| Edwin
Chan |
Godel's second incompleteness theorem implies that consistency of an axiomatic system T can only be proven from within another axiomatic system A. Although inconsistency can certainly be proven by a single counterexample from within T. My question is: is it possible to prove inconsistency also by appeal to another axiomatic system B? Is it possible that system T is provable to be both consistent and non-consistent depending on which axiomatic system A or B is adopted? Thanks. |
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| Dan
Goodman |
Hello again - great question. This raises some interesting points. Does it follow that just because some axiomatic system A has proved T to be consistent that T really is consistent? Nope. Does it follow from B proving T inconsistent that there is an inconsistency in T? Quite different questions oddy. If A proves T consistent, but T isn't consistent then A is itself inconsistent. Because if A can prove T consistent and T is inconsistent (i.e. you can prove p and not-p in T) then you can prove in A that T is inconsistent (just translate the proof in T of p and not-p into A). If B proves T inconsistent and T is consistent, then it only follows that B is what is called w-inconsistent. A system is w-inconsistent if for some statement P(n) you can prove that P(1) is false, P(2) is false, P(3) is false and so on, but you can also prove that there is some n such that P(n) is true. This isn't an inconsistency because although you can prove that each particular P(n) is false, you can't prove that EVERY P(n) is false, even though they are. And if that isn't weird, nothing is. |
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| Edwin
Chan |
Thanks Dan, that is very enlightening (at least to me!) It would appear we can take little comfort in proofs of consistency/inconsistency via another axiomatic system because they are predicated on one's confidence in the consistency of the proving system itself. Thus it seems that our confidence in the consistency of our axiomatic systems is to a large extent founded on the fact that in the theories that stem from them, no contradiction has been found ... yet. This reminds me of the manner in which we become confident about scientific laws, theories etc to the extent that we can say they are "proven". Quite ironic isn't it? - considering we are talking about mathematics |
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| Dan
Goodman |
Yes, it is a bit like science in that respect. Some mathematical philosophers have said that maths is "quasi-empirical". There are also movements in mathematical philosophy that try and deal with these problems in an alternative way (they were really worrying less about consistency and more about truth, but it's still relevant). For example, the "intuitionists" refused to allow proof by contradiction, the "logicists" tried to deduce all of mathematics from logic alone. Neither attempts were entirely satisfactory, and neither really avoids the problem that you still don't know if there might be an inconsistency even in these weaker or simpler axiomatic systems. |
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| Edwin
Chan |
Thanks Dan, really appreciate you taking the time. Regards Edwin |