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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.

A non-standard model of arithmetic is, usually, a model of (first-order) Peano Arithmetic which differs from the standard model $\mathbb{N}$, specifically by containing nonstandard numbers. As the standard numbers $0,1,2,\ldots$ are an initial segment of every model of Peano Arithmetic, these nonstandard numbers must be greater than every standard number.

182 questions
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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 a nonstandard amount of time be like?
user223391
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How does Gödel Completeness fail in second-order logic?

So a while ago I saw a proof of the Completeness Theorem, and the hard part of it (all logically valid formulae have a proof) went thusly: Take a theory $K$ as your base theory. Suppose $\varphi$ is logically valid but not a theorem. Then you can…
Red
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What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily practicing mathematicians) I am curious as to what…
Dennis
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Exponentiation and a weak fragment of arithmetic

The title is slightly misleading, since the theory I'm looking at is an extension of PA; however, the question is "morally" about very weak arithmetic. Specifically, consider the following theory PA': The language of PA' consists of the usual…
Noah Schweber
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Tennenbaum's theorem without overspill

While trying to clean up Wikipedia's proof sketch for Tennenbaum's theorem (there is no computable non-standard model of Peano Arithmetic), the following strategy occurred to me. Since it seems to be simpler than the ones presented in Kaye's 2006…
hmakholm left over Monica
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Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By virtue of his Incompleteness Theorem, the…
logically challenged
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How can induction work on non-standard natural numbers?

When we consider the Peano axioms minus the induction scheme, we can have strange, but still quite understandable models in which there are "parallel strands" of numbers, as I imagine in the picture below: $\quad\quad\quad$ This mental image makes…
M. Winter
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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 the zero polynomial, with their usual…
Russell Easterly
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"Natural" non-standard models of Peano.

The standard model of Peano is particularly natural, being (among other things) the unique model that embeds into any other model of Peano. It's well known that there are many other models of Peano, for example, ones in which there exist…
Beren Gunsolus
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Primes in nonstandard models of PA

What is known about prime numbers in nonstandard models of PA? Restricted to true natural numbers the sets are identical, but does there always exist nonstandard primes? Can we explicitly define one in some model?
user16697
9
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Is this halting time a non-standard integer?

I'm looking to validate (or invalidate) whether this intuition about what non-standard integers can mean is correct. In $ZFC$, we can define a Turing machine $M$ that enumerates all proofs of $ZFC$ and only stops if it finds one of $0=1$. Because of…
Uretki
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Why do models of ZF which are not $\omega$-models have non-standard formulas whose length is "infinitely large natural numbers"?

In his popular book Set Theory: An Introduction to Independence Proofs, Kunen gives the following definitions on the bottom of page 145: Let $\mathcal{A} = \lbrace A, E \rbrace$ be a structure for the language of set theory. Let also $\mathcal{A}…
Pellenthor
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Relationship between nonstandard analysis and nonstandard models of arithmetic

So I've been reading through some introductory texts on nonstandard analysis, basically through the ultrafilter construction of hyperreals and the transfer principle. It seems that much of the time, the notation is directly copied over from the…
Christoph Sachse
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Why aren't there any first-order sentences which have the property of being true in all non-standard models of PA and false in the standard one?

I'd like to know where the following result comes from (that is, whether there is a more general result from which it follows or else how it can be proven): There is no first-order sentence which is true in all non-standard models of arithmetic but…
Cassidy
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Is the standard model for the language of number theory elementarily equivalent to one with a nonstandard element?

On page 89 in A Friendly Introduction to Mathematical Logic, the author writes that the standard model $\mathfrak{N}$ for $\mathcal{L}_{NT}$ is elementarily equivalent to a model $\mathfrak{A}$ that has an element of the universe $c$ that is larger…
Katie Johnson
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