[This occasionally comes up, so I thought I'd explain it, but it's not particularly important.]
Much woo has been spilled over Godel's incompleteness theorem. Playing a bit fast and loose, it is this:
Any sufficiently advanced formal system of reasoning must be either inconsistent, in that it can prove mutually contradictory theorems, or incomplete, in that some true statements in its domain are inaccessible to it. (Or both)
Thus a perfect system of axioms from which all true statements in mathematics flows, and no false statements, is impossible. Oh well. Life goes on.
The woo comes in when people try to apply this to the human brain. A typical claim is that the human brain is not limited in this way, because it is not a formal system. This is false:
The limit is not some particular limitation of a particular kind of thing. "Formal system" in this context means any system of reasoning that can be lawfully realized at all. The axioms need not be written on paper; they can be implied in the construction of a flesh and spirit mind. If it both reasons and exists, it is a formal system as far as we are concerned here. Anything that isn't an imaginary oracle construct made up to cheat in a thought experiment is subject to this limitation. We can stretch our minds to "imagine" a reasoning system without this limit, but Godel's theorem is just that no such system can exist in any lawful reality.
Though I don't believe in it, and I admit I can't even properly understand it, I think cognitive nonmaterialism doesn't get out of this either. If the mind is lawful, that is, obeys the basic consistency rules of mathematics and of any possible universe, it must follow these rules. The mind is useless if it is not lawful. The whole point of its existence is to perceive the laws of the universe, and follow them to a desired outcome, for which it must be essentially lawful. Thus the rules. We will see that this is not a real limitation in practice:
That out of the way, why does it seem that the human mind, as a reasoning system, is so much more powerful than our toy paper and silicon reasoners? There are two interesting properties of the human mind that might seem to get around Godel's theorem, but don't actually:
In mathematics, we are concerned mostly with consistency, so all the usual formal systems we use are rigidly consistent, and thus incomplete. The human brain takes the opposite side of Godel's fork; it is not consistent. We can and do believe many silly things that are inconsistent with our experience. This ability to consider absurdities on insufficient evidence is what allows us to think the things we need to think quickly and efficiently, even if we are sometimes wrong. Thus, if we seem more "complete" than our mathematical systems, it is only because we are inconsistent and error-prone. In practice, the "true but unproveable" set is mostly self-referential proof theory trivia. Taking the "inconsistent" side of the fork is done more for efficiency than Godel compliance.
With inconsistent formal systems, we can typically devise simple attacks that twist them up into knots and get them to prove absurdities and thus demonstrate their inconsistency. It is very hard to do this with a human because he can see what you are trying to do and outsmart simple logic attacks. Being not bound by simple consistency, even if you can get someone to separately agree to
~P, they can realize what you're up to and pull the plug on your logical chain as soon as you try to then derive the absurdity
Q. Thus we don't have many simple and robust examples of inputs that can tie the human brain in knots and easily demonstrate the inconsistency in high-level beliefs. That said, scam sales, absurd foreign-installed beliefs, "red pill" argument sequences, and various illusions definitely exist.
So the interesting property of human judgement is not that it evades Godel. It rarely if ever even comes close to proof-theoretic limits. Despite the hype, Godel basically doesn't matter unless you are doing weird provability stuff. The interesting property of the human brain is that ability to outsmart the world around it; that it is a material and in-time approximation of time-looping and teleological future-driving; that is, that it is intelligent. Intelligence is really interesting, but it does not escape or violate proof theory, information theory, computability theory, or any other mathematical necessities.