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
37
questions
10
votes
0answers
211 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 ...
5
votes
0answers
89 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 ...
2
votes
0answers
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 ...
14
votes
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 ...
4
votes
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 $\exists\forall$-...
8
votes
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 $\exists\forall$-...
10
votes
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 ...
10
votes
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 ...
3
votes
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 ...
4
votes
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 ...
5
votes
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 ...
8
votes
0answers
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 ...
1
vote
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 ...
10
votes
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 ...
8
votes
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, ...
5
votes
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.
...
4
votes
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 ...
1
vote
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 ...
6
votes
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 ...
8
votes
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 ...
30
votes
2answers
1k views
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}...
8
votes
0answers
333 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 ...
3
votes
2answers
200 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. ...
16
votes
2answers
920 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?
8
votes
1answer
358 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?
...
7
votes
1answer
611 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 ...
0
votes
1answer
244 views
Definability of arithmetic functions and relations
Motivation: Many "weak" arithmetic functions and/or relations ("relations" for short) are equivalent with relations explicitly definable by relations which were recursively defined by them beforehand (...
6
votes
1answer
261 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 (=...
3
votes
1answer
160 views
Outer Definability of a Class
Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ...
2
votes
1answer
281 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~~\...
11
votes
2answers
590 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(...
20
votes
1answer
888 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 \...
14
votes
1answer
260 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 ...
8
votes
1answer
303 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\...
9
votes
3answers
302 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 ...
3
votes
3answers
647 views
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 ...
30
votes
0answers
2k views
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 ...
