# Tagged Questions

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 ...

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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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### 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 ...
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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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### What is an efficient nesting of mathematical theorems?

Various mathematical areas of research evolved from a wide and diverse range of questions. Many are physical in nature or come from informatics/computer science, some are procedural or optimization ...
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### $2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S,<)$

This is (a translation of) an excerpt from a model theory textbook that shows that $2\mathbb Z$ is not a definable set in the structure $(\mathbb Z, 0, S, <)$, where $S$ is the successor function. ...
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### Number of Non-isomorphic models of Set Theory

Assume that the meta theory allows for model theoretic techniques and handling infinite sets etc (The meta theory itself is, informally, "strong as ZFC"). Also assume that I'm studying ZFC inside this ...
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### Confusion of the decidability of $(N,s)$

In some context the PA has only the successor operator $'s'$, but in logic we always refer the structure of PA is $(\mathbb{N},0,1,s,+,\times)$. I believed the theory of the two sturctures are ...
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### FO-definability of the integers in (Q, +, <)

With $Q$ the set of rational numbers, I'm wondering: Is the predicate "Int($x$) $\equiv$ $x$ is an integer" first-order definable in $(Q, +, <)$ where there is one additional constant symbol ...
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### Comparing countable models of ZFC

Let us consider the class $\cal C$ of countable models of ZFC. For ${\mathfrak A}=(A,{\in}_A)$ and ${\mathfrak B}=(B,{\in}_B)$ in $\cal C$ I say that ${\mathfrak A}<{\mathfrak B}$ iff there is a ...
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### Gödel's Completeness Theorem and logical consequence [closed]

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
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### Does the existence of a $\mathbb{Q}$-basis for $\mathbb{R}$ imply that choice holds up to $\frak c$?

The axiom of choice is, for ZF, equivalent to the statement that every vector space has a basis. The implication of AoC by the existence of a basis for any vector space is shown in this paper. The ...
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### Fraïssé limits and groups

I was recently reading up on Fraïssé limits in Hodges' "A shorter model theory." I was trying to think of some examples and wanted to see if I could take the Fraïssé limit on the category of finite ...
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### Show that binary < “less than” relation is not definable on set of Natural Numbers with successor function

After reading about the question, I've come to believe that it would suffice to exhibit and automorphism of that is not order preserving. However, I'm unsure of how to construct such an automorphism, ...
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### A first order sentence such that the finite Spectrum of that sentence is the prime numbers

The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ ....
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### dense linear orders DLO

I am asked to prove that if I have two models of dense linear orders DLOs WITH the minimum and maximum. must be izomorpic to each other by fining direct izomorphy. I seem to always get stuck ...
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### Show that if U is a principal ultrafilter then the canonical inmersion is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$

Show that: If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$
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### Union of definable sets

I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable. What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ ...
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### Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
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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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### Comparing Category Theory and Model Theory (with examples from Group Theory).

The following question eats my brain: The standard definition of a "category" and a list of examples following this definition confuses me. My question is simple: (Q1)If someone write "the category ...
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### Most astonishing applications of compactness theorem outside logic

The compactness theorem has a lot of applications to logic and model theory. I'm looking for applications. I'm looking for theorems in other areas of mathematics which seem at first sight to have ...
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### Applications of model theory to analysis

Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from ...
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### Do isomorphic structures always satisfy the same second-order sentences?

I know that if two mathematical structures are isomorphic, then they satisfy the same first-order sentences. The converse is false. This is probably a completely obvious question, but is it true that ...
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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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### Complete calculus of first-order logic working for empty structures too

Usually, in model theory, one presupposes that structures (models) are non-empty. I don't like this (related: What's the deal with empty models in first-order logic?). So let us explicitly permit ...
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### coproducts of structures

Suppose $S$ is a family of $L$-structures where $L$ is some collection of constant symbols, relation symbols, and function symbols. Does the coproduct of elements of $S$ exist? If not, how does one ...
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### What is the real meaning of Hilbert's axiom of completeness

According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) ...
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### Weak categoricity in first order logic

In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second ...
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### Models of the empty theory

so throughout my reading of model theory the idea of the "empty" theory has been put down as trivial, however I am curious as to why. Let us look at the following. Suppose We have $L_=$, the language ...
Let $\kappa$ be an regular uncountable cardinal carrying a $\tau$-structure for some countable language $\tau$. What can be said regarding the existence of ordinals $\alpha <\kappa$ carrying ...