# 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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### Natural functions outside $\sf PA$?

Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?

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### Complexity of definable global choice functions

It is well-known that $L$ has a $\Sigma_{1}$-definable global choice function; it is also known that there are other transitive class models of ZFC with this property.
I wonder about the complexity ...

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### Definability in pure-second order logic

Are the universe and the empty set the only definable sets without parameters in the language of pure second-order logic? TIA

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### 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 ...

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### 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 ...

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### 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?

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### 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_\...

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### 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{...

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### 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:
$$\...

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### 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 (...

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### 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 $\...

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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:
$...

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### 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 ...

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### 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_{...

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### 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 $\mathcal L_{\omega_1, \omega}$...

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### 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 ...

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### 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 \...

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### 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,.., \...

2
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### 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 \...

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### 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 < ...

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### 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. ...

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### 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 ...

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### 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., ...

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### 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 $P$ ...

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### 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?

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### 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) ...

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### 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 $\varphi(\dot R)$, ...

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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 ...

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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)...

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### 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 ...

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### 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 ...

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### 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 \\\...

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### 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(\...

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### 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^\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\exists\forall$-...

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### 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 $\exists\forall$-...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...