Refutation of Godel and Extension of Ideas about Infinity.

Discussion in 'The Science Forum' started by Excognito, Mar 19, 2011.

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  1. Godel's papers have been the subject of much review and evaluation. I do not wish to disbelieve you without allowing you the opportunity to present your hypothesis for review. The floor is yours.
  2. Chroist - I've just looked up Godel's Wikki page out of curiosity, having never heard of him, and tried to read the second paragraph.

    I understand most of the individual words, but the sentences? Dear me!!

    Respect to you both if you can actually have an argument about that stuff, without boffin walting of course.

    I'm off back to the NAAFI, cheers boys.
  3. HB initial response is posted in the 'Are you religious thread' in this message. It does not contain any formal comment on or extension of Godel's Incompleteness Theorems.
  4. I have always understood that infinity means that there is no god. If there is a 'god' in a religious sense then infinity ends... The characteristics applied to 'god the unknown' are man this sentence is true, the one true god, etc.. it goes on... Cogi, You have no idea what exists outside of the system. 'Outside' is an endless paradox...

    In other peoples words:

    "Gödel's Incompleteness Theorem demonstrates that it is impossible for the Bible to be both true and complete."

    Gödel's First Incompleteness Theorem applies to any consistent formal system which:

    Is sufficiently expressive that it can model ordinary arithmetic
    Has a decision procedure for determining whether a given string is an axiom within the formal system (i.e. is "recursive")
    Gödel showed that in any such system S, it is possible to formulate an expression which says "This statement is unprovable in S."

    If such a statement were provable in S, then S would be inconsistent. Hence any such system must either be incomplete or inconsistent. If a formal system is incomplete, then there exist statements within the system which can never be proven to be valid or invalid ("true" or "false") within the system.

    Essentially, Gödel's First Incompleteness Theorem revolves around getting formal systems to formulate a variation on the "Liar Paradox." The classic Liar Paradox sentence in ordinary English is "This sentence is false."

    Note that if a proposition is undecidable, the formal system cannot even deduce that it is undecidable. (This is Gödel's Second Incompleteness Theorem, which is rather tricky to prove.)

    The logic used in theological discussions is rarely well defined, so claims that Gödel's Incompleteness Theorem demonstrates that it is impossible to prove (or disprove) the existence of God are worthless in isolation.

    One can trivially define a formal system in which it is possible to prove the existence of God, simply by having the existence of God stated as an axiom. (This is unlikely to be viewed by atheists as a convincing proof, however.)

    It may be possible to succeed in producing a formal system built on axioms that both atheists and theists agree with. It may then be possible to show that Gödel's Incompleteness Theorem holds for that system. However, that would still not demonstrate that it is impossible to prove that God exists within the system. Furthermore, it certainly wouldn't tell us anything about whether it is possible to prove the existence of God generally.

    Note also that all of these hypothetical formal systems tell us nothing about the actual existence of God; the formal systems are just abstractions.

    Another frequent claim is that Gödel's Incompleteness Theorem demonstrates that a religious text (the Bible, the Book of Mormon or whatever) cannot be both consistent and universally applicable. Religious texts are not formal systems, so such claims are nonsense.
  5. I'd suggest that Godel's incompleteness theorem is the mathematical equivalent of someone walking up to you and saying "I am lying" i.e. it is a paradox. If you accept the theorem then you have shown that all supposedly logical mathmatical systems are inherently illogical. If you need order and certainty and want to construct systems to demonstrate that you have achieved this the idea is thus a bit of a b****r. If you're happy with uncertainty then it's no big deal.
  6. "Let me remind you that although we conventionally use a Gödel-undecidable mathematical structure (including integers with Peano's recursion axiom, etc.) to model the physical world, it's not at all obvious that the actual mathematical structure describing our world actually is a Gödel-undecidable one."
  7. If you're going to quote Tegmark, you might at least have the courtesy to name him and perhaps also give some context within which the quote is embedded.
    You might also note that the statement that "Only Gödel-complete (fully decidable) mathematical structures have physical existence" is also a hypothesis and has problems of its own: Wiki
    The quote might not be as "great" as you think.
  8. When reading material everything should be taken into context and this must be a lesson for all of us... just like reading scriptures.... One can take and interpret bits to suit ones self... we live in a chaotic system.

    What I take from this are the infinite possibilities...infinite numbers, infinite universes and probably infinite answers. We can never know the ultimate answer because there probably isn't one but even if there is we can never prove it with language or maths....or can we? At the moment we don't know...Godel has proved this or has he?...see the problem?.

    I'm happy to live with uncertainty because there is no choice...certainty is an illusion and in some cases a delusion. Those who claim to have found the answer from the infinite amount of information that is available and live their lives by it should re-examine because they are probably wrong. Plucking any idea out of thin air is just as credible but this is not a good way to enjoy this existence...IMHO.