# Tagged Questions

Questions about Gödel's incompleteness theorems and related topics.

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### Why a system becomes incomplete once it's capable of doing arithmetic?

For a Formal axiomatic system to obey Godel's incompleteness theorems, It has to be powerful enough to incorporate Peano Axioms. Why It does not apply to say, Presburger arithmetic or the axioms of ...
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### show the set of valid second-order $\emptyset$-sentences is not R-enumerable

show the set of valid second-order $\emptyset$-sentences is not R-enumerable this would have the empty symbol set i.e. $S = \emptyset$ so it would be sentences that are universally or existentially ...
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### diagonalization about incompleteness

In the Enderton's logic book(page 186-187), he writes 'diagonalization approach' for proving undefinability of $\#Th\mathfrak{N}=\{\#\tau |\vDash_{\mathfrak{N}}\tau \}$ where $\#\tau$ is the godel ...
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### Why is Gödel's Second Incompleteness Theorem important?

Given that the consistency of a system can be proven outside of the given formal system, Gödel says, It must be noted that proposition XI... represents no contradiction to the formalities ...
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