# Tagged Questions

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### 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 ...
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### Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
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### particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
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### Does second-order arithmetic (Z2) prove soundness and uniform reflection for first-order arithmetic (PA)?

(1) Does full second-order arithmetic (Z2) prove soundness and uniform reflection schemas for first-order arithmetic (PA)? That is, do we have for all formulas $\phi$:  \underset \phi \forall \; ...
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### Non-self-referential undecidable sentences in arithmetic

Are there any known undecidable sentences for PA are neither "self-referential" (like a sentence equivalent to its own nonprovability) nor imply consistency of PA (like in the Paris Harrington ...
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### Have I understood the whole Incompleteness business correctly?

I am reading Gödel-Escher-Bach and a good dialogue by Eliezer Yudkowsky and I think I might have understood the nature of the Completeness and Incompleteness theorems (at least regarding Peano ...
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### Are the Godel's incompleteness theorems valid for both classical and intuitionistic logic?

I am studying an undergraduate text about math logic. The proofs of the two Godel's incompleteness theorems are not completely formal: they are admittedly simpler that the real proofs. For what I ...
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### How is the standard model of number theory specified, and why can't we use that specification to prove any number theoretical sentence of interest?

According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, ...
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### Sequent calculus and first incompletness theorem

Wikipedia says that sequent calculus is sound and complete (http://en.wikipedia.org/wiki/Sequent_calculus#Properties_of_the_system_LK). Godel first incompleteness theorem says that system capable of ...
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### Is the negation of the Gödel sentence always unprovable too?

The incompleteness theorem says that certain theories+deduction system contain at least one sentence (the Gödel sentence "$G$"), which can't be proven (in the system in which it holds). (i) Is ...