Articles of model theory

Morley rank (with an unusual definition)

For a definable set $X \subseteq \mathbb{U}^n$, let us denote $\text{RM}(X)$ the Morley Rank $\text{RM}(\varphi(\bar{x}))$, with $\varphi(\bar{x})$ the formula defining $X$. Show that, for $X \subseteq \mathbb{U}^n$ and $Y \subseteq \mathbb{U}^m$, we have $\text{RM}(X \times Y)= \text{RM}(X) + \text{RM}(Y)$. I know this should be some kind of standard result. My problem is we defined $\text{RM}$ […]

Gluing together mathematical structures, how?

By structure, I mean that which is defined here: What I’m looking for is a way of gluing together structures so that each structure used is embedded within the whole glued-together object. (Each–meaning not just “most”) I don’t need this embedding to be elementary; just something that “preserves” function, relation, and constant symbols. I […]

ACF universal is the theory of integral domains

When studying David Marker’s “Model Theory: An Introduction” book trying to understand the proof of Lemma 3.2.1 which says: $ACF_{\forall}$ is the theory of integral domains, I couldn’t understand the last line of the proof which comes as followa: (I copy the whole proof) Proof. The axioms for integral domains are universal consequences of $ACF$. […]

Formally proving the consistency of a formal theory

This is based on a related question. I did some further reading to understand the problem. I’m just posting this question to check whether I have correctly understood the solution. Let’s say we are using ZFC as our meta-language (in standard first order logic). We have proven that a theory is consistent iff it has […]

Is for every ultrahomogenous structure M the theory Th(M) model complete?

A structure M is ultrahomogenous if every isomorphism between finitely generated substructures of M can be extended to an automorphism of M. A theory is model complete if every embedding between models of the theory is elementary. I can’t think of any ultrahomogenous structure, which doesn’t have a model complete theory. Is there such a […]

Why is completeness theorem true?

Who can teach me completeness theorem? Thanks! Recommending a book is also welcome. More specifically, it says that if a statement is true in all models of a theory, then it has a proof from this theory.

What is finite in a finite model

I am studying some theorems of model theory in an introductory text of mathematical logic. I know that a model is a way of associating the relationary symbols of a signature $\Sigma$ to $k$-ary relations ($k\ge 0$) and the $k$-ary functional symbols ($k>0$) and constants (which are 0-ary functions) respectively to $k$-ary functions ($k>0$) and […]

Prove $\langle \mathbb{N}, x \mapsto x +1, 1 \rangle$ is a Peano system without circular reasoning

How can we show that the structure $\langle \mathbb{N}, x \mapsto x +1, 1 \rangle$ is a model of a Peano system? How can we show that it satisfies the axiom of induction? Don’t we have to implicitly rely on this axiom, and thereby make our reasoning circular? And if we do, then what has […]

embdedding standard models of PA into nonstandard models

Maybe it’s well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the axiom schema of induction for each formula. I’m stuck at the step how to show $f$ is well defined. We can […]

A finitely axiomatizable consistent second-order theory without a model

The completeness theorem fails for second-order logic. This question has some nice examples of consistent second-order theories without models. But non of them is finitely axiomatizable, at least those examples use infinitely many axioms. Are there consistent finitely axiomatizable second-order theories without models, or is it possible to prove a completeness theorem for these theories?