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- How are existential quantifiers present in the internal logic of regular categories? March 28, 2024Intuitively speaking, how do existential quantifiers appear? I'm just starting to get familiar with these definitions. Top and conjunctions appear because of finite products. (Plus, I assume, something that makes them work nicely with the existential quantifiers.) But my understanding of existential quantifiers in categorical terms is by way of them being left adjoint to […]Julián
- Introducing function symbols in (first-order) set theory March 28, 2024Suppose we want to formalise some parts of mathematics within set theory, which is itself formalised in a first-order language (in the standard way). Logic is assumed to be classical. In particular, this means that – at the level of semantics – every term must have a denotation, which in turn means that every function […]wiktoria
- Question regarding the completeness theorem and ZFC March 28, 2024In order to prove the completeness theorem we obviously need a framework such as ZFC (I'm aware that ZFC isn't the only possibility) so that we can talk about a language $\mathcal{L}$ and also about models of $\mathcal{L}$. Now the completeness theorem makes perfect sense to me in so far as the language which we […]Gergő Kelemen
- How to translate this statement into $A \land B$? March 28, 2024A friend of mine showed me an SAT question today in which one of the choices is: No genetic variations that were common to those finches that used technique Q were not common to the finches that did not use technique Q. I just learned a bit about logic so I tried breaking down the […]ten_to_tenth
- Referencing a statement with quantifiers in two separate lines March 27, 2024I want to show that a statement with several quantifiers, e.g., "$f(a, b)Ruth
- What is the matter in defining the necessity operator internally $\Box: \Omega \to \Omega$? March 27, 2024I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the distributive under implication $(\Box(\phi \to \psi)\to (\Box \phi \to \Box \psi))$. The reason why is I would like to study categories in […]Miviska
- Topological properties vs homeomorphisms March 27, 2024I'm studying general topology and a question has come to my mind. We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it. I can see the ambiguity […]Amanda Wealth
- What is the technical difference between $\longleftrightarrow$ and $\iff$? [duplicate] March 27, 2024I understand that they can both be used to mean "if and only if" or similarly link two statements together, but in the general sense, what is the difference between the two symbols ($\longleftrightarrow$ and $\iff$)? Thank you!MWQOJYNWQA
- Rearranging propositional equations March 27, 2024so I wanted to ask if there is anything that allows for this case: I have $3$ statements $A, B$, and $C$ and I wish to change them from the form: $(A \operatorname{xor} B)$ $(A \operatorname{xor} C)$ Into $A \operatorname{xor} B \operatorname{xor} C$ Is this possible and how does one do this? Preferably with references […]Lawrence Page
- A countable inductive first-order theory which has non existentially-closed (e.c) model has a non e.c. model of any inifinite size March 27, 2024I'm trying to prove Lindström's theorem, which says that a countable inductive first order theory $T$ which is $\kappa$-categorical for some $\kappa\geq \aleph_{0}$ is model-complete. The strategy I follow is proving each model has an existentially-closed extension of any bigger cardinality, which I proved using transfinite induction. Now, I try to prove an analogous result […]Oria
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