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3
votes
1answer
132 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
211 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 ...
10
votes
2answers
401 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 : ...
19
votes
1answer
702 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
194 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 ...
7
votes
1answer
221 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 ...
6
votes
3answers
162 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 ...
1
vote
3answers
308 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 ...