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Results tagged with lo.logic
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user 83901
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
10
votes
0
answers
235
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Is there an expansion of $(\mathbb{N},+,<)$ by a pairing function that is still NIP?
I was told once that there is a theory consisting of just a pairing function that is stable, although I cannot find a reference for it. This motivated my question, which is essentially the title, alth …
- 13.2k
2
votes
0
answers
89
views
'Algebraic Skolemization' of (neo-)stable theories
A slightly obtuse way to say that a first-order theory $T$ has Skolem functions is to say that for any $M\models T$ and $A\subseteq M$, $\mathrm{dcl}(A)\preceq M$.
This suggests a similar condition w …
- 13.2k
8
votes
1
answer
322
views
Is there a stable structure on $[0,1]$ that approximates every continuous function?
The $n$-dimensional form of the Weierstrass approximation theorem is the statement that polynomial functions are dense under the $\ell_\infty$-norm in the space of continuous functions on $[0,1]^n$ fo …
- 13.2k
1
vote
1
answer
137
views
Does every uncountably categorical theory have a $\varnothing$-definable strongly minimal im...
An uncountably categorical theory always has a strongly minimal set definable over its prime model, but sometimes this set needs parameters to define.
By a $\varnothing$-definable imaginary I mean th …
- 13.2k
4
votes
1
answer
173
views
A Beth definability style result for imaginaries?
Let $T$ be a theory and let $I$ be an imaginary sort of $T$. We'll let $T_I$ denote the induced theory on $I$, by which I mean specifically the theory of all $\varnothing$-definable relations on $I$.
…
- 13.2k
5
votes
0
answers
99
views
How many parameters are needed to define strongly minimal sets in $\aleph_0$-categorical, $\...
There's a well-known result of Cherlin, Harrington, and Lachlan that gives a very precise structural understanding of $\aleph_0$-categorical, $\aleph_0$-stable theories. One thing in particular is tha …
- 13.2k
7
votes
0
answers
104
views
Can superstability of a countable theory be characterized in terms of not 'weakly trace inte...
The concept that I want to think about in this question is very similar to trace definability, introduced here and also independently by Guingona, but is somewhat different from it. There's a very goo …
- 13.2k
7
votes
0
answers
98
views
How similar can a model of $I\Delta_0$ be to the intersection of all of its definable cuts?
Let $M$ be a model of $I\Delta_0$. Recall that a definable cut is a definable (possibly with parameters) subset $I$ of $M$ that is non-empty, downwards closed, and closed under successor.
If we consid …
- 13.2k
7
votes
1
answer
187
views
A mixed preservation theorem for two-sorted structures?
A well known model theory fact is that for any first-order theory $T$ the collection of universal consequences of $T$, written $T_\forall$, is a precise axiomatization of the class of substructures of …
- 13.2k
2
votes
1
answer
278
views
A generalization of Vaught's two cardinal theorem
I'm trying to determine whether or not the following generalization of Vaught's two cardinal theorem is true.
Let $T$ be a complete theory in a language with a unary predicate $U$ and a binary predic …
- 13.2k
6
votes
0
answers
99
views
Does stable embeddedness improve two-cardinal behavior?
Let $T$ be a first-order theory with two designated unary predicates $P$ and $Q$. We'll say that $P$ is bounded over $Q$ if for every $\kappa$, there is a $\lambda$ such that if $M \models T$ and $|Q^ …
- 13.2k
6
votes
0
answers
82
views
Is there an unstable NSOP theory in which all invariant types are coheirs?
There are some first-order theories $T$ with the property that for any $M \models T$, the only $M$-invariant global types are those that are coheirs over $M$ (i.e., finitely satisfiable in $M$). Two e …
- 13.2k
3
votes
1
answer
147
views
Consistency and consistency strength of certain special cuts in $I\Delta_0$
Recall that $I\Delta_0$ is the theory in the language of arithmetic that consists of the axioms of $\mathsf{PA}$ with induction restricted to $\Delta_0$ formulas (i.e., formulas where all quantifiers …
- 13.2k
3
votes
0
answers
187
views
Consistency of a tighter bound than given by Erdős–Rado in a special case
Let $0 < n, \ell$ be natural numbers with $n < \ell$. We will define a function $F_{n\ell}(\kappa)$ on infinite cardinals indexed by $n$ and $\ell$.
To compute $F_{n\ell}(\kappa)$ let $A$ be a set of …
- 13.2k
7
votes
0
answers
151
views
Is there a complete characterization of hyperimaginaries in $\mathsf{DLO}$?
Recall that in a first-order theory $T$, a hyperimaginary is an equivalence class of some type-definable equivalence relation $E(x,y)$ (with $x$ a possibly infinite tuple of parameters). The $E$-equiv …
- 13.2k