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
42
questions
6
votes
0
answers
214
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 ...
- 5,298
11
votes
1
answer
389
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 \\\...
- 5,298
1
vote
1
answer
170
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(\...
- 5,298
2
votes
0
answers
123
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^\...
- 361
0
votes
1
answer
106
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 ...
- 1,219
9
votes
0
answers
263
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 ...
- 576
10
votes
0
answers
195
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 ...
- 1,569
2
votes
0
answers
289
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 ...
- 5,298
14
votes
2
answers
522
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 ...
- 2,033
4
votes
1
answer
255
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 $\exists\forall$-...
- 3,979
8
votes
1
answer
225
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 $\exists\forall$-...
- 3,979
10
votes
1
answer
340
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 ...
- 3,979
10
votes
0
answers
291
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 ...
- 4,071
4
votes
0
answers
356
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 ...
- 1,219
5
votes
1
answer
296
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 ...
- 3,979
6
votes
1
answer
303
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 ...
- 451
9
votes
0
answers
314
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
1
answer
246
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 ...
- 5,298
10
votes
2
answers
378
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 ...
- 101
8
votes
1
answer
254
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, ...
- 1,479
5
votes
0
answers
138
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.
...
- 921
4
votes
0
answers
44
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 ...
- 1,457
1
vote
1
answer
195
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 ...
- 1,605
6
votes
1
answer
367
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 ...
- 754
8
votes
1
answer
460
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 ...
- 195k
30
votes
2
answers
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}...
user40023
8
votes
0
answers
338
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 ...
- 2,033
2
votes
2
answers
205
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. ...
- 12.4k
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?
- 163
8
votes
1
answer
369
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?
...
- 10.2k
6
votes
1
answer
657
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 ...
- 195k
0
votes
1
answer
273
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 (...
- 9,100
6
votes
1
answer
272
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 (=...
- 2,033
3
votes
1
answer
162
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 $n_{R}$ - ...
user43940
2
votes
1
answer
282
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~~\...
user42090
12
votes
2
answers
612
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(...
- 9,100
20
votes
1
answer
904
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 \...
- 195k
14
votes
1
answer
274
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 ...
- 2,991
8
votes
1
answer
327
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\...
- 21.3k
9
votes
3
answers
316
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 ...
- 10.2k
4
votes
3
answers
817
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 ...
- 1,335
30
votes
0
answers
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 ...
user16974