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Godel -> Gödel; TeX fixes
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LSpice
LSpice
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Generalize the GodelGödel sentence requires a fixed point theorem

I am trying to generalize the GodelGödel sentence as follows. Define a pair of sentence $A$ and $B$ such that: $$ A := \lnot Prov(\hat B) $$ $$ B := Prov(\hat A) $$\begin{gather*} A := \lnot \operatorname{Prov}(\hat B) \\ B := \operatorname{Prov}(\hat A) \end{gather*} where $\hat A$ and $\hat B$ are the GodelGödel numbers of $A$ and $B$ respectively.

From the definition above I derive the following equation $\hat A$ = The Godel number of $ (\lnot Prov(\hat Prove(\hat A)) $$\hat A = \text{the Gödel number of $\lnot \operatorname{Prov}(\widehat{\operatorname{Prov}(\hat A)}) $}$. My question is for a minimal system that the incompleteness theorems apply, whether this equation always has a fixed point?

Generalize the Godel sentence requires a fixed point theorem

I am trying to generalize the Godel sentence as follows. Define a pair of sentence $A$ and $B$ such that: $$ A := \lnot Prov(\hat B) $$ $$ B := Prov(\hat A) $$ where $\hat A$ and $\hat B$ are the Godel numbers of $A$ and $B$ respectively.

From the definition above I derive the following equation $\hat A$ = The Godel number of $ (\lnot Prov(\hat Prove(\hat A)) $. My question is for a minimal system that the incompleteness theorems apply, whether this equation always has a fixed point?

Generalize the Gödel sentence requires a fixed point theorem

I am trying to generalize the Gödel sentence as follows. Define a pair of sentence $A$ and $B$ such that: \begin{gather*} A := \lnot \operatorname{Prov}(\hat B) \\ B := \operatorname{Prov}(\hat A) \end{gather*} where $\hat A$ and $\hat B$ are the Gödel numbers of $A$ and $B$ respectively.

From the definition above I derive the following equation $\hat A = \text{the Gödel number of $\lnot \operatorname{Prov}(\widehat{\operatorname{Prov}(\hat A)}) $}$. My question is for a minimal system that the incompleteness theorems apply, whether this equation always has a fixed point?

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Thuc Hoang
Thuc Hoang
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Generalize the Godel sentence requires a fixed point theorem

I am trying to generalize the Godel sentence as follows. Define a pair of sentence $A$ and $B$ such that: $$ A := \lnot Prov(\hat B) $$ $$ B := Prov(\hat A) $$ where $\hat A$ and $\hat B$ are the Godel numbers of $A$ and $B$ respectively.

From the definition above I derive the following equation $\hat A$ = The Godel number of $ (\lnot Prov(\hat Prove(\hat A)) $. My question is for a minimal system that the incompleteness theorems apply, whether this equation always has a fixed point?