One thing the questioner seems to want to know is in what kinds of theories the first incompleteness theorem can itself be proved. As Peter says, the proof of the theorem is fine given informally, as almost all actual mathematical results are. But still, one might want to know: Where is this theorem provable? The answer is: Not in Q, but in fairly weak theories. It can certainly be proven in PA---full Peano arithmetic---though that hardly counts as a "weak" theory. I am fairly certain, though, that it can be proven in the theory known as "I-Sigma-1"---though I'd have to check that before betting my life on it. Part of the reason I'm sure about this, though, is that I-Sigma-1 is susceptible to the second incompleteness theorem. What you get in such cases is (e.g.) that PA |- Con(PA) --> G PA ; but then, if PA |- Con(PA), then PA |- G PA , and PA is inconsistent, by the first incompleteness theorem, and this works for any theory T that extends Q and is capable of proving: Con(T) --> G T . ...

Better: There's a fairly simple proof that this is false. Just consider the theory consisting of the sentence letters p1, p2, .... This theory is clearly consistent. It'd be an amusing question just how easy the proof is, i.e., exactly what sorts of theories are needed for the proof.

I take it that by "bivalence", you mean the principle that every proposition is either true or false. And if we take that principle in unrestricted form---we really do mean every proposition---then, well, it's hard to see how it could fail to imply that the proposition expressed by "There will be a riot in London on 13 January 2076" is either true or false. If you don't like that conclusion, then you have to abandon bivalence---or, perhaps, the claim that the sentence in question expresses a proposition, though that seems rather worse. But note that you do not have to abandon bivalence, so to speak, across the board. You might still think that every mathematical proposition is either true or false, or that every proposition about the past is either true or false, or.... Perhaps there is something special about the future here. As you probably know, Michael Dummett argued that one way to understand debates over "realism" takes them to turn upon our attitude towards bivalence regarding...

I've generally become pretty good at detecting fallacious reasoning, both my own and that of others. I'm not perfect, to be sure, but pretty good. So, overall, it seems that, in any given case, my chances of having committed an undetected fallacy are fairly small. How small depends upon the complexity of the reasoning. If I'm trying to prove a complicated theorem, I may have to go over the proof repeatedly, spell out all the details, etc, before I can convince myself that the proof is correct. So maybe in those cases, at least initially, the chances I'd committed a fallacy are higher than they would be otherwise. But that seems all right. It just means that, to be justifiably sure that I've not committed a fallacy in any given case, I may have to do different things: a lot in some cases, maybe nothing in others. But if I have done those things and come to the belief that my reasoning is correct, then I see no particular reason the belief should not count as knowledge. Of course, one can push...

A dialethist would also make this claim. If p is a true contracdition, then it is both true and false. Hence it's negation is both true and false, but it's negation is a tautology.

Recent epistemology has made a lot of the distinction between justification and something else that goes by the name warrant or entitlement , though some philosophers use "warrant" as an umbrella notion that covers both justification and entitlement. And of course the distinction gets drawn in different ways. But the basic idea is that being justified in some sense involves being able to appreciate that justification, which will, in central cases, take the form of an argument: If my belief that p is justified, then what justifies it must be something I know or at least could know. In that sense, justification, as philosophers in the tradition I'm exploring use the term, is more an "internalist" notion. An agent's being entitled to a belief, on the other hand, need not involve h'er appreciating the nature of that entitlement: I could be entitled to a belief that p even though I have no idea what it is that entitles me to that belief. And so entitlement is more of an "externalist"...

The idea that there is a hierarchy of statements, each saying something about the level below, but none of the lower ones saying anything about the higher ones, is central to formal work on truth. It originates where such work originates, with Alfred Tarski's great paper "The Concept of Truth in Formalized Languages". In Tarski's work, it doesn't take quite the form you mention. You suggestion regarding (3), in particular, sounds more like the form the hierarchy takes in Saul Kripke's treatment in "Outline of a Theory of Truth". On such treatments, the hierarchy certainly does not end with (3): There are also statements about statements about statements about statements, and so forth; and the hierarchy does not even end with all the finite levels.

No, I don't see any mistake---other than that you dismiss the problem simply on the ground that there is self-reference. Self-reference isn't always a problem. In fact, some times it's essential. Consider this phrase: (*) yields a sentence when preceded by its own quotation This is a perfectly sensible verb phrase, and some phrases yield a sentence when preceded by their own quotations---e.g., "is a sentence" does so---whereas some others---e.g., "Bill is"---do not. As it happens, (*) too does so. That is, (**) "yields a sentence when preceded by its own quotation" yields a sentence when preceded by its own quotation There's self-reference there, too, but what (**) says is true, and I'm not sure how else you'd propose to report that truth. Or should we just not report it? or try very hard not to think about it? or what? Now, among the phrases that do yield a sentence when preceded by their own quotation, some yield true sentences---e.g., "contains three words" does---and some do...

Unfortunately, I think the only concrete thing I can suggest is that you look for other texts on critical reasoning. (You don't say which two you have, and I'm not familiar with many, so I'm afraid I can't give more specific advice.)

In the example you give, it seems obvious that there is another interest people have of which you are not taking account, namely, the desire to be "the first on the block" to have the new toy. I suppose one could question whether it is rational to have such an interest, but lots of people clearly do have it, rich or otherwise.