Questions tagged [model-theory]
Model theory is the study of (classes of) mathematical structures (e.g. groups, fields, graphs, universes of set theory) using tools from mathematical logic. Objects of study in model theory are models for formal languages which are structures that give meaning to the sentences of these formal languages. Not to be confused with mathematical modeling.
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(When) are recursive "definitions" definitions?
This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
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Invariance under $\operatorname{Aut}(N / \{M\})$
Let $M\preceq N$ where $N$ is $|M|^+$ saturated.
Let $p(x)$ be a partial type over $\le|M|$ parameters.
What do we know (if anything at all) about when $p(N)$ is invariant under the action of $\...
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Is the class of linearly-orderable rings first order axiomatizable?
A linearly ordered ring is a commutative ring $R$ with unity equipped with a linear order $\leq$ that is compatible with addition, and such that the set of nonnegative elements are closed under ...
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Proving a simple consequence of the Compactness Theorem
I am self-learning logic, and trying to prove the following exercise using the Compactness Theorem:
Suppose $T$ is a theory for language $L$, and $\sigma$ is a sentence of $L$ such that $T \models \...
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Does Löwenheim-Skolem require Foundation in any way?
As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
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Is quantifying over natural numbers non first order?
I was reading here that
Note that ‘x is an infinitesimal’ is not first order, because it requires you to quantify over the naturals.
Whats's non first order about quantifying over natural numbers?
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Is there a theory in which all types can be omitted?
Is there a natural example of a first order complete (consistent) theory $T$ in which every 1-type can be omitted? or is there always some isolated type? In that case, why?
Of course there are plenty ...
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Proving that the set of sentences that are true using the symbols $+,<,=$ is the same over all ordered fields
I am interested in whether the set of formulas that one can prove true for a concrete ordered field using the symbols $+,<$ and $=$, depends on the field. In particular, I am interested in the set ...
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Functional completeness over a structure
The set of propositional connectives $\{\wedge,\vee\}$ is of course not functionally complete; correspondingly, the logical vocabulary $\{\forall,\exists,=,\wedge,\vee\}$ is not sufficient for ...
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in definition of assigment, what's means 'except possibly a'?
in frist-order logic,
part of assignments practice represent like this
"if 𝜙is ∀𝛼𝜓, where 𝛼 is a variable, then ⊨vℳ 𝜙 iff for every assignment 𝑣' that agrees with 𝑣 on the values of every ...
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Skolemization in Marker's Proof of Theorem 2.3.7 (Downward Löwenheim-Skolem)
For convenience, I'll re-state Lemma 2.3.6 and Theorem 2.3.7 in Marker's Model Theory:
$\textbf{Lemma 2.3.6}$: Let $T$ be an $\mathcal{L}$-theory. There are $\mathcal{L}^\ast \supseteq \mathcal{L}$ ...
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Why do theories extending $0^\#$ have incomparable minimal transitive models?
This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows my emphasis):
[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^...
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Two families of isomorphic structures have isomorphic ultraproduct.
I am trying to prove the following result:
Let $(\underline{M}_i)_{i\in I}$, $(\underline{N}_i)_{i\in I}$ be two families of structures such that, for all $i\in I$, $\underline{M}_i \cong \underline{...
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What does Feferman-Vaught say $\textbf{exactly}$ about definable subsets of a direct product of two structures?
Below I reproduce a consequence of the Feferman-Vaught theorem, taken from Wilfrid Hodges' book Model Theory:
Corollary 9.6.4: Let $L$ be a first-order language, let $A$ and $B$ be $L$-structures and ...
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Definability of acyclic graphs
I think you should be able to encode the axioms of a directed, acyclic graph by introducing a strict partial order. Say E(a, b) represents there is an edge from a to b. We introduce a strict partial ...
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