading coefficient, plus ...
1answer
104 views

### Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
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### Does Tennenbaum's theorem apply to Modular Arithmetic?

I recently asked on math.stackexchange Are the algebraic numbers recursive? I had assumed the field of algebraic numbers is a model of a theory I call Modular Arithmetic. I also assumed Tennenbaum's ...