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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understanding provability and more about Löb's theorem

This question is an additional question for my previous question,one week ago. Link : understanding provability Fortunately, some persons kindly commented for my question. However, I think I still ...
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2answers
141 views

Is a topological space a structure?

In model theory, a structure (or "model") is typically defined as a set together with some finitary relations and/or operations on that set. For instance, a group can be viewed as a pair $(G,*),$ ...
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329 views

What lessons have mathematicians drawn from the existence of non-standard models?

So, as someone whose knowledge of mathematics has always come from studying it with an eye towards philosophical/foundational issues and studying it with other philosophers (who are not primarily ...
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1answer
125 views

Question regarding inexpressibility results over finite models using compactness and the Löwenheim–Skolem theorem

In the book Elements of finite model theory by Leonid Libkin, they show that the parity query for structures over an empty vocabulary is not first order definable. They do this by constructing two ...
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1answer
84 views

Two homogenous structures realizing the same types are isomorphic

Let $M$ and $N$ be two countable, homogeneous structures, and assume that they both realize the same types with a finite number of variables. Does it follow that $M$ and $N$ are isomorphic? What if ...
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1answer
126 views

understanding provability

I am still confused about provability. . . Let a statement P is, sort-of-says like this. P: ( "X is provable" ∧ "P is provable" ) If ( X is provable ∧ P is provable ) is provable → (P is ...
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1answer
352 views

SHOW that there are infinitely many equivalence classes of formulas

Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the ...
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3answers
259 views

Axiom of Choice and Determinacy

In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
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1answer
113 views

Spectrum and sentences

Is there an example of a language with only one spectrum equal to the even numbers? Also, is there an example of a language with only relation which has spectrum equal to the set of non-primes? A ...
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2answers
136 views

Standard model and formulas in logic

If $\phi(v)$ is a formula defining the prime numbers in the standard model, then are there nonstandard elements satisfying $\phi(v)$? What if $\phi(v)$ is a formula for powers of $2$? Also, what if ...
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2answers
98 views

Non-Archimedan Groups

I'm trying to think of an explicit example of a non-Archimedian group. The definition of Archimiedean is s.t. if for all $x$ and $y$, there is some $m$ a Natural number s.t. $mx = \underbrace {x + x ...
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1answer
66 views

Model THeories not equivalent

How can we show that Th(Rationals) does not equal Th(Integers) does not equal Th(Naturals)? Where Th(M) is a the set of all L sentences which are true in a model, M. I know that L is defined as = ...
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190 views

What is the correspondence between the semantic compactness of logics and the compactness of Stone spaces?

According to the answer by François G. Dorais, we know that a logic $\mathfrak L$ is compact iff its Stone space of the Lindenbaum–Tarski algebra of the empty theory (w.r.t the deductive system) is ...
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1answer
47 views

Is there a property that isolates the formulas that remain valid as you move from the natural numbers to the integers?

Is there a property that isolates the formulas that remain valid (or whose translations into the expanded language remain valid) as you move from the theory of the natural numbers to the theory of the ...
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1answer
97 views

What is the difference between First-Order Structures and Kripke Structures?

In the SEP article on Model Theory by Wilfrid Hodges (here), he writes: Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use ...
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126 views

Applying the compactness theorem

Using a Hilbert system: L is a FOL (First order language) with R, where R is a single binary predicate symbol. Suppse A = ⟨V,E⟩ is a structure for this language domain V = |A|. Suppose also that E = ...
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2answers
85 views

Non-isomorphic structures with equal cardinality

Let $\mathfrak{A}=(\mathbb{N},S,0)$ be a structure where $S$ is the sucessor function. Let $\mathfrak{B} =(\mathbb{N}\times \{0\} \cup \mathbb{Z} \times\{1\} ,S, 0)$ with $0 = (0,0)$ and $$ S(k,i) ...
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331 views

A sentence false in a field of characteristic $0$ but true in all fields of positive characteristic?

Consider the language $L=\{+,\cdot, 0, 1\}$ of rings. It is easy to show using compactness that if $\sigma$ is a sentence that holds in all fields of characteristic $0$, there is some $N\in \mathbb N$ ...
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1answer
58 views

Specturm 3b in a link

Can someone help with #3b in this link: http://homepages.math.uic.edu/~marker/math502f09/ps3.pdf I am trying to practice idea of spectrum, but cannot quite understand whatt they are asking for. ...
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3answers
302 views

Is Induction Independent of the Other Axioms of PA?

I am trying to come up with a model of first order Peano Arithmetic (PA) where induction fails. Let $PA^{-IND}$ have the same axioms as PA except the first order induction axiom schema is replaced ...
10
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1answer
198 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...
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1answer
76 views

L-sentences and spectra [duplicate]

Is there an example of a sentence with spectrum {p|p is prime}? I may need to resort to some theorem of Algebra, but not sure if it will help. Thanks
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1answer
119 views

Spectrum of elements in a set

Suppose that $X$ is a spectrum. Is $\mathbb{N}\setminus X$ a spectrum? By spectrum, we mean that it is the set containing all natural numbers $n$ s.t. there is a model of $\phi$ with exactly $n$ ...
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1answer
159 views

Definition of Spectrum in Logic

So I want to practice with understanding the definition of Spectra. Basically, I understand it is the set consisting of all natural numbers n such that there is a model of phi with exactly n elements. ...
6
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166 views

Is there a useful Galois connection between Languages and Grammars?

I've just beginning to learn logic and proof theory - and the following rather vague and perhaps ill-formed question occurred to me. Given an alphabet it's straightforward to construct the Language, ...
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1answer
61 views

Showing a model does not have a particular substructure and understanding satisfaction relations.

Out of Winfried Just and Martin Weese's Set theory book: Show that the model $\mathfrak B=\langle \Bbb Z, +, \le, 0 \rangle$ does not have any substructure whose universe is $\{-1,0,1\}$. In a ...
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2answers
113 views

Use of forcing to real line to make elements countable

Can we use forcing techniques to force the set of elements of the real line to be countable? If not can anyone show why it is not possible?
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1answer
155 views

Proof of Robinson's test

I have been working with Tent and Ziegler's Model Theory. I am on the Quantifier elimination chapter, and there they mention Robinson's test. It says that, for an $L$-theory $T$ three statements ...
5
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2answers
228 views

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 ...
3
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0answers
106 views

The Logic of Satisfiability?

I am aware of some study into the logic of provability. It is generally taken to be intermediate in strength between S4 and S5 modal logics. Is there corresponding study into something like the logic ...
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1answer
325 views

Why are the rational numbers an elementary substructure of the reals?

I read that the natural numbers ($\mathbb{N}$) are not an elementary substructure (ES) of the integers ($\mathbb{Z}$) , the integers ($\mathbb{Z}$) are not an ES of the rational numbers ...
3
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2answers
208 views

how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
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2answers
79 views

structure in context of ultraproduct

Ultraproduct is defined as $$\prod_{i \in I} M_i $$ I know that structure is usually of form $(A, \sigma, I)$, but in this context, what exactly is structure, and how do we get the cartesian product? ...
2
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2answers
80 views

Proving theorems about ZFC by proving them for an arbitrary model.

To prove that a statement follows from the group axioms, we typically write: Let $G$ denote an arbitrary group... Then... Thus, it s a theorem of the group axioms that... Presumably, this form ...
8
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1answer
164 views

what does it mean that constructible universe is definable from ordinals?

I know how constructible universe is created, but I also separatedly read that the universe is definable from ordinals - so I am wondering what it really means.
5
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3answers
234 views

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 ...
2
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1answer
79 views

What does it mean $Φ^M = Φ$, if $Φ$ is a primitive formula?

$Φ$ is a primitive formula in the language of set theory, while $Φ^M$ is the relativisation of $Φ$ to the class $M$. I can't understand why $Φ^M = Φ$. Let $Φ$ be $0 \in x$, it seems to me, ...
2
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2answers
383 views

Proof of Lowenheim-Skolem theorem

For each first-order $\sigma \,$-formula $\varphi(y,x_{1}, \ldots, x_{n}) \,,$ the axiom of choice implies the existence of a function $f_{\varphi}: M^n\to M$ such that, for all $a_{1}, \ldots, ...
2
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2answers
119 views

$V_k$ transitive model of ZFC when $k$ is inaccessible?

Is $V_k$ transitive model of ZFC when $k$ is inaccessible? I know that $V_k$ is a model of ZFC, but not sure if it's transitive one. If it is, why is it?
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1answer
111 views

$\infty$ as inaccessible cardinal and relation of inaccessible cardinal to second-order ZFC

(1) It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (Vα, ∈, U ∩ Vα) is only required to be 'elementary' with respect ...
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1answer
39 views

What are objects in the substructure referring to?

Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ Vκ, there exists α < κ such that ...
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1answer
232 views

How does one define a standard model of ZFC?

I sometimes come across the phrase, "a standard model of ZFC." Is this a rigorous concept? If so, how does one define it?
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1answer
123 views

why is first-order logic strongest?

I get how first-order logic has Lowenheim-Skolem, compactness theorem, but I am not sure why this leads to first-order logic being strongest. All Lowenheim-Skolem seems to say is that for first-order ...
3
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1answer
93 views

application of Lowenheim-Skolem theorem

So if minimal model of ZF exists, it is said that it is countable set by Lowenheim-Skolem. So, is Lowenheim-Skolem saying that for any countable theory with existence of infinite model there exists ...
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244 views

$\mathcal U$ Grothendieck universe. Is $\mathcal{P(U)}$ a model for NBG?

Suppose we are in ZFC, let $\mathcal U$ be an uncountable Grothendieck universe and consider the set of its parts $\mathcal{P(U)}$. (I will index axioms as $(\mathcal U.n)$) Note that if $x \in ...
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248 views

How can we tell if a set of axioms uniquely determines an algebraic structure?

Up to isomorphism. For instance, the group axioms are verified by an infinite number of non-isomorphic algebraic structures. But the Peano axioms, I think (my proof may lack some formality due to my ...
2
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2answers
141 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
4
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1answer
114 views

Proof of Compactness Theorem

I'm going through Enderton's Mathematical Logic text and have encountered a problem that I'm having trouble solving. After searching this website I've found that another user had the same problem (you ...
8
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1answer
185 views

If a theory has a countable $\omega$-saturated model does it need to have only countable many countable models?

If a theory has countably many countable models (up to isomorphism) then it has at countably many types, and it follows that there exists a countable $\omega$-saturated model of such theory. If a ...
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2answers
195 views

Is there a definition of “truth” without interpretations?

I know that given a sentence or formula of a formal system, this formula is a logical truth if it is true under all interpretations. Is it possible to define this same concept of logical truth ...