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

Filter by
Sorted by
Tagged with
4 votes
0 answers
130 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
1 answer
242 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
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
1 vote
0 answers
154 views

Can ordinal definability be defined using no more than one ordinal parameter?

This answer shows that one can indeed define ordinal definable this way: $\begin{align} \textbf{Define: } & \operatorname {OD} (X) \iff \\& \exists \theta \, \exists \varphi: X= \{y \in V_\...
Zuhair Al-Johar's user avatar
3 votes
1 answer
242 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
1 vote
1 answer
166 views

Can we state $\sf V=HOD$ using a single ordinal parameter(other than the formula code)?

$\sf V=HOD$ is stated as: $\forall X \, \exists \theta \, \exists \alpha < \theta \, \exists \varphi : X=\{y \in V_\theta\mid V_\theta\models\varphi(y,\alpha)\}$ This use two ordinal parameters (...
Zuhair Al-Johar's user avatar
3 votes
2 answers
257 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
12 votes
1 answer
586 views

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
0 votes
1 answer
142 views

How do these two principles of Foundation written in $\mathcal L_{\omega_1,\omega}$ compare?

This comes in continuation to posting titled "Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories?". Here, an attempt at a stronger notion of Foundation, yet ...
Zuhair Al-Johar's user avatar
1 vote
1 answer
125 views

Can we have ZF + Definability + True Foundation + True Finiteness? Can it be Categorical?

Lets extend $\mathcal L_{\omega_1, \omega_1}$ with axioms of equality and of: $\sf ZF + Definability+Ture$-$\sf Foundation+True$-$\sf Finiteness $ Where $\sf ZF$ is written, as usual, in $\mathcal L_{...
Zuhair Al-Johar's user avatar
2 votes
1 answer
131 views

Does adding definability axiom expressed in infinitary language to ZF, let all models be pointwise definable?

This posting is generally related to a prior posting titled "Are all constructible from below sets parameter free definable?" If we work in infinitary language $\mathcal L_{\omega_1, \omega}$...
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
5 votes
2 answers
366 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
  • 491
-2 votes
1 answer
196 views

Does cardinal definable choice imply AC?

Recall the definition of cardinal definable sets, to re-iterate: $Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda_1,.., \...
Zuhair Al-Johar's user avatar
2 votes
1 answer
162 views

Which fragment of ZF does the class of all hereditarily predicatively definable sets capture?

$\begin{align}x \text { is predicatively} &\text{ definable } \iff \exists x_1,..,x_n \exists \varphi:\\ & \rho(x_1) < \rho(x) ,.., \rho(x_n) < \rho(x) \ \land \\&\forall y \, (y \...
Zuhair Al-Johar's user avatar
4 votes
1 answer
231 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
3 votes
1 answer
152 views

If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?

If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
Zuhair Al-Johar's user avatar
1 vote
1 answer
141 views

Is there a model of each of the following kinds of theories in the first transitive model of ZFC?

The question of whether the minimal transitive model $M$ of $\sf ZFC$ have a model of each consistent extension $\sf T$ of $\sf ZF$, is answered to the negative, basically because $M$ is countable and ...
Zuhair Al-Johar's user avatar
1 vote
1 answer
83 views

Are externally pointwise definable models of ZFC subject to the same limitations of the internally pointwise definable ones?

By pointwise definable models, it's meant that every element of those models is definable after a formula in a parameter free manner, but that defining formula is in the language of that model, i.e., ...
Zuhair Al-Johar'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
  • 2,063
1 vote
0 answers
51 views

Can we define the notion of ordinal\cardinal definable set in Z + Ranks?

Working in Zermelo + Ranks, can we define the notions "ordinal definable" set, "cardinal definable" set? Or does it beg Replacement\Reflection to be defined?
Zuhair Al-Johar'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
12 votes
3 answers
847 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
22 votes
1 answer
1k views

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
44 votes
2 answers
4k views

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
1 vote
0 answers
92 views

Is definability in $V$ in $\sf Ack+MK$ expressible in its language?

Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$, while $V$ is the primitive constant symbol ...
Zuhair Al-Johar'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
11 votes
1 answer
496 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
1 vote
1 answer
195 views

Is every set being cardinal definable consistent with ZF?

$Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda_1,.., \lambda_n <^\rho \kappa \ \exists \phi : \\ X=\{ y \in V_{\rho(\...
Zuhair Al-Johar's user avatar
2 votes
0 answers
127 views

Quantifierisation of maps

I will rewrite my question using Matt F. suggestion. Consider the logical structure $L = (\mathbb{R}, +, *, 1, 0, =)$ and a function $f:\mathbb{R}\to\mathbb{R}$. Consider the map $Q:2^\mathbb{R}→2^\...
A.Skutin's user avatar
  • 319
0 votes
1 answer
121 views

An infinite Leibnizian structure in a finite language with precisely $n$ definable elements

This question was inspired by Joel David Hamkins's excellent question on Leibnizian structures with no definable elements. Let $n$ be a positive integer. Is there an infinite structure in a finite ...
user107952's user avatar
  • 2,063
10 votes
1 answer
444 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
  • 1,989
2 votes
0 answers
311 views

Can we have $\sf V=HTD$? How it relates to $\sf V=HOD$?

In my search for some type-set motivated line of thought that might prove axiom of choice, I was thinking of a concept that looks like $\sf V=HOD$, but in terms of types instead of ordinals, that is ...
Zuhair Al-Johar's user avatar
14 votes
2 answers
623 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
4 votes
1 answer
278 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
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
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
0 answers
315 views

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
4 votes
0 answers
522 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
  • 2,063
5 votes
1 answer
394 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
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
  • 479
9 votes
0 answers
319 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
1 vote
1 answer
259 views

Does choice always hold in a model of ZF with point-wise parameter-free definable sets?

If one add to ZF the rule that all sets are parameter free definable. Would that prove the axiom of choice? More specifically. IF we add the following omega rule to inference rules of the language of ...
Zuhair Al-Johar's user avatar
10 votes
2 answers
467 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
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
  • 1,498
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
  • 1,184
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
  • 1,591
1 vote
1 answer
221 views

In NBG (ZFC + classes) if $P$ nonempty predicate of classes must $P$ have definable solution?

Let $A, B, C..X, Y$ range over potentially proper classes in NBG set theory (or adding class quantification to ZFC in obvious way) and $x,y,z$ range over sets. Since I don't know the proper symbols ...
Peter Gerdes's user avatar
  • 2,551