Godel's Incompleteness Theorem and God
#25
After attending the intro course to Logic on coursera, I got a vague idea of Godel's incompleteness theorem and would like to share my understanding here.

Quote:First incompleteness theorem:
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250).

One of the key term to note here in the above statement is provable. In general language this can mean a lot more than what is implied in the statement. In logic provability has a very very specific meaning and the statement has to be understood ONLY in that context. For any axiomatic system, one can have a "proof system". The reason for having the proof system is as follows:
  • A logic system is defined by its axioms, i.e. some propositions which are considered to be true as given.
  • Now a statement's truth value can be evaluated substituting the truth values of these proposition. For example consider an axiomatic system which assumes A and ~B (not B) propositions to be true and defines standard logic operators | and &, i.e. "or" and "and". Then the truth value of statement "A | B" can be evaluated as True and "A & B" as false by mere substitution of truth values of A and B.
  • But its not always easy to do so as number of such propositions or variables can be huge and infact infinite (like all numbers in a number system). So in such cases an alternate approach is followed to derive the truth values of statements.
  • This is done via defining a "proof system" for that logic system and then following its rules to derive the truth value of the statement.

Now a complete system is a logic system which has a proof system by which every true statement can be proven so.
And a consistent system is a logic system whose axiomatic propositions (the ones that are assumed to be true as given) don't have any inconsistency (one proposition being true doesn't lead to other proposition being false).

Now what Godel proved that was for a sufficiently complicated logic system it cannot be complete and consistent at the same time. i.e. There doesn't exist a proof system such that every true statement in that logic system can be proved so, or if such a proof system exists, it is guaranteed that the axiomatic propositions are inconsistent.

Now this part is absolutely not applicable to Physical sciences because they don't have the so called "proof system" as used in logic. Infact physical sciences don't deal with ONE logic system but with TWO logic systems. One being the laws of universe and second being human's formulation of those. So what scientists actually do is check if these two logic systems match by comparing their outputs. The truth value for scientists is thus "comparative'" and not based on some proof system.

Quote:Second incompleteness theorem:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

Second should be pretty straightforward to understand. It merely states that a sufficiently complicated logic system cannot prove consistency of its axiomatic propositions. If it is able to do that then it is certainly inconsistent.

In here as well its important to note that its about provability in the sense defined above and nothing in the sense what physical sciences (empirical evidence or inference to best explanation) use.
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#26
(07-Jan-2012, 09:34 AM)Kanad Kanhere Wrote:
(06-Jan-2012, 09:03 PM)somnath mazumder Wrote: Here I would like to account for the Universe statement
Sorry for usage of technical language in a popular science column, to my readers. It is just to explain the raised objections.
1. Universe means the paradigm that generates a theory. It presupposes a space where a set of operations are defined. This obviously means that the very structure must be closed under the given operations defined in the space.
2. 2nd theorem do not say what you have interpreted viz. "2nd theorem just says a sufficiently complicated axiomatic systems cannot prove the consistency of its axioms.". But formally the 2nd theorem states "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent." This means if a formal effectively generated theory contains a proof about its own consistency then it will be inconsistent.

I think your explanation is perfect till here.

(06-Jan-2012, 09:03 PM)somnath mazumder Wrote: With my atheist example I've said that "if an atheist claim[s] that he has got a complete explanation of the universe using his "postulates" of atheism..." this means he is including a (claimed) proof of the consistency of his stance universally (refer to the universe definition) then via the 2nd theorem of Godel he is bound to make his paradigm inconsistent provided a atheist believes the basic algebraic truths are legitimate and sufficiently complete. What I was trying to say that if an theist claims that his view of the world is complete, then he must part with the universal consistency of his paradigm.

This doesn't make sense. A complete worldview doesn't necessarily mean inclusion of sentences that assert truthness of the "postulates". "Complete explanation of universe" just means all observable facts can be explained using the proposed postulates. But this in no way mandates proof for the postulates.

(06-Jan-2012, 09:03 PM)somnath mazumder Wrote: 3. No, things are not the same with theism (at least some versions of it). If he [theist] adopts that certain states are unknowable (which he says divine interference) then he makes his theory incomplete by default. He can always vary the degree of completeness but in doing so he gets a theory that is consistent, which is very important for science. For example, adoption of the unknownability of simultaneous position and momentum states of microscopic particles (Heisenberg's Uncertainty Principle) thereby eliminating unobservable aspects of the theory subject to Occam's razor, makes quantum theory consistent with wide range of observations.

This seems to be totally reversing logic. Just because a theory is not complete doesn't mean it is not inconsistent.

What I understood of your logic as you have given above ~(~p) = p. You have failed to counter my proposition. For your statement "Just because a theory is not complete doesn't mean it is not inconsistent." doesn't make a sense.
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#27
(26-Nov-2013, 01:46 AM)somnath mazumder Wrote: What I understood of your logic as you have given above ~(~p) = p. You have failed to counter my proposition. For your statement "Just because a theory is not complete doesn't mean it is not inconsistent." doesn't make a sense.

Looks like you didn't understand anything. Kindly go through the post again. And basics of logic therory. An incomplete system CAN be inconsistent as well.
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#28
I concur with the points about information theory. Information is indeed an inherent property of time, space and matter.
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