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That is, can a Turing machine take a formal system, S, as its input and decide if S is Turing complete?

I think this is an undecidable problem, am I right?

If it is undecidable, why can we (as humans) decide Turing completeness?

Jack McKay Fletcher
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    We (as humans) can also provide proofs for algorithms halting. You need to realise that "Can solve in some cases does not imply that you can solve in ALL cases". I could also write a program that would be able to decide the halting problem for some subset of cases, but what it means to be undecidable is that there does not exist a general algorithm that works for arbitrary algorithms. – Cruncher Mar 27 '14 at 16:36

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Hmm :-) deciding turing completeness is not central to determining whether or not human brain is TM complete; one can go through mechanical steps to determine turing completeness; that is not an issue.

The key issue is, whether or not human brain is hyper-computational[1] or hyper-turing.

One test would be to have an answer of "Yes" to one of the following questions: can a human being predict when a turing machine will halt? (i.e. solve the halting problem) Or is a human brain not subject to Rice's Theorem.

Trivially the answer to both question seems to be No in the general case because one can imagine a TM with infinitely long tape and just jumping around, we never can tell when it will hit a cell that tells it to stop.

The seeming hypercomputing capability comes from the fact that we mistake normal computers/software/mechanical processes etc with TMs.

Rice's Theorem can be side-stepped in the "special case" of systems that exhibit Markov Property and has finite number of Place-Transition net representation. Our general environment has these "special cases" in abundance so it may seem as if human brain is capable of Hyper-computing because it tends to jump to general conclusions from special cases, however it probably is not since we as human beings have yet to experience an interaction with a Turing Machine.

Ahmed Masud
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