# 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.

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### Interpretation of the transfer principle.

I am reading an article written by W.A.J Luxemburg about nonstandard analysis (https://www.jstor.org/stable/3038221). My question is: Why does the transfer principle transform sentences and predicates ...
• 55
1 vote
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### Confusion about the model in Robert’s Nonstandard Analysis

I’m working through Alain M. Robert’s “Nonstandard Analysis”. I’m intrigued by his suggestion that his axiomatic approach, rather than adding elements to ℕ to create a nonstandard model *ℕ, “discerns ...
• 5,777
1 vote
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### Gödels incompleteness theorem false for natural numbers

Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then ...
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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 ...
• 6,469
1 vote
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### Does anyone describe Henkinization as closing under Skolem Functions?

When I'm constructing a nonstandard model by by adding in a new constant symbol and invoking compactness, it feels much more like I'm adding in $c$ and "closing under Skolem functions" (at ...
• 2,922
1 vote
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### If I use ZFC as a metatheory for FOL, doesn't it make it weaker?

Basically, there is a theorem of first-order logic that says that no L-structure whose domain is infinite can be axiomatized up to isomorphism, so in particular there is no set $\Phi$ of formulas that ...
• 75
133 views

### Extending a Model of $T + \operatorname {Con} ( T )$ to a model of $T + \neg \operatorname {Con} ( T )$

Let $T$ be a recursively axiomatizable extension of $\mathsf {PA}$ and $\mathfrak M$ be a model of $T + \operatorname {Con} ( T )$. Is it true that there must exist a model $\mathfrak N$ ...
• 6,683
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### Is there a "closed form" expression for the powerset of a complement?

Let us consider some consistent subset of naïve set theory, in which a universal set $U$ exists, the power set $\mathcal{P}(A)$ and complement $A'=U-A$ exists of any set $A$, on top of the usual ...
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