# Tagged Questions

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### Possible independence of a generic sentence?

Let the generic sentence $P :$ "$\exists z \in Z \text{ such that }p(z) \text{ is true }$". In addition $Z$ is recursively enumerable, and for a given $z_0$ in $Z$, "$p(z_0)$ or $\lnot p(z_0)$" is ...
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### Are there finitely many interesting theorems?

I'm not a logician, I read, a long time ago, about Gödel etc... A theorem is a provable proposition under a system of axiom. Let's take the usual system of ZFC. Of course, the notion interesting is ...
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### Non-Standard Arithmetic - order

Recently I try to figure out some facts about one specific way of "constructing" a non-standard model for (peano) arithmetic. I guess there are answer to my question already out there, but somehow I ...
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### Every Countable Model of PA is Recursive?

I am interested in any obvious flaws in the following argument. Assume we have a countable model of Peano arithmetic in a meta-theory like ZFC. Assume this model has a set of ordered triplets, ...
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### Reformulation of Theories

Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) ...
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### Not Skolem's Paradox - Part 3

This is a follow up to a previous question: Not Skolem's Paradox - Part 2. Assume we have a countable, non-standard model of Peano Arithmetic in ZFC. This ZFC model must include a set of ordered ...
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### Why is the Ehrenfeucht theory complete?

I am looking at the theory T of Dense linear orders without endpoints, extended with the set $\{c_i<c_j|i\in\omega\}$ and am asked to prove that this theory is complete. I know that it has three ...
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### Can A Decidable Theory Have Nonrecursive Models?

Tennenbaum's theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA. ...
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### Curve in $(\mathbb{R},<)$ going to infinity

My question is the following: Given the structure $(\mathbb{R},<)$ and $t \in \mathbb{R}$, can I have a definable function $f$ over a finite set of parameters, with domain $(-\infty, t)$ and with ...
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### Does model theory extend to partial functions?

I have been reading a bit about effect algebras and d-posets recently, sets $M$ on which you have a single partial binary operation (partial here meaning partially defined, i.e. the domain of this ...
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### Questions on a sentence in ZFC asserting that ZFC has a model

Question 1. Let $\operatorname{con}(\mathsf{ZFC})$ be a sentence in $\mathsf{ZFC}$ asserting that $\mathsf{ZFC}$ has a model. Let $S$ be the theory $\mathsf{ZFC}+\operatorname{con}(\mathsf{ZFC})$. ...
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### Prove that the set of $\mathcal{L}$-terms has size $\max\{\aleph_0, \#\text{ of constant sym plus the$\#$function sym}\}$.

Let $\mathcal{L}$ be a first-order language. Prove that the set of $\mathcal{L}$-terms has size $\max\{\aleph_0,\text{ the number of constant symbols plus the number of function symbols}\}$. I know ...
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### Prove that $\Phi_{eq}$ has continuum many closed complete extensions.

Full question: Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation and let $\Phi_{eq}$ be the axioms for an equivalence relation. Prove that $\Phi_{eq}$ has continuum many closed complete ...
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### Embedding models of ZF into another model

I had some ideas regarding models of ZF. My ideas (phrased as questions) are: Given two models of ZF, what are the condition for a model containing both models (in the sense of embedding) to exist? ...
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### Do metatheoretic results carry between mutually interpretable theories?

If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., ...
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### Prove the elementary equivalence of the two models

There are two models $\mathfrak A$ and $\mathfrak B$ in class $K$. $\mathfrak A = <P(\omega), \subseteq>$ $\mathfrak B = <P(\omega), \supseteq>$ Is the $Th(K)$ of a full theory of ...
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### truth of a sentence to the linearly ordered set

Let Φ - sentence of signature σ = <≤> such that for any infinite linearly ordered set A satisfies A ⊨ F. Prove that there exists n ∈ N such that for every linearly ordered set B power greater than ...
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### Set of formulas in Model Theory

I'm reading the book Model Theory by Chang and Keisler and there is one thing that always bugs me. Very frequently we have something like $\Sigma(x)$ representing the set of all formulas in a language ...
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### Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding to this ...
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### 2-type not-realised in Q

my question is the following: given the additive group of rational numbers, i.e. $Q = \langle {\mathbb Q},+,0\rangle$ and $T$ the theory of $Q$, how can I find (explicitly) a 2-type which is not ...
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### Finitely many countable models implies decidability

Suppose $T$ is decidably axiomatizable first order theory and has no finite model. We shall focus on countable models. If $T$ has just one countable model (up to isomorphism), which means $T$ is ...
I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean ...
### Showing a Theory $T$ is Substructure Complete
Let $T$ be a (complete and consistent) theory. Suppose $T$ exhibits the following two properties: (1) model-completeness: if $\mathcal{M} \models T$ and $\mathcal{A} \subseteq \mathcal{M}$ s.t. ...