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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How to verify satisfialibility in a model? (Confusions with Gödel's Completeness Theorem)
I just cannot believe that Gödel's Completeness Theorem is right.
Let say we fixed some first order logic with some structure. Theorem claims that for any sentence $P$ in this logic we have that
$$\...
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Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$
I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227.
A partial ordering $R$ is said to be dense iff whenever $xRz$, ...
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Clarifying the definition of an axiomatic system
I'm a really beginner in Mathematical Logic.I'm currently reading Shoenfield Mathematical's Logic and i'm having a hard time trying to relate the concept of Formal Systems with the concept of Axiom ...
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In what sense of "structure" do group homomorphisms "preserve structure"?
It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia:
The purpose of defining a group homomorphism as it is, is to create functions that preserve ...
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Model of concatenation theory with left-cancellation but no right-cancellation
The theory of concatenation (TC) can be equivalently expressed as the following assumptions:
Closure of strings under concatenation $+$.
Existence of an empty string $e$, namely $e+x = x = x+e$ for ...
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Connection between interpretation, variable assignment and truth valuation.
Let us have some formal language $\mathcal{L}$ and an $\mathcal{L}$-structure $\mathcal{U}=(A,\mathcal{I})$. Where $A$ - non-empty set, called domain, and $\mathcal{I}$ - interpretation.
I know that ...
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What axioms need to be added to second-order ZFC before it has a unique model (up to isomorphism)?
What axioms need to be added to ZFC2 (second-order ZFC) before the theory has a unique model (up to isomorphism)?
I was thinking: adjoin the generalized continuum hypothesis (GCH) and a statement ...
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(Why) is topology nonfirstorderizable?
Is it the right point of view to say, that topology is nonfirstorderizable (only) because the union of arbitrarily many open sets has to be open? And if "arbitrarily many" was relaxed to "finitely ...
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How can there be genuine models of set theory?
I know that this a beginner's question asked too many times, but I still didn't get an answer which lets me quit asking:
Given that a model/interpretation of a theory (in the Tarskian sense) is a ...
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Example of non-isomorphic structures which are elementarily equivalent
I just started learning model theory on my own, and I was wondering if there are any interesting examples of two structures of a language L which are not isomorphic, but are elementarily equivalent (...
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Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?
Or more importantly, is it independent of the axiom of choice. The compactness theorem states the given a set of sentences $T$ in a first order Language $L, T$ has a model iff every finite subset of $...
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Is there a bijection between the reals and naturals?
I found this pop math article saying that there was a paper published last year that proved that the cardinalities of the reals and naturals are equal. Is this true or is it a misinterpretation of the ...
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Confusion over the definition of "model"
In my question yesterday I asked about the definition and usage of the word "model", for which I was told the following definition:
A formula of propositional logic is true under an interpretation ...
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Learning Model Theory
What books/notes should one read to learn model theory? As I do not have much background in logic it would be ideal if such a reference does not assume much background in logic. Also, as I am ...
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Is this a characterization of well-orders?
While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
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