Paradox,shortest proof

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I have read somewhere that the shortest proof of a certain formula in the language of natural numbers contains some kind of paradox. I cannot remember what this paradox was nor where I've read it. It may well be something like that
it is formally provable that even shorter proof exists.
Any help please?




number-theory predicate-logic proof-theory paradoxes




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  • 2




    That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
    – alexod
    Jul 19 at 14:48










  • That's it. Do you please have a non-wikipedia, pdf like detailed reference?
    – user122424
    Jul 19 at 14:49











  • JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
    – Mauro ALLEGRANZA
    Jul 19 at 14:56










  • Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
    – user122424
    Jul 19 at 15:50










  • But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
    – Mauro ALLEGRANZA
    Jul 19 at 19:07














up vote
2
down vote

favorite












I have read somewhere that the shortest proof of a certain formula in the language of natural numbers contains some kind of paradox. I cannot remember what this paradox was nor where I've read it. It may well be something like that
it is formally provable that even shorter proof exists.
Any help please?




number-theory predicate-logic proof-theory paradoxes




share|cite|improve this question















  • 2




    That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
    – alexod
    Jul 19 at 14:48










  • That's it. Do you please have a non-wikipedia, pdf like detailed reference?
    – user122424
    Jul 19 at 14:49











  • JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
    – Mauro ALLEGRANZA
    Jul 19 at 14:56










  • Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
    – user122424
    Jul 19 at 15:50










  • But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
    – Mauro ALLEGRANZA
    Jul 19 at 19:07












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I have read somewhere that the shortest proof of a certain formula in the language of natural numbers contains some kind of paradox. I cannot remember what this paradox was nor where I've read it. It may well be something like that
it is formally provable that even shorter proof exists.
Any help please?




number-theory predicate-logic proof-theory paradoxes




share|cite|improve this question











I have read somewhere that the shortest proof of a certain formula in the language of natural numbers contains some kind of paradox. I cannot remember what this paradox was nor where I've read it. It may well be something like that
it is formally provable that even shorter proof exists.
Any help please?





number-theory predicate-logic proof-theory paradoxes





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 19 at 14:44









user122424

9521614




9521614







  • 2




    That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
    – alexod
    Jul 19 at 14:48










  • That's it. Do you please have a non-wikipedia, pdf like detailed reference?
    – user122424
    Jul 19 at 14:49











  • JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
    – Mauro ALLEGRANZA
    Jul 19 at 14:56










  • Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
    – user122424
    Jul 19 at 15:50










  • But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
    – Mauro ALLEGRANZA
    Jul 19 at 19:07












  • 2




    That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
    – alexod
    Jul 19 at 14:48










  • That's it. Do you please have a non-wikipedia, pdf like detailed reference?
    – user122424
    Jul 19 at 14:49











  • JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
    – Mauro ALLEGRANZA
    Jul 19 at 14:56










  • Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
    – user122424
    Jul 19 at 15:50










  • But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
    – Mauro ALLEGRANZA
    Jul 19 at 19:07







2




2




That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
– alexod
Jul 19 at 14:48




That is Rosser's trick an improved version of Gödel's theorem. See en.wikipedia.org/wiki/Rosser%27s_trick
– alexod
Jul 19 at 14:48












That's it. Do you please have a non-wikipedia, pdf like detailed reference?
– user122424
Jul 19 at 14:49





That's it. Do you please have a non-wikipedia, pdf like detailed reference?
– user122424
Jul 19 at 14:49













JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
– Mauro ALLEGRANZA
Jul 19 at 14:56




JBarkley Rosser, Extensions of Some Theorems of Gödel and Church, JSL (1936)
– Mauro ALLEGRANZA
Jul 19 at 14:56












Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
– user122424
Jul 19 at 15:50




Yes. And is there an even modern treatment of this paradox ? Either in a book or paper.
– user122424
Jul 19 at 15:50












But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
– Mauro ALLEGRANZA
Jul 19 at 19:07




But it is not a paradox: it is an improved proof of Gödel's incompleteness theorems theorem.
– Mauro ALLEGRANZA
Jul 19 at 19:07















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