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.
4,441
questions
1
vote
1
answer
55
views
Condition for dense isolated types
I need some help with proving the following:
Theorem. Let $T$ be a complete theory in a countable language and let $M \models T$.
If $|\mathcal{S}_n^\mathcal{M}| < 2^{\aleph_0}$ then the isolated ...
- 31
6
votes
1
answer
207
views
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 ...
- 63
3
votes
1
answer
64
views
What are other examples of $\aleph_1$-categorical theories?
In model theory, $\aleph_1$-categorical (first order) theories (in a countable language) are very important, and I am studying them at the moment. However, it seems that the only examples I can find ...
- 492
2
votes
1
answer
80
views
Can equivalence relations have extra non-trivial properties?
The theory of equivalence relations can be axiomatized by 3 equality-free universal sentences, namely:
1.$xRx$
2.$xRy \rightarrow yRx$
3.$(xRy \land yRz) \rightarrow xRz$.
Now, certainly, we can add ...
- 21.3k
4
votes
1
answer
79
views
Why are extensions of countable models of ZFC better behaved than extensions of arbitrary models of ZFC?
This answer hints that certain kinds of extensions are only guaranteed to exist for countable models of ZFC. Why?
One intuitive reason i can think of is that the metatheory might not have enough new ...
- 457
6
votes
0
answers
359
views
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 ...
- 249k
1
vote
1
answer
53
views
Need to check this proof that the class of models of ZC that fail replacement is not axiomatisable.
Here ZC is ZFC minus Axiom of Replacement.
My proof is as follows:
Suppose $M$ was axiomatized by a theory $H$.
For non-zero limit ordinal $\alpha$, let $T_\alpha$ be the set of the replacement axioms ...
- 335
1
vote
1
answer
47
views
Is the interpretation of a constant symbol an injective map?
The context
Im trying to show that the reduct to the luanguage $\frak{L}$ of any model of the complete diagram $D(\frak{M})$ of an $\frak{L}$-structure $\frak{M}$ is an elementary extension of some ...
- 535
-3
votes
1
answer
76
views
On pseudofiniteness and injective-surjective
In many literatures it is noted that “let M be pseudofinite and f a definable function, then f is injective if and only if it is surjective.” Let's break it down into parts, Let A --- M be a ...
0
votes
1
answer
118
views
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 $\mathcal{L}^\ast \supseteq \mathcal{L}$ ...
- 17
2
votes
1
answer
104
views
In what sense is forcing "impossible" in $L$?
I just saw an interesting video from Hugh Woodin about Ultimate $L$. In it, he says one of the reasons $L$ is so interesting is because it not only settles many natural set theory questions, but is ...
- 7,950
2
votes
1
answer
53
views
Last Bits of Proof of the Compactness Theorem in Propositional Logic
I am reading the proof of compactness theorem for the propositional logic and the last part of the proof is left as exercise 2 of section 1.7 in the book by Enderton, A Mathematical Introduction to ...
- 15k
1
vote
1
answer
59
views
Linear disjointness of normal algebraic extension
In [Tent & Ziegler] "A Course in Model Theory" section B.3, it mentions that for ring $R,S$ contained in a common field extension with a common subfield $k$, if $S$ is an algebraic ...
- 403
3
votes
1
answer
59
views
Algorithm for Determining Truth of First-Order Sentences in Complex Numbers
Following my previous question Decidability in Natural Numbers with a Combined Function, I realized that there is a spectrum regarding the hardness of deciding whether a first-order sentence is true ...
- 87
-1
votes
1
answer
87
views
Decidability in Natural Numbers with a Combined Function [closed]
It is well known that there is no algorithm to determine whether a given first-order sentence is true in the structure of natural numbers with both addition and multiplication. In contrast, Presburger ...
- 87