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The incompleteness theorem is true for sufficient complex systems. Is it known if there must be at least one such sufficient complex system which is consistent or could it be that every such system is inconsistent?

Thank you very much

Kevin Meier
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  • Arithmetic is consistent; see Gentzen's consistency proof. – Mauro ALLEGRANZA Dec 23 '16 at 13:48
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    @MauroALLEGRANZA Well, Gentzen didn't prove that arithmetic is absolutely consistent; he reduced the problem to consistency of another formal system. – lisyarus Dec 23 '16 at 13:51
  • Arithmetic has a model : the natural numbers; thus, it is consistent. – Mauro ALLEGRANZA Dec 23 '16 at 13:52
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    @MauroALLEGRANZA Doesn't this, again, assume consistency of the theory where the model lives (say, ZFC)? – lisyarus Dec 23 '16 at 13:54
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    See http://math.stackexchange.com/questions/446084/does-some-proof-of-arithmetics-consistency-exist – mfl Dec 23 '16 at 14:06
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    Gentzen's proof relies on transfinite induction. Goedel's second theorem does not refute this proof because it does not rely on a system containing PA, nevertheless, we cannot be sure whether transfinite inudction is consistent. Nice : PA is consistent if and only if every Goodstein sequence terminates. – Peter Sep 04 '19 at 09:21

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