# 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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### What actually constitutes a *definition* for a function?

I'm reading Enderton's text on logic trying to justify ( to myself ) the use of induction on recursively defined sets. This is of course used repeatedly in trying to prove results about well-formed ...
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### Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
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### Checking understanding on proving uniqueness of identity and inverse elements of a group.

Sorry for such a trivial question, but just wanted to check my understanding. When proving a statement, for example, that the inverse of a group element is unique (in elementary group theory) one ...
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### Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
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### How much set theory is necessary for serious logic?

I'm currently studying logic at my university and I have been trying to squeeze in as much set theory on the side as possible. Considering that I am spending quite a lot of time studying set theory ...
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### Formulating a recursive definition

Let Σ(k) = 1 + 3 + 5 + ... + (2k+1) be the sum of all odd natural numbers from 1 up to and including (2k+1). Formulate a recursive definition for Σ including both the base case Σ(0) = 1 and a (k+1)th ...
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### Manifolds of Non-Standard Dimension

Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many ...
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### Blue Eyes: A Logic Puzzle, has a puzzling solution (a.k.a. What does common knowledge have to do with it?)

In Blue eyes: a logic puzzle (specifically, the follow up questions), the most common answer is that it needs to be common knowledge that someone has blue eyes for all the blue-eyed people to leave. ...
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### How to prove the following using direct proof

$[(\sim p \vee q) \wedge p ] \Rightarrow q$ What should be done next in order to apply direct proof to the example above? The following process has been already done but seemingly it's incorrect: ...
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### FO-axiomatizable class?

I came across this question while preparing for my logic exam. Can this class be (finitely) axiomatizable, where the class contains all structures $\mathfrak{A} = (A, <, f)$, and for no $a \in A$ ...
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### Questions regarding Universal Quantifiers

The question is to show that: $$\exists x:(P(x) \implies Q(x))\qquad\equiv\qquad\forall x:P(x) \implies \exists x:Q(x)$$ First I use double negation to get to the universal quantifier since it ...
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### Proving $\vdash \exists x (x=c)$ for each term $c$.

I wish to prove that $\vdash \exists x (x=c)$ for each term $c$. It seems quite obvious that this would be the case, for $c$ is such an $x$, but creating a formal proof of this is escaping me. ...
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### Understanding logical form of “Nobody in the calculus class is smarter than everybody in the discrete math class”

I'm self studying How to Prove book and have been working out the following problem in which I have to analyze it to logical form: Nobody in the calculus class ...
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### Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$. The proof is easy enough ...
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### Analyzing Logical Forms involving quantifiers

I have been solving the following problem from How to Prove book: Analyze the logical forms of the following statement: Everyone likes Mary, except Mary ...
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### What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $x^n=1$ $where$ $n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $x^0=1$ Induction Step: Let this holds for all ...
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### How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying ...
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### How and why can a true statement *never* imply something false?

The premise of 'proof by contradiction' is that a true statement can never imply a false statement. In my lectures (intro to logic), this has been brushed aside as 'obvious', but is there a formal ...
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### Are there any “obviously” true propositions in number theory?

After all efforts spent on wrong proofs of famous number theory conjectures and theorems like Goldbach's or Fermat's last theorem, could one find some simple statements (might be correct ones) whose ...