Demystifier said:
The standard Godel sentence can be expressed in plain english as:
"This sentence cannot be proved within a certain closed system of axioms".
This sentence is true and can be proved, but only if one steps out from this closed system of axioms.
It seems to me that my sentences in the posts above are not like this sentence. Instead, they seem to be more like inconsistent sentences such as:
"This sentence is not true."
If I am right, then this seems to resolve the paradox.
I think you should use Penrose instead of Godel, because he's alive, and because he seems to think that Godelian arguments can be used to prove that human mathematical abilities are beyond those of machines.
Let's use "unassailably true", because I think that's the phrase Penrose uses in "The Emperor's New Mind". Something is unassailably true if it is true, and it's possible for a competent mathematician to convince himself that it is true, and that there is no doubt possible. For example: "2+2=4". Then define the Penrose sentence to be:
"Penrose will never accept this sentence to be unassailably true"
I do not agree that this is an antinomy such as the Liar paradox---"This sentence is not true"---because it is possible for it to be true, and it's also possible for it to be false (while the Liar paradox cannot consistently be either). It can be true, and yet Penrose never accepts it (there are true statements that he never accepts as unassailably true). Or it can be false (there might be false statements that he mistakenly believes are unassailable truths).
I guess what you could say about it is that Penrose would never accept it to be unassailably true, since it is an obvious mistake to do so. But suppose we add some "closure" properties to his set of unassailable truths: Suppose we assume the following:
If Penrose considers statements S_1, S_2, ... to all be unassailable, and C follows logically from those statements, then he will eventually come to believe C to be unassailably true, as well. If his unassailable beliefs don't have this property, then they might be logically inconsistent, but still paraconsistent---A paraconsistent theory is one that is still useful even if it's logically inconsistent; the inconsistency doesn't make the whole theory useless. I actually think that humans are paraconsistent, but not consistent.
If we assume that Penrose' unassailable beliefs are logically closed, then he might not realize immediately that other unassailable beliefs of his logically imply the Penrose sentence.