# 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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### Definability of the class of Hausdorff topological spaces using the closure operator in a first-order way.

Recall the Kuratowski closure axioms for general topological spaces: $f(0)=0$, where $0$ is the empty set $x \subseteq f(x)$ $f(x)=f(f(x))$ $f(x \cup y) = f(x) \cup f(y)$ The question I am ...
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### what's means V is model of zfc?

I thought that set of formulas S has a model iff there is an interpretation satisfying that all the formula in S make true. so I understand zfc's model is a kind of assignment that make all the zfc's ...
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### Can you give me an example of an implicit use of Godel's Completeness Theorem, say for example in group theory?

Wikipedia says that the more general form of Godel's completeness is "used implicitly, for example, when a sentence is shown to be provable from the axioms of group theory by considering an ...
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### Is there a theorem that says we don't limit our notion of conservative extension by only focussing on set models?

From the current Wikipedia article on conservative extensions, it states: an extension $T_2$ of a theory $T_1$ is model-theoretically conservative if $T_1\subseteq T_2$ and every model of $T_1$ can ...
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### An Example of a Proof-theoretic Conservative Extension that's not a Model-theoretic One

Coming from here and the Wikipedia page on conservative extension, which defines a proof-theoretic conservative extension like this: a theory $T_2$ is a (proof theoretic) conservative extension of a ...
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### Understanding the witness property in the Henkin construction

I'm trying to get a really good intuition for why the proof of the compactness theorem via the Henkin construction in Marker's model theory uses the witness property, specifically, why use For every ...
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### If the Collatz conjecture is undecidable, then it is true

Suppose that the Collatz conjecture is undecidable in PA. Then, by Godel's completeness theorem, there are models where it is true, and models where it is false. Let M be a model where it is true. The ...
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### Need help with a question about cardinality and perhaps model theory

Let $I$ be the unit interval. Call a collection $\Sigma$ of subsets of $I$ "valid" if it satisfies the following three conditions: $I \in \Sigma$. $\Sigma$ has the same cardinality as $I$. ...
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### Can you define a topology so that all the parametrically definable subsets are open?

Let $\mathcal{L}$ be a first order language with only relation and constant symbols. Suppose $\mathfrak{A}=\langle A,\ldots\rangle$ is a structure for $\mathcal{L}$. Let $\mathscr{D}$ be the set of ...
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### Large invariant types realized in $\cal U$.

Below $\cal U$ is a monster model; $\varphi(x;y)$ a given parameter-free formula; $a\in\cal U^y$. When is the type $p(x)$ = { $\varphi(x;fa):f\in{\rm Aut}({\cal U})$ } realized in $\cal U$? (Assuming ...
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### A first order theory of $\mathbb{R}$

The properties that $\mathbb{R}$ should satisfy can be summarized as "Dedekind complete ordered field". The axioms of ordered fields are first-order sentences. I came up with a first-order ...
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### Difference between a Function and the name of a Function

I am currently studying the book by Loeb and Hurd on Nonstandard Analysis. In the chapter "simple languages for relational systems" they differ between an mathematical object like a function ...
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### General theory of formal variable and algebra extension [closed]

Often in structure construction, a formal variable is introduced into a given structure (possibly with certain generating or constrained relation), resulting in a new structure. Are there any ...
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### Is it possible to find a general algorithm for determining how many models for a given theory there are?

In this context there already exists a model and the theory is consistent. I'm leaning towards no, because of Turing's Halting Problem and the fact that you should be able to mimic a Turing machine ...
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### Proof that any algebraically closed field is homogeneous.

Hi guys I'm currently try to solve one of the exercises in Marker's "Model Theory: An Introduction" and I'm kinda stuck. The exercise asks to prove that any algebraically closed field is ...
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### Horn sentences and power structures

It is well-known that Horn sentences are "preserved" under products (see for instance Show that the direct product of structures satisfies a Horn sentence). I was wondering what happens with ...
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### The well-ordering number of second-order logic

The definitions and referred pages are from (Model-Theoretic Logics, Barwise 1985), primarily from chapter II. A free version is available here: Chapter II Definition (pinning down ordinals): Let < ...
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### In what theory consistency proofs take place?

I am reading Kenneth Kunnen's Set Theory and had this question. For instance, on Page 113 Lemma 2.3 states that: Assume $S,T$ are set of sentences in the language of set theory, and assume that for ...
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### Axiomatization of hyperreal numbers

From wikipedia: The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one ...
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### For what theories are submodels closed for intersection?

The theories coming from a variety in the sense of universal algebra are a known example. Fields and ordered groups are important examples not covered by universal algebra. I also believe models of ...
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### (When) are recursive "definitions" definitions?

This is a "soft" question, but I'm greatly interested in canvassing opinions on it. I don't know whether there is anything like a consensus on the answer. Under what conditions (if any) are ...
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### Is the class of linearly-orderable rings first order axiomatizable?

A linearly ordered ring is a commutative ring $R$ with unity equipped with a linear order $\leq$ that is compatible with addition, and such that the set of nonnegative elements are closed under ...
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### Does Löwenheim-Skolem require Foundation in any way?

As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
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I am self-learning logic, and trying to prove the following exercise using the Compactness Theorem: Suppose $T$ is a theory for language $L$, and $\sigma$ is a sentence of $L$ such that $T \models \... • 541 3 votes 1 answer 229 views ### Is quantifying over natural numbers non first order? I was reading here that Note that ‘x is an infinitesimal’ is not first order, because it requires you to quantify over the naturals. Whats's non first order about quantifying over natural numbers? • 6,025 5 votes 1 answer 425 views ### Is there a theory in which all types can be omitted? Is there a natural example of a first order complete (consistent) theory$T$in which every 1-type can be omitted? or is there always some isolated type? In that case, why? Of course there are plenty ... 1 vote 1 answer 82 views ### Proving that the set of sentences that are true using the symbols$+,<,=$is the same over all ordered fields I am interested in whether the set of formulas that one can prove true for a concrete ordered field using the symbols$+,<$and$=$, depends on the field. In particular, I am interested in the set ... • 6,025 -2 votes 2 answers 113 views ### in definition of assigment, what's means 'except possibly a'? in frist-order logic, part of assignments practice represent like this "if 𝜙is ∀𝛼𝜓, where 𝛼 is a variable, then ⊨vℳ 𝜙 iff for every assignment 𝑣' that agrees with 𝑣 on the values of every ... • 33 3 votes 0 answers 68 views ### Two families of isomorphic structures have isomorphic ultraproduct. I am trying to prove the following result: Let$(\underline{M}_i)_{i\in I}$,$(\underline{N}_i)_{i\in I}$be two families of structures such that, for all$i\in I$,$\underline{M}_i \cong \underline{...
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 ...