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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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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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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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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90 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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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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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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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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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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281 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 ...
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186 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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73 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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114 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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144 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. ...
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161 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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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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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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137 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 ...
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215 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 ...
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105 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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291 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 ...
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197 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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78 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? ...
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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 ...
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158 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.
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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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78 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, ...
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356 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, ...
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103 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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107 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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38 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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215 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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111 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 ...
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91 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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235 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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234 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 ...
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2answers
139 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 ...
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1answer
103 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 ...
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175 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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193 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 ...
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245 views

Is there a well defined “intended” model of the real numbers in the same sense as there is one for the natural numbers?

Background: We know that PA has more models than the intended model, N, because it is not strong enough and is also satisfied by non-intended models, known as non-standard models of arithmetic. When ...
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75 views

Is there a formal notion of equivalence between structures with potentially different signatures?

What is the appropriate notion of equivalence between structures with potentially different signatures? Consider an example from abstract algebra. Whether a group is defined as a pair $(X,*),$ or as ...
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208 views

What are the formal properties of Godel numbering that are required to make it 'work'?

Godel numbering assigns a number to every formula. It appears to me that any encoding will do. However its also apparent, though I'm not sure how, that certain properties of the encoding used in Godel ...
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78 views

Construct countable Boolean algebra

How can I construct a countably infinite Boolean algebra with $n$ atoms, for $n \in \mathbb{N}$?
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331 views

Is “reflexive transitive closure of relation $R$” a first-order property?

Suppose I have a language with two binary relation symbols $R$ and $R^\ast$. Suppose I have a first-order theory $T$ which says some things about $R$, but nothing about $R^\ast$. Is there a set of ...
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Axiomatizable classes

Are the statements below true or false: The class of finite sets is axiomatizable The class of infinite sets is axiomatizable The class of infinite sets is finitely-axiomatizable The class of fields ...
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Complete n-types of the theory of atomless Boolean algebras

I have to answer the next questions: What is the number of complete 1-types of the theory of atomless Boolean algebras? What is the number of complete 2-types of the theory of atomless Boolean ...
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56 views

Uncountable models for a language $L_Q$

$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$. Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of ...
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What techniques are available for “surgical adjustment” of models of set theory?

Suppose I have a model $M$ of set theory (ZFC, or whatever). Let's say that I want to take a set $a$ out of it, and still have a model of set theory. For the sake of argument, say $a$ is one of the ...
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The ultraproduct $\mathbb{N}^\mathbb{N} / \mathcal{F}$ is uncountable

I have to prove: Let $\mathcal{F}$ be a non-trivial ultrafilter on $\mathcal{P}(\mathbb{N})$. Prove that the ultraproduct $ \mathbb{N}^* = {\mathbb{N}^{\mathbb{N}}}/{\mathcal{F} } $ (I don't know if ...
6
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1answer
233 views

Non-isomorphic countable Boolean algebras

I'm trying to solve the next exercise: Construct a sequence $\mathcal{B}_0,\mathcal{B}_1, \ldots$ of countable Boolean algebras such that for all $m \neq n$ then $\mathcal{B}_m \ncong \mathcal{B}_n$. ...