1
vote
0answers
87 views

Do we know that if $\pi$ is normal then there is a proof of it?

We do not know whether $\pi$ is normal or it is not and many other weaker statements, e.g. (*) $\pi$ contains infinitely many $0$s. Inspired by the Godel's incompleteness theorem that there are some ...
15
votes
1answer
273 views

Could it be that Goldbach conjecture is undecidable?

The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach ...
1
vote
1answer
169 views

Is Douglas Hofstadter's version of Godel's proof utter nonsense?

Is Douglas Hofstadter's version of Godel's proof, which he offers in his book Godel, Escher, Bach, utter nonsense? Hofstadter goes to great length to disguise the fact that there are two distinct ...
2
votes
1answer
42 views

Completeness of a Sub Theory of Number Theory

When I was an undergraduate I had a philosophy professor tell me that if you took the axioms of number theory with only addition or only multiplication then the resulting theory was decidable. It is ...
2
votes
1answer
189 views

A qualitative, yet precise statement of Godel's incompleteness theorem?

I read online a statement to the effect that (I'm paraphrasing): Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers. I am ...
9
votes
1answer
276 views

Freeman Dyson's example of an unprovable truth

Freeman Dyson has claimed that $$\nexists m,n \operatorname{Reversed}(2^n) = m ^ 5 $$ (where $\operatorname{Reversed}(l)$ just is the reverse of the digits of $l$ in base 10), is probably an ...
5
votes
1answer
165 views

Is there any recursive definition, using only addition, of the set of values of $x^2+y^2$?

There is a recursive definition of the set of squares which uses only addition: $CS(x,y) := IS(x) \wedge IS(y) \wedge x \lt y \wedge \forall z: (x \lt z) \wedge (z \lt y)⇒\neg IS(z)$ $IS(x)⇔ x=0 ...
12
votes
1answer
425 views

What's the theory in which incompleteness of PA is proved?

Maybe this is a dumb question, but I have to admit that it is not really clear to me what the theory is, in which incompleteness of PA and stronger theories is proved. The texts I did study so far are ...