# Tagged Questions

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the ...

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### How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$...
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### complete, finitely axiomatizable, theory with 3 countable models

Does it exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models? A few relevant comments: There is a classical example of a complete theory ...
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### Spanning the reals with a small set - choicelessly

Working in ZF (so, no choice): is it possible that there is a set of reals $X$ such that $\vert X\vert<\mathbb{R}$, but $X$ generates $\mathbb{R}$ as a subgroup under addition? This seems ...
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### Non-axiomatisability and ultraproducts

Let $T$ be a first-order theory over a language $L$, and let $\mathcal{M}$ be a subclass of the class of models of $T$. As I understand it, if there is no theory $\hat{T}$ over $L$ whose class of ...
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### References on filter quantifiers

This post is primarily a reference request. In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is ...
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### Model theory in terms of type spaces/Lindenbaum algebras

Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
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### Gödel's Incompleteness Theorem in “Gödel, Escher, Bach”

Ok, so I'm reading the chapter on Gödel's Incompleteness Theorem in "Gödel, Escher, Bach" and I want to make sure I'm getting this right: the idea of the book's proof is to form the sentence "There ...
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### Type-definable Forcing or forcing in a non-first order setting

Roughly speaking, in set forcing the forcing notion is a set from ground model's perspective and in class forcing its a definable subset of the ground model given by solutions of some formula with ...
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### Functorial first order theories interpretation

Question will be a bit naive, so please, be kind. Consider first order theories, $\Gamma, \Gamma'$ . Let $\mathcal{M}$ be the category of models for $\Gamma$ and $\mathcal{M}'$ be the category of ...
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### which branch of maths studies Standard Logical Matrices

In classical logic you have truthtables like: & | T | F ---|---|--- T | T | F F | F | F In many valued logic you have tables like: (this one is ...
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### Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
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### Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ...
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### Is there a schsim between classical logic and categorical logic?

I've been trying to learn a little bit more about the foundations of mathematics, and it has strike me that there seems to be two competing points of view about what the foundations should be. While ...
### The monadic second order theory with $<$ and Presburger arithmetic
Consider the monadic second order logic over the natural numbers with $<$ as a predicate, i.e. the second order logic over $(\mathbb N, 1, <)$, where we can quantify over sets and individual ...