So then here's a smaller lemma: for all x and all q:

If(not(x))

Then provable(if x then q): by definition of if-then

So replace x by Provable(P) and q by p.

Where's the flaw?

Oh, that's what I've been failing to get across.

I'm not saying if not(p) then (if provable(p) then q). I'm saying if not provable(p) then (if provable(p) then q)

So the statement (if not(p) then (if p then q)) is not provable in PA? Doesn't it follow immediately from the definition of if-then in PA?

That doesn't actually answer my original question--I'll try writing out the full proof.

Premises:

1. P or not-P is true in PA

2. Also, because of that, if p -> q and not(p)-> q then q--use rules of distribution over and/or

So: 1. provable(P) or not(provable(P)) by premise 1

2: If provable(P), provable(P) by: switch if p then p to not p or p, premise 1

3: if not(provable(P)) Then provable( if provable(P) then P): since if p then q=not p or q and not(not(p))=p

4: therefore, if not(provable(P)) then provable(P): 3 and Lob's theorem

5: Therefore Provable(P): By premise 2, line 2, and line 4.

Where's the flaw? Is it between lines 3 and 4?

I made a plot of the entropy and the (correct) energy. Every feature of these plots should make sense.

Note that the exponential turn-on in E(T) is a common feature to any gapped material. Semiconductors do this too :)

Well, there is, unless i misunderstand what meta level provable(not(provable(consistency))) is on.

so temperature is in the mind

I am not quite sure in which way this statement is useful.

"..and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing." -- Douglas Adams

Your reasons were that not(provable(c)) isn't provable in PA, right? If so, then I will rebut thusly: the setup in my comment immediately above(I.e. either provable(c) or not provable(c)) gets rid of that.

[Spoiler alert: I can't find any 'spoiler' mode for comments, so I'm just going to give the answers here, after a break, so collapse the comment if you don't want to see that]

.

.

.

.

.

.

.

.

.

.

For the entropy (in natural units), I get

and for the energy, I get

Is this right? (upon reflection and upon consulting graphs, it seems right to me, but I don't trust my intuition for statistical mechanics)

I downloaded the paper you linked to and will read it shortly. I'm totally sympathetic to the "didn't want to make a long comment longer" excuse, having felt that way many times myself.

I agree in the single-world case, I wouldn't want to do it. That's not because I care about the single world without me per se (as in caring for the people in the world), but because I care about myself who would not exist with ~1 probability. In a multiverse, I still exist with ~1 probability. You can argue that I can't know for sure that I live in a multiverse, which is one of the reasons I'm still alive in your world (the main reason being it's not practical for me right now, and I'm not really confident enough to bother researching and setting something like that up.) However, you also don't know that anything you do is safe, by which I mean things like driving, walking outside, etc. (I'd say those things are far more rational in a multiverse, anyway, but even people who believe in single world still do these things.)

Another reason I don't have a problem with discontinuity is that the whole problem seems only to arise when you have an infinite number of worlds, and I just don't feel like that argument is convincing.

I don't think you need infinite knowledge to know whether x=0 or x>0, especially if you give some probability to higher level multiverses. You don't need to know for sure that x>0 (as you can't know anyway), but you can have 99.9% confidence that x>0 rather easily, conditional on MWI being true. As I explained, that is enough to take risks.

If I wake up after, in my case that I laid out, that would mean that I won, as I specified I would be killed while asleep. I could even specify that the entire lotto picking,noise generation, and checking is done while I sleep, so I don't have to worry about it. That said, I don't think the question of my subjective expectation of no longer existing is well-defined, because I don't have a subjective experience if I no longer exist. If am cloned, then told one of me is going to be vaporized without any further notice, and it happens fast enough not to have them feel anything, then my subjective expectation is 100% to survive. That's different from the torture case you mentioned above, where I expect to survive, and have subjective experiences. I think we do have some more fundamental disagreement about anthropics, which I don't want to argue over until I hash out my viewpoint more. (Incidentally, it seemed to me that Eliezer agrees with me at least partly, from what he writes in http://lesswrong.com/lw/14h/the_hero_with_a_thousand_chances/:

"What would happen if the Dust won?" asked the hero. "Would the whole world be destroyed in a single breath?"

Aerhien's brow quirked ever so slightly. "No," she said serenely. Then, because the question was strange enough to demand a longer answer: "The Dust expands slowly, using territory before destroying it; it enslaves people to its service, before slaying them. The Dust is patient in its will to destruction."

The hero flinched, then bowed his head. "I suppose that was too much to hope for; there wasn't really any reason to hope, except hope... it's not required by the logic of the situation, alas..."

I interpreted that as saying that you can only rely on the anthropic principle (and super quantum psychic powers), if you die without pain.)

I'm actually planning to write a post about Big Worlds, anthropics, and some other topics, but I've got other things and am continuously putting it off. Eventually. I'd ideally like to finish some anthropics books and papers, including Bostrom's, first.