Minimal axioms necessary to prove the incompleteness theorems?

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What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-verifying theory?



What about the second theorem? Given that this is a stronger result I would expect the axiomatic systems to be different.



EDIT (clarification):
What's the minimal (plausibly) consistent theory needed to prove the First Incompleteness theorem about some theory T, including a sufficient fragment of Robinson arithmetic Q,




axioms incompleteness




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  • See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
    – Mauro ALLEGRANZA
    Jul 30 at 12:11










  • What do you mean with "a self-verifying theory" ?
    – Mauro ALLEGRANZA
    Jul 30 at 12:11






  • 1




    a more interesting question is the minimal axioms for the second incompleteness theorem.
    – Rene Schipperus
    Jul 30 at 12:12










  • I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
    – user519216
    Jul 30 at 12:15







  • 1




    @ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
    – user519216
    Jul 30 at 12:18















up vote
2
down vote

favorite












What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-verifying theory?



What about the second theorem? Given that this is a stronger result I would expect the axiomatic systems to be different.



EDIT (clarification):
What's the minimal (plausibly) consistent theory needed to prove the First Incompleteness theorem about some theory T, including a sufficient fragment of Robinson arithmetic Q,




axioms incompleteness




share|cite|improve this question





















  • See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
    – Mauro ALLEGRANZA
    Jul 30 at 12:11










  • What do you mean with "a self-verifying theory" ?
    – Mauro ALLEGRANZA
    Jul 30 at 12:11






  • 1




    a more interesting question is the minimal axioms for the second incompleteness theorem.
    – Rene Schipperus
    Jul 30 at 12:12










  • I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
    – user519216
    Jul 30 at 12:15







  • 1




    @ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
    – user519216
    Jul 30 at 12:18













up vote
2
down vote

favorite









up vote
2
down vote

favorite











What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-verifying theory?



What about the second theorem? Given that this is a stronger result I would expect the axiomatic systems to be different.



EDIT (clarification):
What's the minimal (plausibly) consistent theory needed to prove the First Incompleteness theorem about some theory T, including a sufficient fragment of Robinson arithmetic Q,




axioms incompleteness




share|cite|improve this question













What's the minimal sufficient (plausibly) consistent system of axioms to prove the First incompleteness theorem? More interestingly can the First incompleteness theorem be proved in a consistent self-verifying theory?



What about the second theorem? Given that this is a stronger result I would expect the axiomatic systems to be different.



EDIT (clarification):
What's the minimal (plausibly) consistent theory needed to prove the First Incompleteness theorem about some theory T, including a sufficient fragment of Robinson arithmetic Q,





axioms incompleteness





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited yesterday
























asked Jul 30 at 11:51









user519216

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164











  • See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
    – Mauro ALLEGRANZA
    Jul 30 at 12:11










  • What do you mean with "a self-verifying theory" ?
    – Mauro ALLEGRANZA
    Jul 30 at 12:11






  • 1




    a more interesting question is the minimal axioms for the second incompleteness theorem.
    – Rene Schipperus
    Jul 30 at 12:12










  • I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
    – user519216
    Jul 30 at 12:15







  • 1




    @ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
    – user519216
    Jul 30 at 12:18

















  • See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
    – Mauro ALLEGRANZA
    Jul 30 at 12:11










  • What do you mean with "a self-verifying theory" ?
    – Mauro ALLEGRANZA
    Jul 30 at 12:11






  • 1




    a more interesting question is the minimal axioms for the second incompleteness theorem.
    – Rene Schipperus
    Jul 30 at 12:12










  • I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
    – user519216
    Jul 30 at 12:15







  • 1




    @ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
    – user519216
    Jul 30 at 12:18
















See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
– Mauro ALLEGRANZA
Jul 30 at 12:11




See Robinson arithmetic as well as Beeson's Lecture 14 : Robinson’s Arithmetic and Church’s theorem.
– Mauro ALLEGRANZA
Jul 30 at 12:11












What do you mean with "a self-verifying theory" ?
– Mauro ALLEGRANZA
Jul 30 at 12:11




What do you mean with "a self-verifying theory" ?
– Mauro ALLEGRANZA
Jul 30 at 12:11




1




1




a more interesting question is the minimal axioms for the second incompleteness theorem.
– Rene Schipperus
Jul 30 at 12:12




a more interesting question is the minimal axioms for the second incompleteness theorem.
– Rene Schipperus
Jul 30 at 12:12












I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
– user519216
Jul 30 at 12:15





I mean self-verifying theory as a theory to which incompleteness theorems don't apply, i.e. can prove its own consistency.
– user519216
Jul 30 at 12:15





1




1




@ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
– user519216
Jul 30 at 12:18





@ReneSchipperus indeed that's an interesting question, but I am interested in the first as afaik it's a weaker result, hmm thinking about it I am interested in results on both.
– user519216
Jul 30 at 12:18
















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