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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 the elements of a domain $D$.

I thought that a model if finite when the set of the $k$-ary relations and $k$-ary functions ($k\ge0$), including the elements of $D$, is finite, but I have just read a theorem which is shaking my convinctions: Löwenheim-Skolem-Tarski theorem says that if a theory has a model of a given infinite cardinality then its has models of any greater cardinality. If a theory is in the form $\{P_1,P_2,P_3,\ldots\}$ where $P_i,i\in\mathbb{N}$ are 0-ary relational symbols, I would say that it has a countable models where the $P_i$ are propositions, but I do not see how the model could be made uncountable.

- Upper and Lower bounds on proof length
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- Big Bang Theory Reference to Formal Logic
- Justification of ZFC without using Con(ZFC)?
- Fraïssé limits and groups
- Undecidable conjectures

What am I misunderstanding? I thank you very much for any clarification…

- Undecidable conjectures
- What is the Tarski–Grothendieck set theory about?
- Why is recursion theory suffering from terminological bloat?
- Proof by Contradiction, Circular Reasoning?
- How to find the logical formula for a given truth table?
- Concrete example for diagonal lemma
- Prove that formula is or is not a tautology
- What's an Isomorphism?
- What is exactly the difference between $\forall x \neg P(x)$ and $\neg \forall xP(x)$?
- Composition of 2 involutions

The cardinality of the model always refers to the cardinality of its universe.

Models, or to be precise structures of a language can be seen as a pair $\langle M,I\rangle$, where $M$ is a non-empty set (whose cardinality is “the cardinality of the model”, so a finite model means that $M$ is finite) and $I$ is an interpretation function, taking symbols from the language and returning their interpretation as elements (constants) or functions or $k$-ary relations on the set $M$ according to each symbol’s designation.

When we say model, we often have a specific list of sentences in the language, also called a theory sometimes, that is assumed to be true in the structure. So a finite model for a theory $T$ means that $M$ is a finite set, and that $T$ is a list of sentences which are true in the structure. Saying that the model is countably infinite means that $M$ is countably infinite, and so on.

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