# 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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I'm not sure if this is a paradox or a nonsense or neither of both. Anyway this is the "problem" if we can call it like that: A: B is True B: A is False How can ...
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### Applications of ultrafilters

I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters? I'm ...
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### When we say, “ZFC can found most of mathematics,” what do we really mean?

ZFC works as a foundation because it can prove many sentences that are "translations" of theorems from "standard" mathematics into the language of ZFC. But there's a subtlety. When we say, "ZFC can ...
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### Can multiplication be defined in terms of divisibility?

Peano Arithmetic has two axioms which use the multiplication symbol: ∀x:x*0=0 and ∀x:∀y:x*Sy=x+x*y. The 2-term relation "x divides y" can be expressed as D(x,y) := ∃z:z*x=y. Multiplication is a ...
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### Do you need the Axiom of Choice to accept Cantor's Diagonal Proof?

Math people: It is my understanding that set theorists/logicians compare cardinalities of sets using injections rather than surjections. Wikipedia defines countable sets in terms of injections. ...
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### Odd proof method

I read on the wikipedia article for the Riemann Hypothesis that some theorems have been proved by assuming the hypothesis to be true and then false and proving the certain theorem from both cases. I.e....
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### How does one show a set of axioms is independent? (of each other?)

I am being asked to show that the groups axioms of existence of an identity, of inverses and of associativity are independent. Does this mean that none of two of them imply the third? That is: do I ...
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### Set of All Groups

In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist. Are there any group ...
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### Is there a “tree-like” proof of compactness theorem in the case of uncountably many variables?

I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, ...
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### Do you need the Axiom of Choice to assert that every real vector space has a norm?

Math people: This question is 95% answered (the first answer) at Does every $\mathbb{R},\mathbb{C}$ vector space have a norm? and Vector Spaces and AC . The questions, answers, and links found there ...
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### Primitive recursive function which isn't $\Delta_0$

What is the simplest/cutest example (and/or example with the most student-friendly proof that it is an example) of a primitive recursive function which isn't representable by a $\Delta_0$ wff?
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### What is an example of a Transfinite Argument?

I was in a discussion today with a philosopher about the merit of the technique of "proof by contradiction." He mentioned the Law of Excluded Middle, wherein we (typically as mathematicians) assume ...
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### Are sets constructed using only ZF measurable using ZFC?

Suppose $S$ is a subset of $\mathbb{R}$ which can be defined without using the axiom of choice, i.e. which can be proved to exist using only the axioms of ZF. Does it follow that $S$ is measurable? ...
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### In axiomatization of propositional logic, why can uniform substitution be applied only to axioms?

I'm reading an introductory book about mathematical logic for Computation (just for reference, the book is "Lógica para Computação", by Corrêa da Silva, Finger & Melo), and would like to ask a ...
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### What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
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### Is quantum logic producing interesting/different mathematics?

Is quantum logic producing interesting/different mathematics? Is it different from the intuitionist approach to mathematics? How?
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### What is the most influential work of Grothendieck in mathematics?

Recently Alexander Grothendieck has passed away but his mathematical wave is still alive and passes its growth ages. It is hard to describe the influence of such a great man in mathematics just in few ...
I remember once hearing offhandedly that in set builder notation, there was a difference between using a colon versus a vertical line, e.g. $\{x: x \in A\}$ as opposed to $\{x\mid x \in A\}$. I've ...