Ohhh, thanks. That explains it. I feel like there should exist things for which provable(not(p)), but I can’t think of any offhand, so that’ll do for now.
Ebthgidr
Karma: 28
To answer the below: I’m not saying that provable(X or notX) implies provable (not X). I’m saying...I’ll just put it in lemma form(P(x) means provable(x):
If P( if x then Q) AND P(if not x then Q)
Then P(not x or Q) and P(x or Q): by rules of if then
Then P( (X and not X) or Q): by rules of distribution
Then P(Q): Rules of or statements
So my proof structure is as follows: Prove that both Provable(P) and not Provable(P) imply provable(P). Then, by the above lemma, Provable(P). I don’t need to prove Provable(not(Provable(P))), that’s not required by the lemma. All I need to prove is that the logical operations that lead from Not(provable(P))) to Provable(P)) are truth and provability preserving
is x or not x provable? Then use my proof structure again.
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:
P or not-P is true in PA
Also, because of that, if p → q and not(p)-> q then q—use rules of distribution over and/or
So:
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?
Well, there is, unless i misunderstand what meta level provable(not(provable(consistency))) is on.
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.
I’ll rephrase it this way:
For all C: Either provable(C) or not(provable(C)) If provable(C), then provable(C) If not provable(C), then use the above logic to prove provable C. Therefore all C are provable.
Wait. Not(provable(consistency)) is provable in PA? Then run that through the above.
Ok, thanks for clearing that up.
That’s an interesting correlation, but I’m curious about the causal link: is it that a certain type of neural architecture causes both predisposition to rationality and asperger’s, or the social awkwardness added on to the neural architecture creates the predisposition—i.e. I’m curious to see how much being social affects rationality. I shall need to look into this more closely.
I forget the formal name for the theorem, but isn’t (if X then Y) iff (not-x or Y) provable in PA? Because I was pretty sure that’s a fundamental theorem in first order logic. Your solution is the one that looked best, but it still feels wrong. Here’s why: Say P is provable. Then not-P is provably false. Then not(provable(not-P)) is provable. Not being able to prove not(provable(x)) means nothing is provable.
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
Not only that—the greater degree of neuroplasticity that I think 16-year olds still have(if I’m wrong about this, someone please correct me) makes it a good deal easier to learn skills/ingrain rationality techniques.
Nice to meet you—it’s rather reassuring to see another member at my age.
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
My suspicion about the thin-tailed risk here is that either congress or the SEC passes landmark regulation about SPACs (which is potentially plausible) and those stocks go to 0, very quickly, as the initial investors who IPOed the SPAC pull their money out. See, ICOs (though those were obviously higher risk)