Questions tagged [model-theory]
Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.
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A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
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Two notions of generalized quotient/substructure
Given a language $\Sigma$ and a $\Sigma$-algebra (in the sense of universal algebra) $\mathcal{A}=(A;\dotsc)$ and a function $f:A\rightarrow A$, let $\mathcal{A}_f$ be the $\Sigma$-algebra whose ...
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Extensions of $PA+\neg Con(PA)$ with large consistency strength
There is a large hierarchy of theories strengthening $PA$ eg $PA+Con(PA)$, $PA+Con(PA+Con(PA))$,..., second-order arithmetic and $ZFC$, ordered by consistency strength.
Is there an extension of $PA+\...
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Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
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Reference for Understanding Shelah's Proof of Vaught's Conjecture for $\omega$-stable Theories
I'm looking for a source to help me better understand Shelah's proof of Vaught's Conjecture for $\omega$-stable Theories (https://shelah.logic.at/files/95409/158.pdf). An obvious candidate is Makkai's ...
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Comparing computable structures via Kleene and Skolem
Below, by "structure" I mean "computable structure in a finite language with domain $\omega$," and by "sentence" I mean "finitary first-order sentence containing no ...
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A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$
Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have
$M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
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Interest in the size of ultrapowers in model theory
It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set ...
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General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
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How is it possible for PA+¬Con(PA) to be consistent?
I'm having some trouble understanding how a certain first-order theory isn't just straight-up inconsistent.
Let $PA$ be the axioms of (first-order) Peano arithmetic and let $C$ be the following ...
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Interpreting group-theoretic sentences as statements about algebraic groups
Suppose we are given a sentence in the language of groups, e.g. $\phi=\forall x\forall y(x\cdot y=y\cdot x)$, and suppose that we are also given the data defining an algebraic group $G/k$. One can ...
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Can a model of "true computation" exist? What would be its consequences?
Analogous to the model of True Arithmetic, the model of "True Computation" is defined to be the set of all true first-order statements about Turing machines i.e. answers to elementary ...
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Symmetric monoidal functors from powers of the natural numbers to Set
Consider the full subcategory of $\mathbf{Set}$ consisting of the singleton $1$ and countable infinite sets. (Originally this came from the powers $\mathbb{N}^{\times k}$ and the morphisms between ...
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For a pure-injective module $M$ does the property "$\operatorname{Hom}(-,M)$ is surjective" commute with certain limits?
$\DeclareMathOperator\Hom{Hom}$Let $M$ be a pure-injective module. Then $\Hom(\varphi,M)$ is surjective for a pure-mono $\varphi$. It is well-known that $\varphi$ is a direct limit of split monos $\...
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Finite (schema) axiomatizability of representable cylindric algebras
If we know that the class of all representable cylindric algebras of dimension $\alpha$ (for any ordinal number $\alpha>2$) is NOT finitely (schema) axiomatizable*, then does it (perhaps trivially) ...
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