Editing Theory of everything (section)

===Gödel's incompleteness theorem===
A number of scholars claim that [[Gödel's incompleteness theorem]] suggests that attempts to construct a theory of everything are bound to fail. Gödel's theorem, informally stated, asserts that any formal theory sufficient to express elementary arithmetical facts and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement that can't be derived in the formal theory.

[[Stanley Jaki]], in his 1966 book ''The Relevance of Physics'', pointed out that, because a "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.<ref>
{{cite book
 |last=Jaki |first=S.L.
 |date=1966
 |pages=127–130
 |title=The Relevance of Physics
 |publisher=Chicago Press
 }}</ref>

[[Freeman Dyson]] has stated that "Gödel's theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. […] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them."<ref>Freeman Dyson, NYRB, May 13, 2004</ref>

[[Stephen Hawking]] was originally a believer in the Theory of Everything, but after considering Gödel's Theorem, he concluded that one was not obtainable. "Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind."<ref>Stephen Hawking, [http://www.hawking.org.uk/godel-and-the-end-of-physics.html Gödel and the end of physics] {{Webarchive|url=https://web.archive.org/web/20200529232552/http://www.hawking.org.uk/godel-and-the-end-of-physics.html |date=2020-05-29 }}, July 20, 2002</ref>

[[Jürgen Schmidhuber]] (1997) has argued against this view; he asserts that Gödel's theorems are irrelevant for [[computable]] physics.<ref>{{cite book
 |last=Schmidhuber
 |first=Jürgen
 |date=1997
 |title=A Computer Scientist's View of Life, the Universe, and Everything. Lecture Notes in Computer Science
 |volume=1337
 |url=http://www.idsia.ch/~juergen/everything/
 |pages=201–208
 |publisher=[[Springer (publisher)|Springer]]
 |isbn=978-3-540-63746-2
 |doi=10.1007/BFb0052071
 |citeseerx=10.1.1.580.1970
 |s2cid=21317070
 |access-date=2008-03-26
 |archive-date=2014-02-27
 |archive-url=https://web.archive.org/web/20140227162350/http://www.idsia.ch/~juergen/everything/
 |url-status=live
 }}</ref> In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose [[pseudo-randomness]] based on [[undecidable problem|undecidable]], Gödel-like [[halting problem]]s is extremely hard to detect but does not prevent formal theories of everything describable by very few bits of information.<ref>
{{Cite journal
 |author=Schmidhuber, Jürgen
 |date=2002
 |title=Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit
 |journal=International Journal of Foundations of Computer Science
 |volume=13
 |issue=4
 |pages=587–612
 |arxiv=quant-ph/0011122
 |bibcode=2000quant.ph.11122S
 |doi=10.1142/s0129054102001291
}}</ref>

Related critique was offered by [[Solomon Feferman]]<ref>{{cite web
 |last=Feferman
 |first=Solomon
 |url=http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf
 |title=The nature and significance of Gödel's incompleteness theorems
 |publisher=[[Institute for Advanced Study]]
 |date=17 November 2006
 |access-date=2009-01-12
 |archive-date=2008-12-17
 |archive-url=https://web.archive.org/web/20081217035530/http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf
 |url-status=live
 }}</ref> and others. Douglas S. Robertson offers [[Conway's game of life]] as an example:<ref>
{{cite journal
 |last=Robertson |first=Douglas S.
 |date=2007
 |title=Goedel's Theorem, the Theory of Everything, and the Future of Science and Mathematics
 |journal=[[Complexity (journal)|Complexity]]
 |volume=5 |pages=22–27
 |doi=10.1002/1099-0526(200005/06)5:5<22::AID-CPLX4>3.0.CO;2-0
 |issue=5
|bibcode=2000Cmplx...5e..22R}}</ref> The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws.

Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", most physicists argue that Gödel's Theorem does ''not'' mean that a theory of everything cannot exist.{{Citation needed|reason=Precarious wording, unclear significance, and any relevance of the purported affidavits necessitate scrutiny|date=August 2021}} On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of [[prime number]]s into a physical question.<ref>{{cite web
 |last=Hawking
 |first=Stephen
 |date=20 July 2002
 |title=Gödel and the end of physics
 |url=http://www.damtp.cam.ac.uk/strings02/dirac/hawking/
 |access-date=2009-12-01
 |archive-date=2011-05-21
 |archive-url=https://web.archive.org/web/20110521123113/http://www.damtp.cam.ac.uk/strings02/dirac/hawking/
 
 }}</ref> This definitional discrepancy may explain some of the disagreement among researchers.