# 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.

108 questions
Filter by
Sorted by
Tagged with
3k views

### 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 $\varphi$ ...
4k views

### 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 ...
579 views

### 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 ...
404 views

### 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 ...
170 views

### 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':...
338 views

### 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 ...
623 views

### 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 ...
177 views

### 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, ...
210 views

### 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 ...
495 views

### 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 ...
363 views

### 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 ...
100 views

### 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 ...
229 views

348 views

### Is the axiom of induction constructively verifiable for a non-standard model of arithmetic?

There exist models of the natural numbers which include infinite numbers. Such models are called non-standard models of arithmetic. Just like the standard model $\mathbb{N}$, the non-standard models ...
211 views

### The satisfaction relation is undefinable, but does it still “exist”?

In Jech's book "Set Theory" chapter 13, he shows how the satisfaction relation $\models_{n}$ for $\Sigma_n$ formulas can be formalized in ZF. As he pointed out previously, the full satisfaction ...
65 views

### On an explicit model for $\mathbf Q$ with the negation of the axiom of induction

I've been thinking for a while about non standard models of Robinson Arithmetic (here on out referred to as $\mathbf Q$), specifically ones in which induction ($\mathbf{AI}$) fails. This could happen ...
246 views

### The order-theoretic structure of the ordinal numbers in non-standard models

I have read the following. Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a ...
509 views

102 views

63 views

352 views

### Non-standard model of arithmetic - why is adding new constants to the model acceptable?

I have a question about the following proof of existence of a model of non-standard arithmetic (taken from wikipedia): The existence of non-standard models of arithmetic can be demonstrated by an ...
363 views

### First order axiomatization of $\mathbb{R}$

Recently I've been trying to come to terms with the seemingly contradictory facts that (1) $\mathbb{R}$ is the only Dedekind complete ordered field up to isomorphism, and (2) $^*\mathbb{R}$, the ...
119 views

### Are there Countable $\omega$- models of ZFC?

By the $\text{L-S}_\downarrow$ theorem we know that there must exist a countable model of ZFC. Suppose that there is an $\omega$ model of ZFC, then would $\text{ L-S}_\downarrow$ theorem entail that ...
227 views

### embdedding standard models of PA into nonstandard models

Maybe it's well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the ...
89 views