XPost: alt.philosophy, comp.theory   
   From: janburse@fastmail.fm   
      
   Hi,   
      
   Actually the BB(5) does also construct machines,   
   and does also look at the code of machines.   
      
   It has an amazing history, since the candidate   
   for the busiest beaver was already found in 1989:   
      
   47,176,870	4098	current BB(5), step champion   
   https://turbotm.de/~heiner/BB/mabu90.html   
      
   They use an amazing simple technique to speed up   
   their search. Realizing macro turing machines, that   
   encode what happens with k cells on a tape.   
      
   Plus heuristics to "prove" that a TM does not halt,   
   which seem to be sufficient for 5 state TMs. Plus   
   heuristics to bring the number of considered 5 state   
      
   TMs down, since without reduction they would be   
   26*10^12 many, but they needed only consider 5*10^7   
   many. So that after about ten days using a   
      
   33 MHz Clipper CPU they got their result.   
      
   Bye   
      
   P.S.: My estimate, with todays laptop can do   
   it in 2.5 hours, or maybe in 2.5 minutes if using   
   an AI accelerator. Not 100% sure. Wasn't even   
      
   thinking about such a modern replica of the   
   problem. Coq used Rust. We could use even something   
   else that would tap in AI accelerators, maybe   
      
   even JavaScript and run it in a browser.   
      
   > Hi,   
   >   
   > Well then get an education. Every Gödel   
   > sentence G, has a size, doesn't it?   
   > The formal analogue of the Liar Paradox,   
   >   
   > except it’s expressed arithmetically:   
   >   
   > G ≡ ∀y¬Proof(y,┌G┐).   
   >   
   > Gödel did explicitly construct a Gödel   
   > sentence G in his 1931 paper. He did not   
   > claim it was astronomically large,   
   >   
   > nor impossible to write. Now you can do   
   > the encoded Liar also with Turing Machines TM:   
   >   
   > 1. Fix a formal proof system S (e.g. PA) and   
   > an effective enumeration of all proofs.   
   > 2. Build a TM M(x) that, given a code x, searches   
   > for an S-proof of the formula with code a; if it finds   
   > M(x) halts <=> exists y Proof(y,x) (i.e. Prov(x)).   
   >   
   > Etc.. etc..   
   >   
   > Bye   
   >   
   > dart200 schrieb:   
   >> this shit makes me feel like i'm stuck in a mad house planet   
   >>   
   >> undecidability has nothing to do with computational complexity and the fact   
   we think the limit to decidability is bounded by how well we can bit pack a   
   self-referential turing machine into a proof is just literal nonsense   
      
   Julio Di Egidio schrieb:   
   > TL;DR: There is no such thing as an irrational reasoner,   
   > i.e. not any more than there exists a married bachelor.   
   >   
   > [ Original subject: "daily puzzle: the rational reasoner".   
   >    Salvaged from the Google Groups archive.]   
   >   
   > On November 19 2023, Rich D wrote:   
   >  > On November 15 2023, Jeff Barnett wrote:   
   >  > >> A rational person believes a finite number of propositions;   
   >  > >> that is, he believes all of them they are true. (if he thought   
   >  > >> any one was false, he'd disbelieve it)   
   >  > >> A rational person also disbelieves in his own perfection.   
   >  > >> He expects to be wrong occasionally.   
   >  > >> This implies that one of the list of the propositions   
   >  > >> referenced above, must be false. And he's aware of this   
   >  > >> implication. Which means he believes he believes   
   >  > >> something false.   
   >  > >> Is this inconsistent? Is he rational? Explain.   
   >   
   > Yes, it/he is: to begin with because incorrect is not   
   > the same as incongruous ("inconsistent", though that   
   > is more of a mathematical term).  Moreover, we indeed   
   > do hypothetical thinking, which means thinking with the   
   > (meta-)knowledge that not (and not ever) all (domain)   
   > knowledge is accurate and available.  And so much more,   
   > we do...   
   >   
   >  > > Rational does not imply perfection in thought.   
   >  >   
   >  > I would not define rational as equivalent to perfection.   
   >  > In order to discuss the concept, one must first define the concept.   
   >  >   
   >  > > You seem, above, to float a definition of a rational person then   
   >  > > move on to ask a question given your definition.   
   >  >   
   >  > Define rational person: he attempts to avoid contradiction,   
   >  > he doesn't knowingly accept any contradiction. He utilizes   
   >  > the precepts of first order logic.   
   >   
   > No, he doesn't (or, shouldn't) use "FOL". To begin with,   
   > do not conflate Logic proper (valid reasoning) with formal   
   > and/or mathematical logic, not to even mention the unbounded   
   > complexity of the real world.  At best Logic proper goes with   
   > symbolic logic, whatever that even means...   
   >   
   >  > He attempts to recognize facts and reality,   
   >  > assuming his perceptions of reality are accurate.   
   >   
   > Again, a rational thinker would never assume full   
   > and fully accurate knowledge.   
   >   
   >  > He notices that no one is perfect. By induction, he presumes   
   >  > himself to be imperfect; that is, he's occasionally wrong. Which   
   >  > means one of his accepted propositions must be false.   
   >   
   > Yes, "(he very well knows that) he may be wrong", put simply.   
   >   
   >  > Therefore, he is aware that he believes a false proposition.   
   >  > Hence is inconsistent. Knowingly.   
   >  >   
   >  > A modest man must therefore be inconsistent, unavoidably.   
   >   
   > It ain't about "modesty", it's a matter of "finiteness",   
   > and, to reiterate, it ain't about consistency either, a   
   > _rational_ (hu)man *is* "consistent", in so far as s/he is   
   > being rational, for what rational even means.   
   >   
   >  > > If the definition was of a abstract system (e.g., something in the   
   > class   
   >  > > of Turing machines) you could ask if such a system could be   
   > defined, not   
   >  > > whether it is consistent.   
   >  >   
   >  > You could frame the original question in regard to an abstract   
   >  > system, it wouldn't change anything pertinent.   
   >  >   
   >  > Here's a workaround: call on information theory. Assign b bits   
   >  > of information to each correct proposition. Then recognize that   
   >  > some of those are false, and strive to maximize the total   
   >  > information. Don't sweat the small stuff, I always say -   
   >   
   > Sure, and we already do that, a logic of confidence instead of one   
   > of certainty: by the lenses of an information-theoretic approach...   
   >    
   > (so to speak).   
   >   
   > But often there is more and better we can do than just "averaging"   
   > "from the past": but indeed, contra widespread belief, inductive   
   > reasoning is *not* probabilistic reasoning, it's not even reasoning   
   > strictly speaking (P.F. Strawson, Introduction to Logical Theory),   
   > it's rather *a premise to any reasoning and reasonability*.   
   >   
   > And here is an evergreen for the broader context:   
   > D.C. Williams, The Evils of Inductive Skepticism.   
   >    
   >   
   > Julio   
      
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