You're right that (if X then Y) is just fancy notation for (not(X) or Y). However, I think you're mixing up levels of where things are being proved. For the purposes of the rest of this comment, I'll use provable(X) to mean that PA or whatever proves X, and not that we can prove X. Now, suppose provable(P). Then provable(not(not(P))) is derivable in PA. You then claim that not(provable(not(P))) follows in PA, that is to say, that provable(not(Q)) -> not(provable(Q)). However, this is precisely the statement that PA is consistent, which is not provable in PA. Therefore, even though we can go on to prove not(provable(not(P))), PA can't, so that last step doesn't work.

You're right that (if X then Y) is just fancy notation for (not(X) or Y). However, I think you're mixing up levels of where things are being proved. For the purposes of the rest of this comment, I'll use provable(X) to mean that PA or whatever proves X, and not that we can prove X. Now, suppose provable(P). Then provable(not(not(P))) is derivable in PA. You then claim that not(provable(not(P))) follows in PA, that is to say, that provable(not(Q)) -> not(provable(Q)). However, this is precisely the statement that PA is consistent, which is not provable in PA. Therefore, even though we can go on to prove not(provable(not(P))), PA can't, so that last step doesn't work.

As a fellow 16-year-old (there really seem to be a lot of us popping up around here recently), I concur. With that said, rationality skills are difficult for anyone to learn, because the human brain did not evolve to be rational, but rather to succeed socially. I would add that a good deal of rationality potential is ingrained in those who find themselves attracted to LW at a young age, particularly since surveys have shown that LW users tend to have a higher incidence rate of Asperger Syndrome, the symptoms of which include social awkwardness. This suggests to me that rational thinking comes more easily to people with certain personality types, which is arguably genetic. As a single data point, I suppose I'll add that I myself have been diagnosed with Asperger's when I was younger, although with how trigger-happy American doctors are with their diagnoses these days, that's not really saying much.

I'm not sure how you're getting from not provable(X) to provable(provable(X) -> X), and I think you might be mixing meta levels. If you could prove not provable(X), then I think you could prove (provable(X) ->X), which then gives you provable(X). Perhaps the solution is that you can never prove not provable(X)? I'm not sure about this though.

A question about Lob's theorem: assume not provable(X). Then, by rules of If-then statements, if provable(X) then X is provable But then, by Lob's theorem, provable(X), which is a contradiction. What am I missing here?

I finished up to the first major plot twist/divergence in the rationalfic(well, sort of. I'll just call it an attempted rationalfic) I've been working on for 3 months or so, and it's now in the top 15 most followed fics in the fandom(Danganronpa). Link: light in despair's darkness

Welcome! I'm also 16. Welcome to the group of people who answer "no" to the "were you alive 20 years ago" question on a technicality. It's really great to know about risk assessment errors and whatnot when we're still teenagers, just because the bugs in our brains are even more dangerous when ignored than normal.

Welcome, Leor! I'm also a 16 year old new member.

Hello. I'm Leor Fishman, and also go by 'avret' on both reddit and ffn. I am currently 16. The path I took to get here isn't as...dramatic as some of the others I've seen, but I may as well record it: For as long as I can remember, I've been logically minded, preferring to base hypotheses on evidence than to rest them on blind faiths. However, for the majority of my life, that instinct was unguided and more often than not led to rationalizations rather than belief-updating. A few years back, I discovered MoR during a stumbleupon binge. I took to it like a fish to water, finishing up to the update point in a matter of days before hungrily rereading to attempt to catch whatever plot points I could glean from hints and asides in earlier chapters. However, I still read it almost purely for story-enjoyment, noting the rationality techniques as interesting asides if I noticed them.
About a year later, I followed the link on the MoR website to LW, and began reading the sequences. They were...well, transformative doesn't quite fit. Perhaps massively map-modifying might be a better term. How to Actually Change Your Mind specifically gave me the techniques I needed to update on rather many beliefs, and still does. Both Reductionism and the QM sequence, while not quite as revolutionary as HtACYM for me, explained what I had previously understood of science in a way that just...well, fit seems to be the only word that works to describe it, though it doesn't fully carry the connotation I'm trying to express. Now, I'm endeavoring to learn what I can. I'm rereading the sequences, trying to internalize the techniques I'll need and make them reflexive, and attempting to apply them as often as possible. I've gone pretty far--looking back at things I said and thought before makes that clear. On the other hand, I've still got one heck of a ways to go. Tsuyoku Naritai

Is this a possible explanation or corollary to the sunk-costs fallacy of economics?