# 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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### 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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229 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 ...
2answers
461 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 ...
1answer
239 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 $\exists\forall$-...
1answer
213 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 $\exists\forall$-...
1answer
303 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 ...
0answers
284 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 ...
0answers
307 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 ...
1answer
250 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 ...
1answer
274 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 ...
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248 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 ...
1answer
237 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 ...
2answers
336 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 ...
1answer
244 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, ...
0answers
132 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. ...
0answers
41 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 ...
1answer
179 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 ...
1answer
328 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 ...
1answer
421 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 ...
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1k views

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1answer
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### 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 ...
1answer
303 views