I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...):
Construct a Turing machine A which sequentially runs on all proofs in ZFC and checks if the claim "A does not halt or ZFC is inconsistent" is proved. If such a proof is found, A halts.
Note that A can know its encoding - see Kleene's recursion theorem - and even if it can't this "obstacle" can be overcome. So it seems to me such A is constructible.
Now, if A halts then it is because it has found a proof for "A does not halt or ZFC is inconsistent". Since A halted, it is implied that ZFC is inconsistent. So either A does not halt, or ZFC is inconsistent.
Now (and my guess is that this is the part where this proof "cheats") we have just proved the claim "A does not halt or ZFC is inconsistent", so such a proof exists in ZFC. So A must halt on this proof - and we have seen that this implies that ZFC is inconsistent.
Where is the mistake?