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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3answers
212 views

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 ...
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1answer
62 views

Constructing an infinitely large natural number such that it is not divisible by any number $n \in \mathbb{N}_{>1}$

Apparently, we can use compactness theorem to construct an infinitely large natural number such that it is not divisible by any (standard) natural number $n \in \mathbb{N}_{>1}$. And I must say ...
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1answer
146 views

Can theories with axioms beyond arithmetic make false promises about integer existence?

In this paper, author Nik Weaver warns that there could be questions of $\Sigma_1$-validity of ${\mathrm{ZFC}}$ set theory. As I understand it, he suggests that the axioms of a set theory might be ...
4
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0answers
84 views

Is there a least standard model of Peano Arithmetics?

Here's the informal part: I'm trying to get a better intuition about the differences between the standard model and the non-standard ones. Some people seem to be really good at imagining what non-...
1
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1answer
23 views

Totally order turing machines by halting

Does there exist some computable relation "<" on turing machines, such that the set of machines that halt is an initial segment. (If A and B are Turing machines, and A halts and B doesn't ...
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0answers
102 views

Can all $\mathsf{Q}$-provably recursive functions be “frequently termlike”?

Now asked at MO. Motivated by this question, I'd like to ask whether in a precise sense there are no "interesting" functions which are provably recursive in Robinson's arithmetic $\mathsf{Q}...
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1answer
55 views

How to add new axioms to classical Peano Arithmetic to obtain a non-standard theory.

What is a simple (the simplest?) axiom which can be added to the usual PA axioms so that the new "non-standard PA theory" no longer has the Standard Model as one of its models? Assuming of ...
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0answers
56 views

Kripke's proof of the incompleteness of PA

In Hillary Putnam's writeup on Kripke's beautiful incompleteness proof of PA, which I learned about from Noah Schweber's answer here, I can convince myself that (see the top of p.56) "$s$ ...
2
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1answer
43 views

Clarification on internal sets

I am trying to better understand a definition given by Robinson on page 823 of the paper "Nonstandard Arithmetic". Fix a field $F$ and Galois extension $\Phi$ of $F$ (possibly of infinite ...
4
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1answer
78 views

Are there countable non-standard models of true arithmetic formulated in the uncountable “full (first-order) language of arithmetic”?

Fix the standard language of arithmetic as, for example, $L_A ≔ ⟨0,1,+,×,<⟩$. Define the full (first-order) language of arithmetic, notation $L_\text{full}$, as the first-order language with the ...
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2answers
87 views

What is the hyperreal multiplicative inverse of $1 + \epsilon$, and how do we show it exists?

What is the multiplicative inverse of $1 + \epsilon$, in the ordered field of hyperreals or surreals? Simple algebra shows it must be equal $1-\epsilon+\epsilon^2-\epsilon^3+\epsilon^4...$ But how do ...
4
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1answer
58 views

Understanding non-standard arithmetic models.

I am reading Enderton's Logic Book and I can't understand the proof that there is a non-standard arithmetic model. The proof is typical using the compactness theorem: First we expand $\mathcal{L}$ ...
4
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0answers
39 views

Automorphism group of poset of models of Peano Arithmetic.

Suppose we put a poset structure on the set of countable models of Peano Arithmetic as follows: for models 𝑃 and 𝑄, let 𝑃≤𝑄 if 𝑃 is isomorphic to a submodel of 𝑄. As described, this is just a ...
7
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2answers
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 ...
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0answers
32 views

Physical/Geometric models for Reals , of different Cardinality (Lowenheim-Skolem, etc)?

The Real line is a model for the Standard Reals. Are there also known geometric or other ways of representing models of the Reals of different cardinality, e.g., the Hyperreals?
3
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1answer
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 ...
4
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2answers
86 views

Can proper elementarily equivalent end extensions ever be definable?

Suppose $M\models PA$. Can there be a tuple of formulas $\Psi$ (possibly with parameters from $M$) such that: $\Psi^M\equiv M$ (or more precisely, $\Psi$ is an interpretation of an $\{0,1,+,\cdot,<...
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1answer
76 views

System in the state space with differentiating links in feedback

My question is a copy of the following topic, but slightly reformulated: Representation in state-space of a system with ideal and real differentiating links in feedback There are two systems in the ...
3
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1answer
37 views

No single axiom stating non-Archimedeanity [duplicate]

An ordered field $K$ with the ordering $<$ is Archimedean if for any $x \in K$ there exists $n \in \mathbb{N}$ (where $\mathbb{N}$ is the copy of the natural numbers in $K$) such that $|x|<n$. ...
5
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1answer
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 ...
3
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0answers
62 views

Strongest tools to detect unsoundness

I am curious about the strongest methods we currently know of that can allow us to discover (arithmetical) unsoundness or nonstandardness in a foundational system $S$ that interprets at least ACA (...
2
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1answer
21 views

Uniformly Bounded Functions of Hyperreals

I am trying to study ultraproducts and non-standard analysis, and I have the following question. Fix a non-principal ultrafilter $\mathcal{F}$ to construct the hyperreal field ${}^*\mathbb{R}$. An ...
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2answers
75 views

Is There a notion of Convergence for the Non-Standard Reals? How can we do Analysis?

I was reading up on Non-Standard Analysis from an Analytical perspective, i.e., how to do Analysis when I got stuck thinking of how we can do Analysis without even having a "traditional" metric , e.g.,...
2
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1answer
71 views

Any set can be easily replaced by a base set with the same size

Article Forcing in nonstandard analysis states in the page 266: "For an infinite ordinal $\alpha$, any set $X$ such that every element of an element of $X$ has rank $\alpha$ is a base set and hence ...
3
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1answer
89 views

Do the properties of the hyperreal numbers change depending on the ultrafilter used?

One of the more common constructions of the hyperreals is $\mathbb{R}^{*} := \mathbb{R}^{\omega}/\mathcal{U}$ where $\mathcal{U}$ is some ultrafilter containing the filter of sequences in $\mathbb{R}^{...
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1answer
55 views

Question about the definition of superstructure

Let $S_n=S_0\cup \mathcal{P}(S_{n-1})$ and $V_n=V_{n-1}\cup \mathcal{P}(V_{n-1})$ with $S_0=V_0$. Defining $\hat{S}:=\bigcup_{n\in\mathbb{N}}S_n$ and $\hat{V}:=\bigcup_{n\in\mathbb{N}}V_n$, can we ...
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1answer
35 views

measurability for function on internal measure space

For a (standard) $(X,A,\mu)$ be a measure space, given a function $f\colon X\to\overline{\mathbb{R}}$, we have the following characterization for its measurability. $f$ is $A$-measurable iff forall $...
2
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1answer
42 views

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}...
2
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1answer
56 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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1answer
43 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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0answers
45 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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1answer
103 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 ...
2
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0answers
69 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 $PA^\omega$ ...
15
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1answer
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':...
8
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1answer
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 ...
5
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1answer
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 ...
3
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1answer
53 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
81 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 ...
4
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2answers
256 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
81 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
124 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 (...
3
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2answers
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 ...
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0answers
74 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
69 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 $\mathbb{N}$...
3
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3answers
148 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 $...
9
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2answers
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, ...
4
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1answer
322 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 ...
5
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0answers
110 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
158 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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0answers
42 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 (...