All Questions
Tagged with model-theory definability
30
questions
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?
- 309
3
votes
2
answers
261
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 $\...
- 10.2k
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. ...
- 10.2k
1
vote
1
answer
143
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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 ...
- 10.2k
1
vote
1
answer
86
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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., ...
- 10.2k
7
votes
1
answer
448
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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 $P$ ...
- 2,063
19
votes
1
answer
1k
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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) ...
- 230k
12
votes
3
answers
865
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 $\varphi(\dot R)$, ...
- 230k
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 ...
- 1,768
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)...
- 230k
0
votes
1
answer
124
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 ...
- 2,063
10
votes
1
answer
481
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 ...
- 783
4
votes
1
answer
284
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 $\exists\forall$-...
- 4,527
8
votes
1
answer
247
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 $\exists\forall$-...
- 4,527
10
votes
1
answer
400
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 ...
- 4,527