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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Why isn't $\mathcal{P}(^*A)\subseteq {^*}\mathcal{P}(A)$?

I'm reading Goldblatt's Lectures on the Hyperreals, and he provides the following proof that $^*\mathcal{P}(A)\subseteq \mathcal{P}(^*A)$: Given sets $A,\mathcal{P}(A)\in\mathbb{U}$, the statement $\...
Numeral's user avatar
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Finite field with infinitesimals / nonstandard analysis over finite fields

I have two questions, which are really the same question phrased in two ways: Has there been any research on adjoining infinitesimal elements to finite fields? Has anyone considered extensions to ...
Jim's user avatar
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Ascertaining whether "absolute" standardness of $\omega$ is actually possible in ZFC

Books on set theory seem to at least strongly imply that there can exist some $\mathsf{ZFC}$ universe whose $\omega$ is "standard" in an "absolute" sense (order-isomorphic to the ...
NikS's user avatar
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Consistent theories $T$ such that $T+\mathrm{Con}(T+\mathrm{Con}(T))$ is inconsistent

In this question, it is asked whether there is a theory $T$ such that $T$ is consistent but $T+\mathrm{Con}(T)$ is inconsistent. The answer is yes: for instance, $T=\mathsf{PA}+\neg\mathrm{Con}(\...
Joe's user avatar
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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 ...
DAGO's user avatar
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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 ...
Rivers McForge's user avatar
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83 views

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 ...
Axel Bregnsbo's user avatar
3 votes
3 answers
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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 ...
WillG's user avatar
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1 answer
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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 ...
TomKern's user avatar
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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 ...
Xenônio's user avatar
7 votes
1 answer
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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 $ ...
Mohsen Shahriari's user avatar
4 votes
1 answer
59 views

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 ...
Graviton's user avatar
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Nonstandard analysis and hyperreals

Hyperreal numbers and nonstandard analysis are intimately connected. First, the nonstandard analysis consists of an extension of the axioms for the real numbers, which introduces a new predicate $\...
Davius's user avatar
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standard model of $\mathbb{N}$ and true $\Pi^0_1$ sentences

Does the fact that provability of some true $\Pi^0_1$ sentences is equivalent to the existence of particular (I do not know from the top of my head to which ones, does someone know how they are called)...
user122424's user avatar
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Which theories of arithmetic have non-standard computable models?

From this answer: In particular, while PA is still overkill, there are theories of arithmetic much stronger than arithmetic with successor which are too weak for the Tennenbaum phenomenon to hold for ...
Carla only proves trivial prop's user avatar
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Every nonstandard model of arithmetic has an element which is a multiple of every $n\in \Bbb N$.

Let $\mathfrak N$ be the structure of the natural numbers on the language of arithmetic $\mathcal L=\{0,1,+,\cdot,<\}$. Let $\mathfrak M$ be any nonstandard model of arithmetic. Show that there ...
Addem's user avatar
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Which models of PA can be standard in some model of ZFC?

From wikipedia: For example, there are models of Peano arithmetic in which Goodstein's theorem fails. It can be proved in Zermelo–Fraenkel set theory that Goodstein's theorem holds in the standard ...
Carla only proves trivial prop's user avatar
3 votes
0 answers
28 views

How to use non-standard analysis to prove Baire Category Theorem?

I'm caring about some questions of non-standard analysis. I have found the only book talking about Baire Category Theorem, which is the book of Siu-Ah Ng. But I think the proof in this book is not ...
Sigh酱's user avatar
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Hyper-extensions of Hom space

We fix an ultrafilter $\mathcal{F}$ of $\mathbb{N}$ which contains the cofinite filter. Let $A,B$ be sets and ${}^{*}A,{}^{*}B$ their hyper-extensions. Then is $$ {\rm Hom}({}^{*}A,{}^{*}B) $$ equal ...
M masa's user avatar
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Must a non-well-founded model of Z contain a countable collection that looks like a proper class to the model?

Let's say we have a (not transitive) model, $(M,E)$, of Z. In this post, "class" means an arbitrary collection of elements of $M$. Let $x_0\ni x_1\ni x_2\ni\cdots$ be an infinite descending ...
Ari Herman's user avatar
4 votes
1 answer
129 views

Nonstandard algebraic geometry: Fundamental Theorem of Algebra

I have been trying to study the basics of algebraic geometry using nonstandard analysis and I can't wrap my head around this issue. Let $^*\mathbb{C}$ be the extension of the complex numbers. Now ...
enochk.'s user avatar
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3 answers
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Why isn't there a contradiction between compactness and the Archimedean property when we create non-standard models of real numbers?

I have a question about compactness, non standard models of reals and illusory paradoxes. Now, we know that because of the compactness theorem in FOL there are, for instance, non standard models of ...
WaLuigi's user avatar
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Number of models of the naturals and reals with and without CH

The Wikipedia page on true arithmetic says that it has $2^\kappa$ models for each uncountable cardinal $\kappa$. This refers to the theory of all first-order statements of the naturals. I'm curious ...
Mike Battaglia's user avatar
1 vote
1 answer
86 views

Soundness and Completeness for a single Model Only?

Question modified to hopefully answer the questions (I'm a physicist to all might not be mathematically watertight) In Enderton "A Mathematical Introduction to Logic", logical Implication is ...
user avatar
4 votes
1 answer
178 views

Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?

As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form: $z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$ in which $\omega$ is a formal ...
Mike Battaglia's user avatar
3 votes
2 answers
166 views

"Real-closed" vs "transfer principle"

The hyperreals are "real-closed," which means that any first-order statement that is true of the reals is true of the hyperreals. However, they also satisfy a "transfer principle," ...
Mike Battaglia's user avatar
1 vote
1 answer
42 views

Is $\{\mathcal{B}_T: \mathcal{B}_T$ is isomorphic to $\mathcal{B}_S\}$ countable?

Let $\mathcal{L}=\{0,S,+, \cdot, <\}$ be the language of arithmetic and let $\mathcal{A}$ be the standard model of arithmetic. For each prime $p$, let $\phi_p(v)$ be the formula saying $v$ is a ...
Pascal's Wager's user avatar
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Why do we care about PA proving itself consistent?

"No first-order theory containing a sufficient amount of arithmetic can prove its own consistency". It's the "prove its own" part of Godel's theorem that always confused me. Why do ...
ngc1300's user avatar
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3 votes
2 answers
226 views

(Request for) simple constructive proof of existence of nonstandard model of PA

I know of two straightforward nonconstructive proofs of the existence of nonstandard models of arithmetic. By the existence of the standard model of PA, PA is satisfiable and has an infinite model. ...
Greg Nisbet's user avatar
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1 answer
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Proving that the Yablo's paradox is $\omega$-inconsistent

Jeffery Ketland proved in his Yablo's paradox and $\omega$-inconsistency that the set of Yablo sentences, which leads to Yablo's paradox, is $\omega$-inconsistant, but I do not understand his proof. ...
Constantly confused's user avatar
5 votes
0 answers
106 views

Explicitly defining a nonstandard extension of the reals

Kanovei & Shelah (2004) explicitly constructed a nonstandard extension of the reals. Now, I'm no expert on this but I gather that they first specified a set of free ultrafilters, and then used it ...
phst's user avatar
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1 vote
0 answers
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Enumerating axioms of PA in nonstandard models: trouble in the Kripke-Putnam proof of Gödel's incompleteness theorem

I'm trying to understand the Kripke-Putnam proof of Gödel's incompleteness theorem, somewhat recently cited in this MO question, and can't figure out how to close a gap in the argument. I apologize ...
Jacob Manaker's user avatar
3 votes
1 answer
98 views

PA├ ∃xP(x) but PA$\nvdash$ P(n) for any n?

Problem : Let P(x) be a wff with one free variable x in the language of PA and $PA\vdash \exists P(x)$. And let PA be $\omega$-consistent. Then, is there a natural number $n$ s.t. $PA\vdash P(n)$? ...
정재우's user avatar
  • 117
3 votes
1 answer
128 views

Tennenbaum's theorem generalizations: can we have a computable Collatz map on a nonstandard model of Peano?

Tennenbaum's theorem states that for any (countable) nonstandard model of Peano arithmetic, neither the addition nor the multiplications is computable. I find this result fascinating (and frustrating!)...
Beren Gunsolus's user avatar
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0 answers
127 views

How does second-order arithmetic rule out non-standard numbers?

According to 1 there are axioms of second-order arithmetic which categorically characterize the natural numbers up to isomorphism. The axioms are: $$\forall x\,\neg(x+1=0)$$ $$\forall x\,\forall y(x+1=...
Max's user avatar
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1 answer
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Is this an example of the incompleteness of first-order PA?

Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard ...
Dan Christensen's user avatar
10 votes
2 answers
1k views

"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, ...
Beren Gunsolus's user avatar
1 vote
2 answers
126 views

How would you express, in ZFC, that the number of countable models of Th($\mathbb{N}$), up to isomorphism, is at most $2^{\aleph_0}$?

Intuitively, it is clear to me why, up to isomorphism, there are at most $2^{\aleph_0}$ non-isomorphic models of Th($\mathbb{N}$): I can choose as "representative" of any countable model $\...
Matteo __'s user avatar
1 vote
1 answer
44 views

A non-standard model of PA with total antisymmetric order without induction

I'm looking for a model of PA without induction whose order relation total and Anti-symmetric. to be specific, satisfiying: $\forall x \ (0 \neq S ( x ))$ $\forall x, y \ (S( x ) = S( y ) \...
razivo's user avatar
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0 votes
0 answers
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Definition of finite in set theory [duplicate]

The usual definition of a finite number $n$ in the ZFC set theory is $n\in\mathbb{N}$, which is equivalent to "$n$ is 0 or a successor ordinal, and so are all its elements". But this is not ...
V. Semeria's user avatar
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1 vote
2 answers
139 views

Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms?

Is there a first order formula that is satisfied by $\mathbb{N}$, but not by any other models of Peano axioms? For example, is there a formula that expresses "there is an element that is greater ...
Jiu's user avatar
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1 vote
0 answers
64 views

Are there any significant/meaningful ultrapowers other than the hyperreals?

I have recently begun reading about non-standard analysis. According to this wikipedia article it is possible to construct an ultrapower $M^I/\mathcal{U}$ from any structure $M$ and index set $I$ with ...
Gideon Tveten's user avatar
4 votes
1 answer
121 views

Arithmetic on infinite formal prime factorizations

By the fundamental theorem of arithmetic, we can identify a natural number $x$ with the sequence $(a_n)$ of exponents in its prime factorization $x=\prod_np_n^{a_n}$, where $p_n$ is the $n$th prime. A ...
Karl's user avatar
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0 votes
1 answer
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Non-standard model of naturals and Löwenheim–Skolem

I'm taking a beginner course in mathematical logic. In the proof of some properties of non-standard natural numbers, the lecturer has used the downward Löwenheim–Skolem theorem, which I didn't ...
Albert's user avatar
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3 votes
1 answer
109 views

Extension of the Paris-Harrington principle

Let $[m,n]$ denote the set $\{m,m+1, ... ,n-1,n\}$. $X \to (k)^n_c$ means that whenever $f: [X]^n \to c$ there is a subset $H \subset X$ with cardinality $k$ such that $f$ is constant on $[H]^n$ (The ...
Jori's user avatar
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0 votes
1 answer
142 views

Set-up for the Paris-Harrington Theorem

In his book "Models of Peano Arithmetic" Kaye proves the Paris-Harrington Theorem. He starts off by introducing a "simplification" then proves the short Lemma 14.11 about it (see ...
Jori's user avatar
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1 vote
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63 views

Motivation of indicator construction in Kaye

Kaye says the following in his book about models of $\textbf{PA}$ on p. 198: I have no clue what motivates the definition of $f_n(x, y)$. The reference made to Propositions 14.1, 14.2 don't help me ...
Ibrahim's user avatar
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6 votes
1 answer
160 views

Are there nonstandard $\mathsf{PA}$ models without $\Delta^1_1$ cuts?

My question is the following: Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1_1$-with-parameters-definable nonempty proper successor-closed ...
Noah Schweber's user avatar
6 votes
1 answer
199 views

"$\Sigma_1^1$-Peano arithmetic" - does it pin down $\mathbb{N}$?

Let $\mathsf{PA}_{\Sigma^1_1}$ be the theory in second-order logic gotten by extending the usual first-order Peano axioms to include arbitrary $\Sigma^1_1$ formulas in the induction scheme. My ...
Noah Schweber's user avatar
0 votes
1 answer
72 views

$\text{Sat}_{\Delta_0}(x, y)$ is $\Delta_1(\textbf{PA})$ (Kaye's book)

In Kaye's "Models of Peano Arithmetic" in chapter 9 on satisfaction a lot of effort is expanded to prove that $\text{Sat}_{\Delta_0}(x, y)$, representing truth of $\Delta_0$-sentences, is a $...
Jori's user avatar
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