Questions tagged [definability]

definability by formulas in first-order logic, e.g. as explained at https://en.wikipedia.org/wiki/Definable_set, or as in J. Robinson's first-order definition of the integers in the field of rationals

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Is multiplication implicitly definable from successor?

A relation $R$ is implicitly definable in a structure $M$ if there is a formula $\varphi(\dot R)$ in the first-order language of $M$ expanded to include relation $R$, such that $M\models\varphi(\dot R)...
Joel David Hamkins's user avatar
33 votes
0 answers
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Defining $\mathbb{Z}$ in $\mathbb{Q}$

It was proved by Poonen that $\mathbb{Z}$ is definable in the structure $(\mathbb{Q}, +, \cdot, 0, 1)$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by ...
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30 votes
2 answers
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Define $\mathbb{N}$ in the ring $\mathbb{Z}$ without Lagrange's theorem

It is well-known that the set of nonnegative integers $\mathbb{N}$ is definable in the ring of integers $\mathbb{Z}$. Indeed, by Lagrange's four squares theorem we have $\mathbb{N} = \{n \in \mathbb{Z}...
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22 votes
1 answer
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Is the set of primes implicitly definable from successor?

An earlier question by Joel David Hamkins asked whether multiplication is implicitly definable in the structure $(\mathbb{N},S)$ of the naturals with successor. Here $R$ is implicitly definable if ...
Geoffrey Irving's user avatar
21 votes
1 answer
1k views

Is there a subset of the natural number plane, which doesn't know which of its slices are arithmetic?

$\newcommand{\N}{\mathbb{N}}$My question, more precisely, is: Question. Is there a set $B\subset \N\times\N$, such that the set of indices where it is arithmetically definable, that is, $\{ n\in\N \...
Joel David Hamkins's user avatar
20 votes
2 answers
1k views

Non-definability of graph 3-colorability in first-order logic

What is a proof that graph 3-colorability is not definable in first order logic? Where did it first appear?
Leo Marcus's user avatar
19 votes
1 answer
1k views

Is the theory of a partial order bi-interpretable with the theory of a pre-order?

A partial order relation $\leq$ on a set $A$ is a binary relation that is reflexive, transitive, and antisymmetric. A preorder relation $\unlhd$ (also sometimes known as a quasi order or pseudo order) ...
Joel David Hamkins's user avatar
16 votes
2 answers
1k views

Is factorial definable using a $\Delta_0$ formula?

The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula. Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)? If not, why?
Yoni Zohar's user avatar
14 votes
2 answers
624 views

Definability of Gödel's pairing function on ordinals

Given an infinite cardinal $\kappa$, Gödel's function is a well-known bijection $p:\kappa^2\to\kappa$. Is $p$ definable in the structure $\langle\kappa;\in\rangle$? Is $p$ definable in a bigger 2nd ...
Vladimir Kanovei's user avatar
14 votes
1 answer
287 views

Existence property for ordered fields

A theory $T$ has the existence property (EP) if the following holds: Let $\phi(x)$ be a formula with one free variable (and no parameters) such that $T \vdash (\exists x) \phi(x)$. Then there is ...
aws's user avatar
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13 votes
1 answer
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Can $L$ be defined without parameters?

If we omit parameters in the definition of $L$ would the result still be $L$? That is, we define a successor stage $L_{\alpha+1}$ in the constructible universe $L$, without including parameters; as: $...
Zuhair Al-Johar's user avatar
12 votes
3 answers
848 views

Is there a simple instance of intransitivity for implicit definability?

This question continues the theme of some recent questions on implicit definability. A relation $R$ is implicitly definable in a first-order structure $M$ if there is a property $\varphi(\dot R)$, ...
Joel David Hamkins's user avatar
12 votes
2 answers
699 views

Ways to define "definability"

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(...
Hans-Peter Stricker's user avatar
11 votes
1 answer
498 views

Is every set being cardinal definable consistent with ZF + negation of Choice?

Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it: $Define: X \text { is cardinal definable} \iff \\\...
Zuhair Al-Johar's user avatar
10 votes
2 answers
468 views

Definability of the ring of integer in algebraic extensions of $\mathbb Q$

J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due ...
George Peterzil's user avatar
10 votes
1 answer
391 views

Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally ...
Keshav Srinivasan's user avatar
10 votes
1 answer
448 views

Definable constructions in o-minimal geometry

Recently I've been working with o-minimal expansions of $(\mathbb{R},\times,+)$, and I want to work "internally" to the language of o-minimal sets instead of working with "definable ...
Hunter Spink's user avatar
10 votes
0 answers
211 views

Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...
Arno Fehm's user avatar
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10 votes
0 answers
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Definability up to isomorphism versus definability of an isomorphic copy

Question: Is it provable in ZFC that every structure that is ordinal definable up to isomorphism has an ordinal definable isomorphic copy? If not, what are some counterexamples? All structures are ...
Dmytro Taranovsky's user avatar
9 votes
3 answers
385 views

What ordinals are definable relations in Peano Arithmetic?

I am not asking which order types PA proves are well ordered. That would be all up to $\epsilon_0$. Rather I mean, assuming a stronger ambient theory such as Zermelo set theory, which ordinals have ...
Colin McLarty's user avatar
9 votes
0 answers
320 views

Is there a definable model of PA whose domain is a proper class and whose complete theory is not definable?

Assume ZFC. Is there a formula of $\mathcal{L}_\in$ (without parameters) defining a model $\mathcal{M}$ of PA whose domain is a proper class but the complete theory of that model is not definable by ...
Guy Crouchback's user avatar
8 votes
1 answer
246 views

What is the lowest complexity definition of $\mathbb{Z}$ in an infinite extension of $\mathbb{Q}$

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
Keshav Srinivasan's user avatar
8 votes
1 answer
390 views

Are there known ways to posit definable global choice in ZF without positing V=L?

I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this? ...
Colin McLarty's user avatar
8 votes
1 answer
521 views

Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?

This question arose today at Yevgeny Gordon's talk, "Will nonstandard analysis be the analysis of the future?" at the CUNY Logic Workshop. Here is my way of asking it. Consider the ordered real field ...
Joel David Hamkins's user avatar
8 votes
1 answer
723 views

Non-normal numbers definable without parameters in the langauge of differential rings with composition

Background: It is currently unknown whether $e$ is normal. A natural way to approach this question is to find a class to which $e$ belongs, and prove all members of that class are normal. For example, ...
James's user avatar
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8 votes
1 answer
350 views

Iterating definability

An odd -- probably basic -- question about model theory: For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace U\...
Noah Schweber's user avatar
8 votes
0 answers
349 views

Cohen's model yet again

It has been discussed already whether a countable OD set necessarily contains an OD element. See e.g. A question about ordinal definable real numbers . A negative answer was obtained in Archive for ...
Vladimir Kanovei's user avatar
7 votes
1 answer
444 views

Are no infinite subsets of the set of all propositional atoms definable in this structure, even with parameters?

I asked this on Math Stack Exchange, but apparently no one paid attention to it. So, I am asking it again, filling in the background necessary to understand it. Consider a countably infinite set $P$ ...
user107952's user avatar
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6 votes
2 answers
468 views

Reconstructing a model from its definable sets

Let $\mathcal{M}$ be an infinite model of a first-order language, and for each $n$, let $\mathcal{B}_n$ be the algebra of definable sets of $n$-tuples from $|\mathcal{M}|$. Given $\{\mathcal{B}_n\mid ...
Henry Towsner's user avatar
6 votes
1 answer
727 views

Can $V\neq\text{HOD}$ if every $\Sigma_2$-definable set has an ordinal-definable element?

This question arises from an issue arising in user38200's recent question concerning models of set theory in which every definable set has a definable element. In my answer to that question, with ...
Joel David Hamkins's user avatar
6 votes
1 answer
358 views

Definability of ordinals in various signatures

Recently, I've been studying what the definable subsets of the countable ordinals "look like" from the perspective of bare-bones first order logic (not set theory) equipped with various ways ...
exfret's user avatar
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6 votes
1 answer
455 views

Problem with definability in the constructible hierarchy

This is a rather technical question. I cannot find my mistake in a proof of the (obviously wrong) following sentence: Every countable ordinal is $\Sigma_2$-definable in $J_{\omega_1 + 1}$ by a formula ...
Archimondain's user avatar
6 votes
1 answer
286 views

Ordinal definable sets of reals in the Solovay

To be precise, let $\Omega$ be an inaccessible cardinal in $L$ and let N be the Solovay model defined by the Levy-collapse in this case. Then $\Omega$ is $\aleph_1$ in $N$. How many different OD (=...
Vladimir Kanovei's user avatar
6 votes
0 answers
249 views

Can HCD accommodate all known large cardinal axioms?

HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals. However, there is a definable well ordering over HOD, so it cannot satisfy ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
399 views

Is there a complete characterization of ordered fields without definable proper subfields?

$\mathbb{Q}$ has no proper subfields. As a result, all ordered fields elementarily equivalent to $\mathbb{Q}$ have no proper subfields which are first-order definable without parameters. And by the ...
Keshav Srinivasan's user avatar
5 votes
1 answer
245 views

Why include $0$ and $1$ in the signature of Presburger arithmetic?

I recently became interested in some problems concerning decidability of extensions of Presburger arithmetic. However, being a number-theorist rather than a logician, I am confused by some notational ...
Jakub Konieczny's user avatar
5 votes
2 answers
367 views

Terminology for ordinals whose constructible level is the least one satisfying some formula

An ordinal $\alpha$ is "meta-definable" by some formula $\varphi$ without free variables if: $$ \begin{cases} L_\alpha \models\varphi \\ \forall\beta < \alpha \, L_\beta \not\models \...
Johan's user avatar
  • 501
5 votes
0 answers
143 views

Higher-order equivalence of ordinals

I wonder which ordinals are second-order equivalent, and similarly for other logical equivalences. Let the signature be fixed and include only <. For concreteness, let us first ask for the first ...
Alexey Slizkov's user avatar
5 votes
0 answers
140 views

Self-additive posets

We say that a partially ordered set $(P,<)$ is self-additive if the two natural embeddings of $P$ in $P\oplus P$ (the linear sum of $P$ and itself) are elementary. We have the following. ...
tomasz's user avatar
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4 votes
3 answers
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Methods for proving non FO definability

I'm having difficulties to prove that the subset of even numbers is not first-order definable in $(\mathbb{N},<)$. Any hint is welcomed! More generally, what are usual techniques in order to prove ...
mathcounterexamples.net's user avatar
4 votes
1 answer
232 views

Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one?

Working in $\sf ZF$ Define: $W_0 = \emptyset \\ W_{\alpha+1} = H_{\leq |W_\alpha|} =\{x \mid \forall y: y \in \operatorname {trcl} (\{x\}) \ |y| \leq |W_\alpha| \} \\ W_\lambda= \bigcup W_{\alpha < ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
280 views

What is the lowest complexity definition of $\mathbb{Z}$ in an infinite algebraic extension of $\mathbb{Q}$?

In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. And in 2012, Jennifer Park proved a result which implies that $\mathbb{Z}$ is $\exists\forall$-...
Keshav Srinivasan's user avatar
4 votes
0 answers
523 views

A question regarding the "math tea" argument

Joel David Hamkins published a paper where he analyzes the "math tea" argument, namely, the argument that some real numbers are undefinable. He constructed a countable model of set theory ...
user107952's user avatar
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4 votes
0 answers
48 views

Closed and bounded intervals of definably complete ordered groups

True or false? All closed and bounded intervals of definably complete ordered groups are definably compact. Let $G$ be an ordered abelian group. Then, a definable subset $D ⊆ G$ is said to be ...
Ma Joad's user avatar
  • 1,611
3 votes
2 answers
258 views

Is every countable model of ZFC a subset of some parameter free definable pointwise-definable model of ZFC?

Is it consistent with $\sf ZFC + \text{ countable models of } ZFC \text { exist}$, that every countable model of $\sf ZFC$ is a subset of some parameter free definable pointwise-definable model of $\...
Zuhair Al-Johar's user avatar
3 votes
1 answer
494 views

Are all constructible from below sets parameter free definable?

Lets take the intersection of the theory of $L_{\omega_1^{CK}}$ and $\sf ZF + [V=L]$, this is equivalent to the theory of constructability from below + limit stages. Can this theory prove the ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
243 views

Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?

$\sf ZF + Def$ is the theory that extends $\mathcal L(=,\in)_{\omega_1,\omega}$ with axioms of $\sf ZF$ (written in $\mathcal L(=,\in)_{\omega, \omega}$) and the axiom of definability:- $\textbf{...
Zuhair Al-Johar's user avatar
3 votes
1 answer
225 views

Does V=HOD prove all kinds of consistent universal hereditary definability?

Is the following a theorem of $\sf ZF+[V=HOD]$? If $Q$ is a property definable in a parameter free manner, then: $$\operatorname {Con}(\sf ZF + [V=HQD]) \implies V=HQD$$ where $\sf V=HQD$ means: $$\...
Zuhair Al-Johar's user avatar
3 votes
2 answers
235 views

What are the minimal requirements for the definable hyperreal field plus transfer?

It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
Mikhail Katz's user avatar
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3 votes
1 answer
299 views

$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem

Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define: $Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in \mathcal{L}-Form~~\...
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