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The size of a Gödel sentence G (Re: Could AlphaEvolve find the sixth busy beaver ?)

From Mild Shock <janburse@fastmail.fm>
Newsgroups sci.physics, sci.physics.relativity, comp.theory
Subject The size of a Gödel sentence G (Re: Could AlphaEvolve find the sixth busy beaver ?)
Date 2025-12-03 09:00 +0100
Message-ID <10goqnc$1263a$2@solani.org> (permalink)
References <10ghds1$tg19$2@solani.org>

Cross-posted to 3 groups.

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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

Mild Shock schrieb:
> Hi,
> 
> What we thought:
> 
> Prediction 5 . It will never be proved that
> Σ(5) = 4,098 and S(5) = 47,176,870.
> -- Allen H. Brady, 1990  .
> 
> How it started:
> 
> To investigate AlphaEvolve’s breadth, we applied
> the system to over 50 open problems in mathematical
> analysis, geometry, combinatorics and number theory.
> The system’s flexibility enabled us to set up most
> experiments in a matter of hours. In roughly 75% of
> cases, it rediscovered state-of-the-art solutions, to
> the best of our knowledge.
> https://deepmind.google/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/ 
> 
> 
> How its going:
> 
> We prove that S(5) = 47, 176, 870 using the Coq proof
> assistant. The Busy Beaver value S(n) is the maximum
> number of steps that an n-state 2-symbol Turing machine
> can perform from the all-zero tape before halting, and
> S was historically introduced by Tibor Radó in 1962 as
> one of the simplest examples of an uncomputable function.
> The proof enumerates 181,385,789 Turing machines with 5
> states and, for each machine, decides whether it halts or
> not. Our result marks the first determination of a new
> Busy Beaver value in over 40 years and the first Busy
> Beaver value ever to be formally verified, attesting to the
> effectiveness of massively collaborative online research
> https://arxiv.org/pdf/2509.12337
> 
> They claim not having used much AI. But could for
> example AlphaEvolve do it somehow nevertheless, more or
> less autonomously, and find the sixth busy beaver?
> 
> Bye

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Thread

Could AlphaEvolve find the sixth busy beaver ? Mild Shock <janburse@fastmail.fm> - 2025-11-30 13:38 +0100
  An old Busy Beaver ASIC (Application-Specific Integrated Circuit) (Was: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-11-30 13:54 +0100
  An old Busy Beaver ASIC (Application-Specific Integrated Circuit) (Was: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-11-30 13:55 +0100
    What is analog computing nowadays? (Re: An old Busy Beaver ASIC (Application-Specific Integrated Circuit) (Was: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-12-01 11:25 +0100
      Wake-up call until everybody gets ear-bleeding (Re: What is analog computing nowadays?) Mild Shock <janburse@fastmail.fm> - 2025-12-01 12:01 +0100
        BB(745) is independent of ZFC (Was: Wake-up call until everybody gets ear-bleeding) Mild Shock <janburse@fastmail.fm> - 2025-12-01 12:07 +0100
          Write ZFC formulas on a tape (of a Turing machine) (Re: BB(745) is independent of ZFC ) Mild Shock <janburse@fastmail.fm> - 2025-12-02 17:18 +0100
            Turing machines have neurons (Re: Write ZFC formulas on a tape (of a Turing machine)) Mild Shock <janburse@fastmail.fm> - 2025-12-02 17:19 +0100
              A logical calculus in nervous activity [McCulloch & Pitts 1943] (Re: Turing machines have neurons) Mild Shock <janburse@fastmail.fm> - 2025-12-02 17:20 +0100
                Busy Beaver and Theory Consistency (Was: A logical calculus in nervous activity [McCulloch & Pitts 1943]) Mild Shock <janburse@fastmail.fm> - 2025-12-02 17:39 +0100
                Busy Beaver and Theory Consistency (Was: A logical calculus in nervous activity [McCulloch & Pitts 1943]) Mild Shock <janburse@fastmail.fm> - 2025-12-02 17:43 +0100
                Re: Busy Beaver and Theory Consistency (Was: A logical calculus in nervous activity [McCulloch & Pitts 1943]) Mild Shock <janburse@fastmail.fm> - 2025-12-02 23:18 +0100
      Re: What is analog computing nowadays? (Re: An old Busy Beaver ASIC (Application-Specific Integrated Circuit) (Was: Could AlphaEvolve find the sixth busy beaver ?) Maciej Woźniak <mlwozniak@wp.pl> - 2025-12-01 12:09 +0100
        parallel random-access machine (parallel RAM or PRAM (Was: What is analog computing nowadays?) Mild Shock <janburse@fastmail.fm> - 2025-12-01 12:15 +0100
          Re: parallel random-access machine (parallel RAM or PRAM (Was: What is analog computing nowadays?) Maciej Woźniak <mlwozniak@wp.pl> - 2025-12-01 13:23 +0100
            Nope, you can't, because of the CRCW instuction (Was: parallel random-access machine) Mild Shock <janburse@fastmail.fm> - 2025-12-01 17:12 +0100
              Algorithm introduced in Hogwild! SGD (Niu et al., 2011) (Was: Nope, you can't, because of the CRCW instuction) Mild Shock <janburse@fastmail.fm> - 2025-12-01 17:31 +0100
                PRAMs might be closer to physics: Boltzman machines, etc.. (Was: Algorithm introduced in Hogwild! SGD) Mild Shock <janburse@fastmail.fm> - 2025-12-01 18:02 +0100
              Re: Nope, you can't, because of the CRCW instuction (Was: parallel random-access machine) Maciej Woźniak <mlwozniak@wp.pl> - 2025-12-01 17:59 +0100
                PRAMs might be closer to physics: Boltzman machines, etc.. (Re: Nope, you can't, because of the CRCW instuction) Mild Shock <janburse@fastmail.fm> - 2025-12-01 18:05 +0100
                PRAMs might be closer to physics: Boltzman machines, etc.. (Re: Nope, you can't, because of the CRCW instuction) Mild Shock <janburse@fastmail.fm> - 2025-12-01 18:08 +0100
                Physics more difficult than Rasperry LED cube? (Was: PRAMs might be closer to physics: Boltzman machines, etc..) Mild Shock <janburse@fastmail.fm> - 2025-12-01 18:25 +0100
            Re: parallel random-access machine (parallel RAM or PRAM (Was: What is analog computing nowadays?) Thomas Heger <ttt_heg@web.de> - 2025-12-03 07:17 +0100
              Re: parallel random-access machine (parallel RAM or PRAM (Was: What is  analog computing nowadays?) Python <python@cccp.invalid> - 2025-12-03 06:46 +0000
              Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-03 08:02 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-04 07:50 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2025-12-04 09:57 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-06 17:02 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-07 10:22 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-07 10:39 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-07 10:46 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2025-12-07 11:42 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Michael S <already5chosen@yahoo.com> - 2025-12-07 16:26 +0200
                Re: parallel random-access machine (parallel RAM or PRAM) Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2025-12-08 04:25 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-08 08:51 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Michael S <already5chosen@yahoo.com> - 2025-12-08 13:58 +0200
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-09 09:15 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-08 08:21 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Janis Papanagnou <janis_papanagnou+ng@hotmail.com> - 2025-12-08 09:06 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-09 09:19 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-09 11:43 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-10 08:19 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-10 09:56 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-10 10:01 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-11 09:02 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-11 08:48 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-12 01:58 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-11 20:45 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-11 23:07 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-12 00:49 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-14 14:27 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 10:25 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-14 21:22 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 12:52 -0800
                I have a great idea The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 13:11 -0800
                Einstein refrigerator (was: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-15 03:00 +0100
                Re: Einstein refrigerator (was: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 19:46 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 10:27 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-14 10:58 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-15 07:50 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-15 14:05 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-16 08:44 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-16 22:58 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas Heger <ttt_heg@web.de> - 2025-12-17 08:50 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-17 14:00 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-17 10:49 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-17 11:23 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul.B.Andersen" <relativity@paulba.no> - 2025-12-17 14:24 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-18 22:13 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) The Starmaker <starmaker@ix.netcom.com> - 2025-12-19 12:02 -0800
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-17 18:24 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) John Hasler <john@sugarbit.com> - 2025-12-17 11:58 -0600
                Re: parallel random-access machine (parallel RAM or PRAM) Maciej Woźniak <mlwozniak@wp.pl> - 2025-12-12 07:19 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) "Paul B. Andersen" <relativity@paulba.no> - 2025-12-12 11:35 +0100
                Re: parallel random-access machine (parallel RAM or PRAM) Lawrence D’Oliveiro <ldo@nz.invalid> - 2025-12-18 06:49 +0000
                Re: parallel random-access machine (parallel RAM or PRAM) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-06 05:30 +0100
  Re: Could AlphaEvolve find the sixth busy beaver ? Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-11-30 14:06 +0100
    You shouldn't use NPM hacked services (Was: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-11-30 22:27 +0100
      Its a subconscious hypothesis (Was: You shouldn't use NPM hacked services) Mild Shock <janburse@fastmail.fm> - 2025-11-30 22:33 +0100
        What if of the cosmos does a BB dance? (Was: Its a subconscious hypothesis) Mild Shock <janburse@fastmail.fm> - 2025-11-30 22:43 +0100
          Re: What if of the cosmos does a BB dance? (Was: Its a subconscious hypothesis) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-01 23:43 +0100
          Re: What if of the cosmos does a BB dance? Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-01 23:45 +0100
            newsreader where you can see the message source (Was: What if of the cosmos does a BB dance?) Mild Shock <janburse@fastmail.fm> - 2025-12-02 00:00 +0100
              Re: newsreader where you can see the message source (Was: What if of the  cosmos does a BB dance?) Python <python@cccp.invalid> - 2025-12-02 03:10 +0000
                Re: newsreader where you can see the message source (Was: What if of the cosmos does a BB dance?) Mild Shock <janburse@fastmail.fm> - 2025-12-02 11:51 +0100
                Re: newsreader where you can see the message source (Was: What if of the cosmos does a BB dance?) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-02 20:36 +0100
                Re: newsreader where you can see the message source (Was: What if of the cosmos does a BB dance?) Ross Finlayson <ross.a.finlayson@gmail.com> - 2025-12-01 21:42 -0800
                Orbits of planets in the Sol System (was: newsreader where you can see the message source) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-04 03:29 +0100
                Orbits of planets in the Sol System Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-04 03:30 +0100
              Re: newsreader where you can see the message source (Was: What if of the cosmos does a BB dance?) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-02 20:35 +0100
                Spock thinks I am interested in his gibberish (Re: newsreader where you can see the message source) Mild Shock <janburse@fastmail.fm> - 2025-12-02 23:22 +0100
                Is it like "Wirres Mückengelaber" ? (Was: Spock thinks I am interested in his gibberish) Mild Shock <janburse@fastmail.fm> - 2025-12-02 23:28 +0100
            newsreader where you can see the message source (Re: What if of the cosmos does a BB dance?) Mild Shock <janburse@fastmail.fm> - 2025-12-02 00:11 +0100
            Re: What if of the cosmos does a BB dance? Thomas Heger <ttt_heg@web.de> - 2025-12-03 07:22 +0100
              Re: What if of the cosmos does a BB dance? The Starmaker <starmaker@ix.netcom.com> - 2025-12-03 08:27 -0800
                Re: What if of the cosmos does a BB dance? Thomas Heger <ttt_heg@web.de> - 2025-12-04 07:57 +0100
        Re: Its a subconscious hypothesis (Was: You shouldn't use NPM hacked services) Maciej Woźniak <mlwozniak@wp.pl> - 2025-11-30 23:14 +0100
      Re: You shouldn't use NPM hacked services (Was: Could AlphaEvolve find the sixth busy beaver ?) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-01 23:23 +0100
        Re: You shouldn't use NPM hacked services (Was: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-12-02 00:05 +0100
          Re: You shouldn't use NPM hacked services Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-02 20:29 +0100
    What Spock aka Thomas 'PointedEars' Lahn missed (Was: Different Hubble Theories: de Sitter Energy) Mild Shock <janburse@fastmail.fm> - 2025-12-03 01:09 +0100
  The size of a Gödel sentence G (Re: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-12-03 09:00 +0100
  The size of a Gödel sentence G (Re: Could AlphaEvolve find the sixth busy beaver ?) Mild Shock <janburse@fastmail.fm> - 2025-12-03 09:00 +0100
    Attacking the Busy Beaver 5 [1989] (Re: The size of a Gödel sentence G) Mild Shock <janburse@fastmail.fm> - 2025-12-03 09:10 +0100
  Re: Could AlphaEvolve find the sixth busy beaver ? Sylvia Else <sylvia@email.invalid> - 2025-12-04 12:59 +0800

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