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**1 - 3**of**3**### How Subtle is GĂ¶del's Theorem - More on Roger Penrose

"... the form (8n)Q(n) where Q(n) is a computable property of the natural number n and refers to such sentences as "statements of the form [Q]". 2 Again Penrose emphasizes that "we can directly apply the Godelian insight to any algorithm that purports to be generating (say [Q]-type) mathematical truths ..."

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the form (8n)Q(n) where Q(n) is a computable property of the natural number n and refers to such sentences as "statements of the form [Q]". 2 Again Penrose emphasizes that "we can directly apply the Godelian insight to any algorithm that purports to be generating (say [Q]-type) mathematical truths." Thus, for any such algorithm F, Godel's theorem provides us with a corresponding [Q]-type sentence G(F) such that if F is sound (in the sense that all the [Q]-type sentences it generates are true) then G(F) is true although it is not one of the sentences generated by F. Penrose goes on to say: The original algorithm is thus limited in what it is able to achieve: It is unable

### Arithmetic and the Incompleteness Theorems

, 2000

"... this paper please consult me first, via my home page. ..."

### Incomputability after Alan Turing

"... The year 2012 marks the 100th anniversary of the birth of Alan Turing. The following two articles were inspired by the work ..."

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The year 2012 marks the 100th anniversary of the birth of Alan Turing. The following two articles were inspired by the work