Model theory is the branch of mathematical logic which deals with the connection between a formal language and its interpretations, or models.

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Tractability of forcing-invariant statements under large cardinals

It is usual to mention theorems of the kind: Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi ...
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Defining 'free monoid' without Nat?

Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined? The definitions that I have been able to track down all use the natural ...
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Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
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What are the advantages of the more abstract approaches to nonstandard analysis?

This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
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What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...
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Complete theory with exactly n countable models?

For n an integer greater than 2, Can one always get a complete theory over a finite language with exactly n models (up to isomorphism)? There's a theorem that says that 2 is impossible. My ...
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1answer
571 views

What is the precise relationship between o-minimal theory and Grothendieck's “Esquisse d'un programme”?

I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So ...
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“Fraïssé limits” without amalgamation

All structures are countable with countable signature. Given a structure $\mathcal{A}$, the age of $\mathcal{A}$, $Age(\mathcal{A})$, is the set of structures isomorphic to finitely-generated ...
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Is there a forcing closure?

The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
8
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ω-categorical, ω-stable structure with trivial geometry not definable in the pure set

Briefly, my question is the following. does every countable ω-categorical, ω-stable structure with disintegrated strongly minimal sets interpret in the countable pure set? By countable pure set I ...
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1answer
177 views

relatively free groups in $Var(S_3)$

Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free? This question is related to my previous question Relatively free algebras in a variety ...
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1answer
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How do we know if Vaught's Conjecture is Absolute?

Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture. First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a ...
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1answer
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Compactness in Bishop's constructive mathematics

In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information ...
53
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Is there a 0-1 law for the theory of groups?

Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ...
26
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15answers
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What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...
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6answers
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What are some proofs of Godel's Theorem which are *essentially different* from the original proof?

I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof. This is partly inspired by ...
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In model theory, does compactness easily imply completeness?

Recall the two following fundamental theorems of mathematical logic: Completeness Theorem: A theory T is syntactically consistent -- i.e., for no statement P can the statement "P and (not P)" be ...
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10answers
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Completeness vs Compactness in logic

One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
26
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4answers
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Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
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Reasons to believe Vopenka's principle/huge cardinals are consistent

There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
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Intuition for Model Theoretic Proof of the Nullstellensatz

I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is ...
8
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4answers
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Incompleteness and nonstandard models of arithmetic

The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome. Reading Peter Smith's "Gödel Without (Too Many) ...
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Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
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6answers
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A book about model theory

I am looking for a good book about model theory. As this is obviously too vague, let me explain what I am looking for and why. First I am interested about the basics and foundations of model theory. ...
27
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3answers
697 views

Is the field of constructible numbers known to be decidable?

By the field of constructible numbers I mean the union of all finite towers of real quadratic extensions beginning with $\mathbb{Q}$. By decidable I mean the set of first order truths in this field, ...
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1answer
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Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?

By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following: We are considering a ...
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Composite pairs of the form n!-1 and n!+1

It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$. Is ...
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The Closure-Complement-Intersection Problem

Background Let $A$ be a subset of a topological space $X$. An old problem asks, by applying various combinations of closure and complement operations, how many distinct subsets of $X$ can you ...
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3answers
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Is non-connectedness of graphs first order axiomatizable?

A recent question asked for graph properties that are first order axiomatizable but not finitely axiomatizable. Connectedness was mentioned in the context. Connectedness can be axiomatized in ...
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3answers
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Intuitionistic Lowenheim-Skolem?

Is there a version of the Löwenheim-Skolem theorem in intuitionistic logic? I'm particularly interested in the "downward" form. The standard proof I know uses the Tarski-Vaught test for ...
13
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2answers
963 views

When does Vopěnka's principle hold?

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with ...
9
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2answers
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Non-standard models of finite set theory

It is well known how the intended model and how the (countable) non-standard models of arithmetic look like. It's also well known how the intended model of set theory with the axiom of infinity ...
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2answers
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A question about open induction

An old theorem of A. J. Wilkie (Some results and problems on weak systems of arithmetic, Logic Colloquium '77) asserts that a discretely ordered ring $R$ can be extended to a model of open induction ...
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Are the types of nonstandard natural numbers within a Z-chain identical?

Hi, I was wondering how much (if anything) $\mathcal{L}_{PA}$ can express about individual nonstandard elements in a nonstandard model of PA. For instance, presumably it can say that each has ...
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1answer
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Least ordinal not in a countable transitive model of ZFC

Frequently it is useful do deal with countable transitive models M of ZFC, for example in forcing constructions. The notion of being an ordinal is absolute for any transitive model, so certainly if ...
17
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5answers
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What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. ...
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How does categoricity interact with the underlying set theory?

Here's the setup: you have a first-order theory T, in a countable language L for simplicity. Let k be a cardinal and suppose T is k-categorical. This means that, for any two models M,N |= T of ...
8
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Vaught's conjecture for partial orders

In ``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208'' the following is proved: Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...
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1answer
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A totally categorical structure with trivial geometry which is not interpretable in the trivial structure

Among the theorems of early geometric model theory there is one by Lachlan stating that every totally categorical structure with a trivial pregeometry is intrepretable in a dense linear order. That ...
6
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Is there any o-minimal expansion of the real field with functions of growth higher than exponential?

Let $\bar{\mathbb{R}}$ be the structure of the real field, that is $(\mathbb{R},0,1,+,-,*,<)$ . We say that a function $f$ is of growth higher than exponential if for all $N\in \mathbb{N}$ there ...
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Applications of forcing in model theory

What are the major applications of (set theoretic) forcing in model theory?
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1answer
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Is the first-order theory (with =) of real numbers with addition and multiplication complete and decidable?

Due to Andreas Blass's answer to my question "Is the feasibility of a system of nonlinear, non-convex equations (inequalities) decidable?", i have now investigated real closed fields (RCF), because i ...
6
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4answers
670 views

Self-defining structures

The relations $R$ in abstract graphs (with genuinely propertyless vertices) cannot be defined because there is nothing the relations can base on: they have to be presupposed. But consider derived ...
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1answer
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Show that the positive existential theory is undecidable

To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: ...
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1answer
263 views

Self-containing graphs

[Second try, after this question failed.] Let me sketch a notion of self-containing structures by a simple example. Consider the class $\Gamma$ of finite or countable digraphs ("graphs" for short) ...
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1answer
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A Special Pair of Formulas

Consider the first order language ‎$‎‎‎\mathcal{L}=\{\in,\subseteq\}‎$ and ‎$‎‎\{\in\}$-theory ‎$\text{ZFC}$.‎ ‎Is ‎the‎re a formula ‎$‎‎\psi ‎(x,y)‎ \in \{\subseteq\}-Form‎$ ‎with ‎the ‎following ...
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1answer
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Riemann hypothesis for zeta function of definable sets over finite fields

Hi, Consider the zeta function of a definable set over a finite field. More precisely, let $\varphi$ be a formula in the language of fields and let $X$ be a definable subset of $F^n$ given by ...
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1answer
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Completeness of Algebraically Closed Valued Fields(ACVF) Theory

One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,*, U, \mid )$. In ...
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Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...
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3answers
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Categoricity in second order logic

Hi, It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example ...