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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A proper class of formulas with every set-sized (but no proper-class-sized) subcollection satisfiable

What feature(s) must a (non 1st-order) language with proper-class-many formulas have in order to guarantee that: There is a proper class P of formulas such that both (a) every set-sized sub-...
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Kripke models of $HA$

Let $K$ be a kripke model and $k$ be one of its node, then $\mathcal{M}_k$ is classical structure of $k$. What is the strongest theory of arithmetic like $T$ such that for every kripke model $K\...
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Can we use this symbol? [closed]

We consider the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ and the language $L=\{+, \cdot , \frac{d}{dx} , 0, 1\}$. The ring consists of elements of the form $$\sum_{i=0}^N \...
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End Extension models of $I\Delta_0$

Recently I'm thinking about question below, but I can not prove or disprove it. Is it true that for every model $M\models I\Delta_0$ there exists a model $M'\models PA$ such that $M'$ is end ...
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555 views

Why is there a need for ordinal analysis?

Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...
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Infinitesimally small elements in extensions of models of model-complete theories

Suppose that we have a first order language $\mathcal{L}$ that extends the language of rings. Let $T$ a be a topological $\mathcal{L}$-theory of fields in the sense of Pillay.. this means that not ...
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342 views

Is this theory decidable?

It is well-known that both Presburger arithmetic (by contrast with Peano arithmetic) and Tarski geometry are decidable. I was in the shower this morning and wondered whether there exists an elegant ...
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Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ? How can we prove the (...
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What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of $\...
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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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Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...
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Semantic reflection

Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g. let $\ulcorner \varphi \urcorner$ denote the Godel number of $\varphi$. Let $T$ be a first-order arithmetic theory, e....
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How to extend Morley's omitting type theorem to uncountable languages?

In his 1965 paper Omitting Classes of Elements (found in The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley, published by North-Holland Publ. Co., Amsterdam (1965)), ...
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372 views

Extending an infinite simple group

Maybe the question does not fit here. Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For ...
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575 views

Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
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271 views

Can there be computable non-standard models of PA in a weaker sense?

By Tennenbaum's theorem, in the usual sense of computability for models, neither addition nor multiplication can be computable in a non-standard model of PA. Weak version: Can addition or ...
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157 views

From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...
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232 views

Near model completeness

A theory $T$ is called near model complete if every formula is equivalent to a Boolean combination of existential formulas mod $T$. I wonder whether there is an equivalent "semantic" definition of ...
9
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Does non-stablity imply that there is a difference between non-forking and coheir extension

Fix some theory $T$. Let $p$ be a type over some Model M and let $q$ be some global extension of $p$. Note: The number of global coheirs of $p$ is bounded by the number of ultrafilters on $M$. Also ...
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model theory of non-reduced schemes

In model theory one studies Boolean algebras of definable sets of complete theories. For many theories definable sets are in direct correspondence with geometric objects, for example, definable sets ...
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Two cardinal obstructions

Given a theory $T$ and a formula $\phi(x)$ we say that they admit a $(\kappa, \lambda)$ model if there is a model $M$ such that $|M| = \kappa$ and $|\phi(M)| = \lambda$. In all examples that I know ...
3
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Scott Rank of Models of Infinitary Sentences

Let $\mathscr{L}$ be a recursive language. Let $\varphi$ be a $\mathscr{L}_{\omega_1 \omega}$-sentence and $\varphi \in L_{\omega_1^\emptyset}$. (Let $\varphi$ be a computably infinitary formula.) Let ...
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Precise interpretability strength of $\mathcal P_{DF}(\omega)$ and $L_{\omega_1^{CK}}$

I am curious about the relationship between the definable power set of $\omega$ and the $\omega_1^{CK}$th level of the constructible sets $L$. In short, $\omega_1^{CK}$ is the least nonrecursive ...
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Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq \text{Hom}_{\mathcal{...
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240 views

History of unstable formulas [closed]

There are many equivalent definitions for stability, one of them being that being unstable is equivalent to the existence of a formula having the order property. While intuitively it makes sense that ...
3
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258 views

Löwenheim-Skolem for many-sorted theories

Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete ...
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143 views

Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be first-...
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164 views

Do models-and-homomorphisms always form an accessible category?

It's well-known that the category of models of any first-order theory $T$ form an accessible category if we take the elementary embeddings as morphisms. This is true in finitary first-order logic or ...
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94 views

Nice model theoretic properties of a theory after adding predicates

I would like to know what nice model theoretic properties (for example simplicity, NIP, stability, etc) can be preserved when we add a new predicate to the language. Explicitly, if T is an L-theory ...
2
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1answer
226 views

Classification Theory

It is often said that unsuperstable theories do not admit a classification in the sense of Shelah. Why exactly is this so? And also what exactly does in the sense of Shelah mean? It is hand waved in a ...
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Number of non-isomorphic models

I had this question up on Math stackexchange: http://math.stackexchange.com/questions/1349247/number-of-non-isomorphic-models/1350763#1350763 . While it was answered partially there, I'm posting here ...
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Non-Forking and Related Concepts

Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm ...
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Lascar strong types in fragments of arithmetic

Are Lascar strong types (definition below) in models of fragments of arithmetic always type definable? (They trivially are, in models of full induction.) Definition Given a saturated model ${\cal M}$ ...
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231 views

Provability of unprovability

I have three questions (without any real background, this is just something I've been wondering about recently) Can PA prove that PA can't prove nor disprove, say, Goodstein theorem (or any natural ...
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O-Minimal sentences in $L_{\omega_1,\omega}$?

Is there any meaningful sense in which we can talk about o-minimal sentences of $L_{\omega_1,\omega}$? I can give a first attempt, easily; given a countable fragment $F$ and a sentence $\Phi$ in that ...
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Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...
5
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Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory. The facts are stated as follows: Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...
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If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?

Fix a first-order signature $\Sigma$. There is an equivalence relation $\sim$ on the class $\Sigma\mathrm{-Str}$ of all $\Sigma$-structures given by $M \sim N$ iff $M$ and $N$ are elementarily ...
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Is second-order ZFC categorical with regard to its proper class models

Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...
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Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...
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Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic

According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...
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Is there a Leibnizian model with no definable elements, in a finite language?

A first-order structure $M$ is Leibnizian, if any two distinct elements $a,b\in M$ satisfy different $1$-types; that is, if there is some formula $\varphi$ such that $M\models\varphi(a)$ and $M\models\...
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323 views

Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?

I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that ...
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What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?

Suppose that $\phi$ is a formula in the language of set theory such that there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and $\mathrm{Eq}:X^{2}\...
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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 $\...
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When is a formula preserved under taking factors in a reduced product or the stalk in a Boolean product?

I want to know if there is a nice characterization of when a formula is preserved under taking reduced factors. We say that a formula $\phi$ is closed under taking reduced factors if whenever $I$ is ...
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Strongly minimal set with DMP

Recall that a strongly minimal theory $T$ has the Definable Multiplicity Property (DMP) if for all natural $k$, $m$ and $\varphi(\bar{x},\bar{b})$ of rank $k$, multiplicity $m$, there exists a formula ...
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Universal anti-Horn classes?

Is there published work about universal anti-Horn classes? Anti-Horn formulas are also sometimes known as dual Horn. See also related question Is there any research of universal algebras axiomatized ...
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saturated model [closed]

Suppose you have a saturated model N of a complete theory T without finite models. How is it possibile to construct a proper saturated elementary substructure of N of the same cardinality of N ? I ...
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Fixed Points of the Friedman Stanley Jump

Consider the situation of a pair $(X,E)$, where $X$ is a standard Borel space and $E$ is an invariant equivalence relation on $X$*. The Friedman-Stanley jump of this pair is an equivalence relation $...