All Questions
18 questions
6
votes
0
answers
239
views
Existing literature on logics "describing their own equivalence notions"
Say that a regular logic $\mathcal{L}$ is self-equivalence-describing (SED) iff for every finite language $\Sigma$ there is a larger language $\Sigma'$ containing at least $\Sigma$ and two new unary ...
- 20.5k
9
votes
1
answer
410
views
On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces
In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
- 3,011
6
votes
0
answers
189
views
Reference requestion: theorem guaranteeing self-embeddings of expansions of $\mathit{Ord}$
This is an attempt to locate a theorem I vaguely remember but cannot find (which arose in the context of Consistency of non-trivial elementary embedding $j : \mathit{Ord} \to \mathit{Ord}$ and ...
- 20.5k
4
votes
0
answers
182
views
$\mathcal{C}$-filtering of modules inherited by submodules
I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology.
DEFINITION: Let $\mathcal{C}$ be a ...
- 2,231
1
vote
1
answer
1k
views
A new generalisation of dimension? part 2
I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...
- 4,074
27
votes
4
answers
3k
views
What "metatheory" did early set theory/logic researchers use to prove semantic results?
Things like the first-order completeness theorem and the Löwenheim-Skolem theorem are considered foundational in mathematical logic.
The modern approach seems to be, usually, to interpret a "model" ...
- 4,897
1
vote
1
answer
228
views
Is there a 'Constructible Universe' that is a submodel of a non-transitive model of $ZF$?
In Barwise's book, Admissible Sets and Structures, the following statement is made (on pg. 8):
"...if $ZF$ is consistent, so is $ZF$ + "There is no transitive model of $ZF$"
He mentions this fact ...
- 6,132
6
votes
0
answers
137
views
Natural theories for the failure of gap-1 transfer principle
The gap-1 transfer principle says that the transfer principle $(\kappa^+, \kappa) \to (\lambda^+, \lambda)$ holds for all infinite cardinals $\kappa, \lambda.$
It is known that for some sentence $\...
- 32.1k
7
votes
6
answers
3k
views
Looking for a source for Intended Interpretation
Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
- 16.6k
1
vote
0
answers
194
views
Reference request: Models of isomorphic languages result into isomorphic categories
This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory.
Fix an uncountable universe $\...
- 10.5k
16
votes
6
answers
2k
views
Application of Fraïssé construction in set theory
As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...
- 1,872
6
votes
2
answers
495
views
Forcing for Arbitrary First Order Theories
Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...
user45939
3
votes
2
answers
557
views
Can we force with Fraisse filters to solve Vaught's conjecture?
Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...
user36136
8
votes
1
answer
384
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 U\...
- 20.5k
4
votes
2
answers
657
views
Cantor theorem on orders
It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum ...
- 3,630
2
votes
1
answer
181
views
Feferman-Kreisel preservation theorem
I want to show the following theorem from Feferman and Kreisel:
Let $\phi$ be such that there is a theorem $\sigma$ of ZFC so that for every transitive model $M$ of $\sigma$, we have: $\phi^M \to \...
- 121
7
votes
1
answer
994
views
Is this a proper application of the Lowenheim-Skolem Theorem to a proper class?
Please don't get put off by the length, all the questions are quite simple, but given the quasi-mathematical context I tried to be precise with the formulation. The more mathematically interesting ...
- 497
7
votes
1
answer
636
views
Model theory stressing order type of universe.
In Appendix B to their Model Theory, Chang and Keisler list some problems and conjectures that, at the time of publication, were unsolved. A few of them take imperative form, for instance:
"Develop a ...
- 1,081