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comment Generalizations of pregeometries
I don't think your theory is uncountably categorical. One can vary the sizes of equivalence classes and obtain many models.
May
25
comment Functions or relations stable under automorphism
Your claim that invariant subsets in a saturated model are definable is not correct. Say any type definable set is invariant under automorphisms fixing its base.
May
9
comment Characterization of superstability
I am not aware of any such characterisation, I don't think you can do much better than finite.
May
8
answered Characterization of superstability
Apr
26
comment Morley’s Categoricity Theorem for uncountable languages.
Accessible expositions of Shelah's work are hard to find. I personally haven't seen any exposition of Morley's theorem for uncountable languages not by Shelah.
Apr
22
comment Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.
Suppose $\mathcal M \not \equiv \mathcal N$. Then there is a formula $\phi$ such that $\mathcal M \models \phi$ and $\mathcal N \models \lnot \phi$. Now use the hint.
Apr
21
answered Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.
Apr
16
awarded  Custodian
Apr
16
reviewed Approve Are there arbitratily long runs of consecutive integers n that are NOT of the form $n = p^k$ or $n = 2p^k$ for some $k>0$ and $p$ an odd prime?
Apr
16
reviewed Approve Prove: if $a$ and $b$ are algebraic, then $a + b$, $a - b$ and ab are also algebraic
Apr
11
comment Saturated model for Th(Z,+,-,0,1)?
I am not sure, but I think $\hat {\mathbb Z}$ the profinite completion of $\mathbb Z$ would be an $\omega$-saturated model.
Dec
9
awarded  Caucus
Nov
27
answered Number of automorphisms of saturated models
Nov
10
comment Uncountably Categorical Theories and Embeddings
It is basically true. For the details see the section 6.3 of Tent and Ziegler "A course in model theory".
Oct
23
comment Marker Exercise 2.5.10: universal part of a theory and supermodel
Since $\bar a$ does not occur in $T$, the relation $T \models \lnot \phi(\bar a)$ implies $T \models \forall \bar x \lnot \phi(\bar x)$.
Oct
19
comment A fragment of Exercise 1.3.4 in _Shorter Model Theory_ by Hodges
It is not true, precisely because of the reason you stated. E.g. take $\mathcal B = \mathcal A$ and let $\phi$ be the identity on $S$. Then extend $\phi$ arbitrarily to $A$. Then $\phi$ satisfies the condition of your statement, but not the conclusion (in general).
Sep
30
awarded  Explainer
Sep
25
comment Question from Hodges' textbook Shorter Model Theory
@PrimoPetri, yes $\exists x \chi$ is false in the empty structure. Did I say something that contradicts it?
Sep
25
answered Question from Hodges' textbook Shorter Model Theory
Sep
15
comment Indiscernible subsequences
If the theory has order, then the pairs $\{a_i, a_j\}$ are divided into to parts depending on whether $a_i < a_j$ or not. An indiscernible sequence must be homogeneous, showing that $\kappa \to (\kappa)^2_2$. Thus $\kappa$ must be weakly compact. I think using a similar argument we can show that $\kappa$ is Ramsey. I don't know if being Ramsey is sufficient.