# Tagged Questions

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 ...

95 views

### Show that if $L$ is countable and contains a two-place predicate symbol, there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$

We say that a class of structures $K$ is closed under elementary equivalence ($\equiv$) if for all $A, B$, if $A \in K$ and $A \equiv B$, then $B \in K$. How to show that if $L$ (as a set of specific ...
84 views

122 views

285 views

69 views

76 views

### Is first-order logic complete with respect to countable structures?

Question. Let $\Gamma$ denote a first-order signature, and consider a sentence $\tau$ and a set of sentences $\Sigma$ in the language generated by $\Gamma$. If every countable $\Gamma$-structure that ...
75 views

82 views

### Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
171 views

### Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
117 views

### Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
724 views

### What is the real meaning of Hilbert's axiom of completeness

According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) ...
95 views

514 views

### There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
237 views

59 views

### Does having a filter which is not maximal implies the negation of Łoś theorem?

If we have a family $(\mathfrak{M}_i)_{i\in I}$ of $L$-structures, and a filter $\mathcal{F}$ over $I$, we can define the reduced product $\prod_{i\in I}\mathfrak{M }_i/\mathcal{F}$. If $\mathcal{F}$ ...
140 views

### Are surjective polynomial maps injective?

An injective polynomial map $p:\mathbb{C}^n\mapsto\mathbb{C}^n$ is surjective (Ax-Grothendieck theorem). What is known about the reverse implication (surjective implies injective)? Why does the model-...
122 views

### Showing $2$ is not definable in $(\mathbb{Q},+)$.

As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$. I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication ...
183 views

### Goedel's completeness theorem in and/or for intuitionistic first order logic

Warning: I am neither a logician nor a set theorist, just curious about foundations of classical and intuitionistic mathematics. Therefore it might well be that the things to come are plain wrong, and ...
49 views

### Prove that Th(N, +, .) has uncountably many 1-types over some finite set

not sure how to go about answering the above question. Thanks for your help! (alternative: Prove that Th(N, +, .) has uncountably many n-types over the empty set.)
318 views

### Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
108 views

### Showing there does not exist a formal proof of a formula $\phi$.

My problem:Suppose $R$ is a binary predicate and use the soundness theorem to show that there does not exist a formal proof of $$\phi =\forall y\exists xR(x,y)\rightarrow \exists x\forall yR(x,y).$$ ...
315 views

### Finite Model Theory

It seems that finite model theory is regarded (in a sense) as a computer theoretic subject. Is this the case or are there questions of interest that are of interest to mathematical logicians or more ...
106 views

### “Syntactic models” and the proof of the Completeness Theorem

In Computational Complexity by Papadimitriou (page 107), he outlines the basic idea for a proof of the completeness theorem for first-order logic - namely, that given a consistent set $\Delta$ of ...
5k views

### How is the Gödel's Completeness Theorem not a tautology?

As a physicist trying to understand the foundations of modern mathematics (in particular Model Theory) $-$ I have a hard time coping with the border between syntax and semantics. I believe a lot would ...