Refutation of Godel and Extension of Ideas about Infinity.

#1
... Hell I had worked out ideas about infinity when I was twelve years old, before I had ever heard of Gödel....
but that you doubt my understanding of Gödel...well that hurts a bit. I actually have a better understanding of his theorem that he did... how's that for pretentious... :)

Yes...you see he believed in god and had delusions about the unknown. He was absolutely right about incompleteness but failed to see that the infinite unknown and unprovable is unprovable too. He filled in the gaps just like any other religious freak. What exists outside of the system is pure speculation and conjecture...
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
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.
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
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.
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 made...like 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 understand most of the individual words, but the sentences? Dear me!!
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
A great quote from a very clever guy...
"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."
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.
Measure? Gödel?
From Alex Vilenkin, vilenkin@cosmos.phy.tufts.edu, Apr 30, 2005
Q: If all mathematical structures (MS) are given equal weight, and there is an infinite number of them, then shouldn't we expect to find ourselves in some incredibly complicated MS? Also, what about Gödel? If the physical world is isomorphic to a MS, then what is the physical counterpart of Gödel's undecidable propositions?
A: I think this "measure problem" is unsolved at Level IV and, as you know, even at Level II. At Level IV, I can envision three resolutions.
The measure punishes complexity.
We actually live in a very complicated mathematical structure, but perceive very little of this complexity. We've repeatedly been surprised to discover new layers of complexity as our experiments got better (atoms, elementary particles, relativity, quantum mechanics, etc.) -- perhaps there's a vast number of additional layers.
Only Gödel-complete (fully decidable) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity. 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.
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
Consistency with Gödel's theorem
It has also been suggested that the MUH is inconsistent with Gödel's incompleteness theorem. In a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford [8], the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is "out there" is incompatible with the idea that it consists of formal systems." Tegmark's response in [8] (sec VI.A.1) is to offer a new hypothesis "that only Godel-complete (fully decidable) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe." Tegmark goes on to note that although conventional theories in physics are Godel-undecidable, the actual mathematical structure describing our world could still be Godel-complete, and "could in principle contain observers capable of thinking about Godel-incomplete mathematics, just as finite-state digital computers can prove certain theorems about Godel-incomplete formal systems like Peano arithmetic." In [2] (sec. VII) he gives a more detailed response, proposing as an alternative to MUH the more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecidable/uncomputable theorems. Tegmark admits that this approach faces "serious challeges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH".
The quote might not be as "great" as you think.
 
#8
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.
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.
 

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