# Tagged Questions

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### Why is the proof of Gödel's first incompleteness theorem no contradiction?

I consider the following version of Gödel's first incompleteness theorem: Assume $F$ is a formalized system which contains Robinson arithmetic $Q$. Then a sentence $G_F$ of the language of $F$ can ...
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### Is it really impossible to lose the Hydra game?

In a number of blog posts I found the claim that the game described below cannot be lost, which is to say, every possible strategy is a winning strategy. In each case, a sketch proof is given that ...
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### Inequality problem $\frac 1x + \frac 1y + \frac 1z > 5$ prove

Prove that: $$\frac 1x + \frac 1y + \frac 1z > 5$$ where $x+y+z=1$, $x, y, z$ are real numbers not equal $0$ and $x\neq y \neq z$
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### What is a simple example of an unprovable statement?

Most of the systems mathematicians are interested in are consistent, which means, by Gödel's incompleteness theorems, that there must be unprovable statements. I've seen a simple natural language ...
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### Theorems of GL in modal logic

So I've been reading George Boolos' "The Logic of Provability" and he's explaining different systems of modal logic. He's taken as his basic symbols → (implication), □ (necessity), ⊥ (falsehood), a ...
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### Existence(?) of a set whose cardinality cannot be determined in ZFC

(First, I apologize if I display any fundamental misunderstanding of how set theory works.) I had a question regarding the limitations of ZFC (assuming its consistency, of course.) Is there any ...
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### How can Goodstein's theorem be expressed in PA

I understand Goodstein's Theorem and its proof. I'm trying to understand the proof of why Goodstein's Theorem cannot be proved in PA. However, it's not immediately clear to me that Goodstein's Theorem ...
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### True but unprovable?

I would like to ask a question about Gödel's Incompleteness Theorems which I've had in the back of my head for some time. Since I'm a student working in a completely different area of maths (my usual ...
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### Can unprovability unprovable? Is there an $\omega$-fold unprovability?

I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system. We know ...
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### how to prove : there are an infinite number of points on the circle

I think the follow problem is equal to the problem set 1.16.(a) in Principles of Mathematical Analysis (walter ruldin), And we take (a, b) in $R^2$, X in $R^i$ how to prove : there are an infinite ...
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### What are techniques for proving undecidability or unprovability of a sentence?

I asked a question the other day on how to form logical equivalence between a sentence $\phi$ and two other sentences $\psi$ and $\chi$, such that neither $\psi$ nor $\chi$ were on their own as '...
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### Is Riemann Hypothesis provable?

Loosely speaking, there are three kinds of propositions. Those propositions which are true and can be proved to be true. Those propositions which are false and which can be proved to be false....
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### can it be proven that something is “difficult” (prime factoring for example)

I understand that the current state of the art suggests that factoring into primes is a difficult problem. I also understand that a large part of public key cryptography seems to be based on that ...
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### How to prove $\sum\limits_{k=0}^n\ (3k^2+2k+1) = n^3 + 5 \begin{pmatrix}n+1\\2\end{pmatrix}+1$

$\sum\limits_{k=0}^n\ (3k^2+2k+1) = n^3 + 5 \begin{pmatrix}n+1\\2\end{pmatrix}+1$ How would you go on proving this equation? Doesn't have to be induction..
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### How to prove an inequality

$a$, $b$, $c$, $d$ are rational numbers and all $> 0$. $\max \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\} \geq \dfrac{a+c}{b+d}\geq \min \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\}$ Hope someone ...
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### What other unprovable theorems are there? [duplicate]

Gödel's incompleteness theorem presents us with the possibility of having theorems that are neither provable nor disprovable in a given axiomatic set. Already we have the continuum hypothesis which ...
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### On provability within minimal logic

In its most naive form my question boils down to this: when is a proposition that is provable "by contradiction" also provable "directly"? IOW, is it possible to know, a priori, that a ...
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### Proving that there exists something.

When asked to "Prove that there exists such x that y" , is giving such "x" enough as a solution or do you need to find like a general formula or something? For example, if asked to "prove that there ...