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כותרת: Re:
Ok… here´s something fishy about this:We assume function Halt does indeed exist, and set out to prove this results in a contradiction. What we know about the function is that it receives another function and its input as Halt´s input, and outputs whether or not the function halts. But, if this function exists, *it will always halt*, since it must return a result.
Now, let Halt(0) be the Halt iteration called from within Trouble and Halt(1) be the Halt testing Trouble.
What our Trouble function does is essentially reverse the output of the Halt function called from within Trouble. What Turing then claims is that if we feed the Trouble function back into Halt, we will get a wrong result in both cases. Let´s examine the two cases:
1) The initial function (F) is indeed infinite (won´t halt): F is fed into Halt(0), which returns "no", because it is infinite. Trouble in turn returns "yes", and hereby halts. Halt(1) returns "yes", because Trouble has halted. Turing claims this is wrong, because F hasn´t halted and therefore Halt is wrong. Here I disagree. Halt(1) was meant to test *Trouble*, *not* F. Halt(0) is finite by our assumption, and so Trouble is also finite, and Halt(1) is right. No contradiction!
2) The initial function (F) is finite. Halt(0) returns "yes", and so Trouble becomes infinite, and Halt(1) returns "no". Again, Halt(1) was meant to test Trouble, not F, and is therefore right. No contradiction again.
So, what am I missing here?
Oren
