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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.

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Hyperfinite intervals are uncountable but nonstandard models of Peano arithmetic can be countable?

My understanding is, by Lowenheim-Skolem I can find a countable nonstandard model of Peano Arithmetic. On the other hand, I have just encountered the following argument: For $\alpha \in {}^*\mathbb{N}...
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1answer
36 views

First-order Peano arithmetic and (the lack of) implicit definition of addition

I'm trying to show, through the existence of non-standard models of arithmetic, that the first-order Peano axioms (without those of multiplication) don't implicitly define addition in the sense of ...
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37 views

Internal set theory: nonstandard ordinals

This topic concerns Edward Nelson's Internal Set Theory. We know that in 'standard math' (just taking ZFC axioms fully) that $z$ is an ordinal iff $z$ is a transitive set and $z$ is totally ordered ...
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1answer
29 views

Internal Set Theory: extensionality (set equality)

In Internal Set Theory we can extend extensionality (set equivalence) via transfer. I'd like to work this out explicitly and then ask my question. Extensionality: $\forall z(z\in x\iff z\in y)\...
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34 views

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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0answers
25 views

Ultraproduct of a Function versus Functions of Ultraproducts

Let $G$ be a group (or even a set for our purposes here) and consider functions from $G$ to $\mathbb{R}$. Now after choosing a non-principal ultrafilter of $\mathbb{N}$, we can construct ultrapowers $^...
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1answer
56 views

Elementary equivalence of standard and non-standard model of arithmetic

There is the common construction of a non-standard model of arithmetic by adding a constant symbol c to the signature and adding $\{n<c:n\in \mathbb{N}\}$ to the theory PA. Now by adding all ...
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106 views

A model of $PA^\omega$ in ZFC [closed]

It is known that the existence of a model (i.e. consistency) of $PA$ is provable in ZFC, the set $\omega$ equipped with usual operation on natural numbers is indeed a model of $PA$, and we can think ...
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62 views

Generalizing the construction of $PA^\omega$

I'm taking an extremely basic course in mathematical logic where we briefly talked about nonstandard models of arithmetic. The only example we worked through has been the construction of $PA^\omega$ ...
14
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1answer
117 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':...
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0answers
80 views

Applications of the compactness theorem. [closed]

It's well know that the compactness theorem has many aplication in model theory, its main shows existence of nonstandars models of aritmetical and the real numbers, and not elementary of some theories ...
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1answer
203 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 ...
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1answer
133 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 ...
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1answer
44 views

Is the union of non-standard analogs of a family of sets a proper subset of the non-standard analog of the union of those sets?

The book on formal logic I'm using for self-study builds the foundation for non-standard analysis (to illustrate the usage of non-standard models) using the following construction. Consider the real ...
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1answer
47 views

Is there a combined theory of the Reals and the Naturals that has a model where the Naturals and Reals have the same cardinality

The Upward Lowenheim-Skolem theory decrees that there must be a (non-standard) model of the naturals of cardinality the same as that of the standard model of the Reals. For any combined theory of the ...
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2answers
164 views

Non-standard integers?

Edit: Realized almost immediately that this was a stupid question - see comment below. Was about to delete it when an answer appeared that seems like saving... Context: I just saw a presentation of ...
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0answers
72 views

Having Trouble Seeing Why Friedman's Theorem (1973) is true.

I am reading about non-standard models of peano arithmetic, and came across a theorem by Friedman that states the following. Every non-standard countable model of (peano) arithmetic is isomorphic ...
3
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1answer
117 views

Any non-standard halting oracles stronger than $\mathbb N$?

Let $M$ be some model of PA. Let $H_M$ be the set of codes of standard turing machines $X$ such that $M \models X \text{ halts}$. For example, $H_\mathbb N$ corresponds to the regular halting oracle (...
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2answers
235 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 ...
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0answers
72 views

Existence of a structure with the same properties as $\mathbb{N}$ and with an infinite element.

So, we are requested to prove the existence a structure $\mathfrak{A}$ - first order logic - that has the same theory as the usual natural numbers and, moreover, there exists an element $\infty\in|\...
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1answer
54 views

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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3answers
133 views

Can a countable model of arithmetic have mutually non-definable numbers?

Let $M$ be a countable model of arithmetic (that satisfies PA). Can we have two numbers $c_1, c_2 \in M$ such that $c_1$ can not be defined in terms of $c_2$, and $c_2$ can not be defined in terms of $...
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2answers
148 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, ...
2
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1answer
175 views

Model of concatenation theory with left-cancellation but no right-cancellation

The theory of concatenation (TC) can be equivalently expressed as the following assumptions: Closure of strings under concatenation $+$. Existence of an empty string $e$, namely $e+x = x = x+e$ for ...
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98 views

Information about $ZFC + (\neg CH)$

$\underline{\text{Some Background}}$: A classic axiomatic system of set theory in modernity is $ZFC$. Results from modern mathematical logic guarantee the existence of statements undecidable given ...
4
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1answer
135 views

What do non-standard cardinalities look like?

The Von Neumann universe $V$ satisfies ZFC, and there are other models within $V$ that are non-standard and satisfy ZFC. If we look at one of these non-standard models $M$ with a non-standard model of ...
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37 views

Why doesn't the transfer-principle hold for the following statement?

Let $\hat{V}$ be the standard universe constructed as $ \hat{V} = V_0\cup V_1 \cup V_2 \cup ....$ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ For the following (...
2
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1answer
127 views

Do non-standard models of arithmetic add properties to what are “intuitively” their standard numbers?

I know that by Tennenbaum's Theorem, non-standard models of arithmetic "don't know" which of their elements form a standard model of arithmetic. However, often facts that are opaque to the model from ...
3
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2answers
130 views

Does an internal set with countably many infinitesimals contain hyperfinitely many infinitesimals?

Let $\mathscr{A}$ be an internal subset of $^*\mathbb{R}$ that contains a countable family, say $\{a_n\}_{n\in\mathbb{N}}$, of infinitesimal members, that is $a_n\approx 0$ for all $n\in\mathbb{N}$. ...
2
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1answer
195 views

Produce a nonstandard model of Robinson arithmetic

I wanted to know if my proof is correct, please, thanks for reading. Robinson arithmetic, $\mathbf{Q}$, is a first-order theory with the following axioms: $1.\space \neg\left(\exists x \left[S\left(...
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1answer
104 views

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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2answers
267 views

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 $\mathbb{N}$ ...
2
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1answer
70 views

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, ...
4
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1answer
54 views

Bound for Non-standard Model of PA

For some non-standard model $M \models PA$, I recently proved that there is some $a \in M$ such that for all quantifier-free formulas $\varphi(x)$, if $M \models (\exists \, x) \varphi(x)$, then $M \...
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4answers
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Showing that the hyperintegers are uncountable

In class, we constructed the hyperintegers as follows: Let $N$ be a normal model of the natural number with domain $\mathbb{N}$ in the language $\{0, 1, +, \cdot, <, =\} $. Also let $F$ be a fixed ...
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0answers
65 views

How do set theorists deal with that every model of ZF might be nonstandard?

The existence of a standard model of ZF is stronger than the assertion of its consistency, but this is not what I mean. It is impossible to first-order say that there is a standard model of ZF. It ...
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0answers
54 views

Model theory of the naturals with a multiplication by an irrational factor

It is well-known that the theory of the structure $(\mathbb{N},<)$ is not stable, but is NIP and has quantifier elimination in the language $L=\{<,0,S,S^{-1}\}$ where $S,S^{-1}$ are function ...
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2answers
268 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 ...
2
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1answer
188 views

What does Robinson's overspill lemma tell us?

My reference is Richard Kaye's Models of Peano Arithmetic, section 6.1. I am wondering what Robinson's overspill lemma actually tells us, and a bit of motivation on it. I have heard of it before, and ...
2
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1answer
92 views

Nonstandard Model of PA where the carrier set is N and Tennenbaum

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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2answers
200 views

Can additional predicates “eliminate” nonstandard models of true arithmetic?

For languages $\mathcal{L} \subseteq \mathcal{L}',$ say that a set of sentences $\Delta$ in $\mathcal{L}'$ eliminates a model $M$ of $\mathcal{L}$ if no matter how the symbols of $\mathcal{L}' \...
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1answer
128 views

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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1answer
49 views

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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3answers
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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 ...
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1answer
108 views

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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3answers
506 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 ...
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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 ...
6
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3answers
567 views

Prime numbers on a non-standard model

I can't imagine how this is possible: Let $\mathcal{M}$ be a nonstandard model of arithmetic. Show that: There is an element $a\in M$ such that for all prime numbers $p$, we have that $\mathcal{M} \...
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2answers
264 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 ...
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1answer
157 views

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 ...