Godel's Incompleteness Theorem and God
#13
On a lighter note, I never would have imagined that a topic out of topology and category theory would come up in a Theological discussion. Kudos.

@Kanhad Kanhere, yes, Godel did prove that an axiomatic system (with some constraints) cannot prove that it is self consistent.

Aside: The more physics I have been studying the more I feeling that the path to the axiomatization of the physical universe might lie in the why questions that physics avoids trying to explain. It is always the how questions.
For example math may describe simple systems, but the axioms that determine the mathematical laws come from physics (That the constants are what they are or that F = MA and not MV (this CAN be explained at the assumption of some other axiom namely the principle of stationary action)). The math cannot explain WHY these are so, it just explains it to us that it IS so.
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#14
(29-Oct-2011, 06:25 PM)Alan DSouza Wrote: Aside: The more physics I have been studying the more I feeling that the path to the axiomatization of the physical universe might lie in the why questions that physics avoids trying to explain. It is always the how questions.
For example math may describe simple systems, but the axioms that determine the mathematical laws come from physics (That the constants are what they are or that F = MA and not MV (this CAN be explained at the assumption of some other axiom namely the principle of stationary action)). The math cannot explain WHY these are so, it just explains it to us that it IS so.

Is the question of whether, in a manner of speaking, 'The universe is a consistent system.', the same as asking whether Background Independence is true? To start with, 'background independence' is of most concern mainly for theoretical physicists and for a great deal of scientific endeavor in more 'derived' disciplines ("How fast does a myelinated axon conduct?", "Can 10-fold symmetry be observed in realizable crystals?") such a concern does not get in the way. Considering the proposition that "Laws of nature apply across space and time", David Hume noted that this statement is unverifiable but must inevitably be resorted to. Are the current views on this by contemporary philosophers of science very different from Hume's?

The 'why' question is very prone to result in teleological narratives of Nature, unlike the 'how' question which yields phenomenological narratives. Historically, the former have been the preserve of theology and much of scientific progress is owed to concentrated efforts of phenomenological description, demanding as it does accurate measurement and succinct summarization into laws. Are there historical instances where a teleological line of inquiry can contributed to scientific understanding where a sufficiently detailed phenomenological description failed?

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#15
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(01-Oct-2010, 09:00 PM)Lije Wrote: Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.
Godel's theorems are a demonstration of the failure of first order logic to account for inconsistency in sufficiently complete structures like Peano's axioms. Based upon Godel's measure. To cut it short, to eliminate the related difficulties lets put it this way... 1st theorem goes like this: If atheism satisfy basic algebraic truths (that is if an atheist believes 2+2 = 4, 2X3=6 etc) then according to Godel's 1st theorem a statement can be made using the postulates of atheistic paradigm that cannot be proved with atheism. 2nd theorem says: if an atheist claim that he has got complete explanation of the universe using his "postulates" of atheism, then his system will be inconsistent with some other conclusion of his claim. Peace! Big Grin
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#16
(21-Nov-2011, 01:48 AM)somnath mazumder Wrote:
(01-Oct-2010, 09:00 PM)Lije Wrote: Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.
Godel's theorems are a demonstration of the failure of first order logic to account for inconsistency in sufficiently complete structures like Peano's axioms. Based upon Godel's measure. To cut it short, to eliminate the related difficulties lets put it this way... 1st theorem goes like this: If atheism satisfy basic algebraic truths (that is if an atheist believes 2+2 = 4, 2X3=6 etc) then according to Godel's 1st theorem a statement can be made using the postulates of atheistic paradigm that cannot be proved with atheism. 2nd theorem says: if an atheist claim that he has got complete explanation of the universe using his "postulates" of atheism, then his system will be inconsistent with some other conclusion of his claim. Peace! Big Grin

Don't think that last statement is correct. Actually depends on what you define by "universe", but in general the 2nd theorem just says a sufficiently complicated axiomatic systems cannot prove the consistency of its axioms. So the atheist won't be able to prove his "postulates" to be consistent. But that doesn't mean conclusions of his claim will be inconsistent.
By the way why the stress on "atheism" here.. Same is equally true for "theism".
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#17
Here's a recent discussion between two members of the Facebook Nirmukta Science Group, between KK and MCS,

Quote:KK: Godel proved that "The consistency of the axioms cannot be proven within the system". Doesn't this mean that "reducto-ad-absurdum" approach to proving might be incorrect?

MCS: Godel's theorem can be better put as, "any arithmetic system cannot be both consistent and complete." The statement applies to a strong arithmetic system taken as a whole, not to the finite set of assumptions that we seek to contradict in using proof by contradiction.

Proof by contradiction is simply this,

Suppose we want to prove a statement P, we just show that if neg(P) (i.e. P is false) is assumed, then it leads to a 'contradiction'. There are many contradictions in mathematics, unrelated to Godel's theorem. For example, proof by contradiction is used in proving sqrt(2) is irrational. We only need one assumption, sqrt(2) = p/q, 2 = p^2/q^2, p^2 = 2q^2, so p^2 is even, so p is even, which is a contradiction because we assumed p/q is the simplest form. Notice that there are only a few assumptions (among all possible truths in arithmetic) used in this proof, and only those are shown to be contradicted. So proof by contradiction still stands.

KK: There aren't that few assumption... Almost all the axioms related to properties of real/rational numbers get used. For e.g. associate, distributive properties, cancellation properties etc.

MCS: Godel's theorems don't say anything about the correctness of specific statements. They refer to the system of axioms as a whole. If an axiomatic system is consistent, i.e., there are no contradictions, then there will remain theorems that cannot be proven. Further, if all statements and theorems can be proven, then your system is not consistent, which is not an option for mathematics. Consistency is a requirement. Notice that it doesn't speak to the accuracy of specific statements. There are unprovable statements in mathematics since consistency is a necessity for proofs. So violating consistency is not an option, thus enabling us to use reductio-ad-absurdum. Completeness, however is a different question. Examples of unprovable statements are those of the form "Claim A cannot be proven."

KK: You are just quoting the first part of the "Incompleteness theorem". Search on Wikipedia gave this as the incompleteness theorem
1. If the system is consistent, it cannot be complete.
2. The consistency of the axioms cannot be proven within the system.
Its the second part that I am talking about.

MCS: nd refers to any formal system of statements that contain statements about its own consistency.

'Proof by contradiction' is not a statement, it is a methodology based on showing contradictions to the few assumptions related to the statement you are trying to prove. 'Proof by contradiction' is useful only for provable statements. I think almost all mathematical statements that you and I encounter are provable. The structure is such,

Assumption 1, Assumption 2, Assumption 3,..., Assumption N are behind Statement 1. Falsity of Statement 1 contradicts some assumptions, thus Statement 1 must be true.

KK: This is where I am confused, if the axioms are inherently inconsistent then a statement can be false by the very use of those axioms. That doesn't make the statement really false in that axiomatic system. For e.g. suppose in some proof we need to prove a=b. Reducto-ad-absurdum method would be to prove that ab leads to some contradiction. Thus proved a=b. But what if the axioms are fundamentally inconsistent and hence the trichotomy itself is false?


MCS: If you include in your list of statements, the statement 'this system is consistent,' then the system is NOT. smile That's all it says. 'Axioms' (Peano arithmetic, in our case) are always assumed true. Most axioms we encounter are typically consistent with one another. But you cannot prove the consistency within the system itself. Gentzen (1938) for example, proves the consistency of Peano arithmetic from outside the axiomatic system.

MCS: Do we know that all the several statements that can be derived from the axioms are consistent? Yes, due to Gentzen. What Godel says is not that "Peano arithmetic" is inconsistent, but rather that its consistency cannot be proved from within the system.

KK: I think I get it now. Normal deduction based proof is not different from reducto-ad-absurdum in the sense that both are wrong if the system is inherently inconsistent. The confusion probably was because reducto-ad-absurdum approach almost explicitly maintains that axioms are consistent while normal deduction based proofs maintain the same thing implicitly

Hope this conversation is meaningful to the discussion on this thread.
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#18
I would like to highlight that Godel's results are actually statements about the inconsistency of axiomatic systems containing/generating statements about their own consistency. Godel himself fueled speculations about the philosophical implications of his theorem in that he believed that mathematical objects were 'real' (aka Platonism). Wittgenstein, as interpreted by some later philosophers, felt that Godel's results did not have any implications beyond mathematical axiomatic systems. Wittgenstein said,

Quote:Just as we ask: “‘provable’ in what system?”, so we must also ask: “‘true’ in what system?” ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. —Now what does your “suppose it is false” mean? In the Russell sense it means ‘suppose the opposite is proved in Russell’s system’; if that is your assumption, you will now presumably give up the interpretation that it is unprovable. And by ‘this interpretation’ I understand the translation into this English sentence. —If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell’s system. (What is called “losing” in chess may constitute winning in another game.)
(credited to http://www.quora.com/Did-Wittgenstein-mi...ss-theorem )

Some historical background:

Gottlob Frege, Bertrand Russell, Alfred North Whitehead, and later Godel were pioneers in marrying the study of logic to mathematics. Logic had largely remained a mainstay of philosophers, while it was implicitly used in mathematics, as KK's excellent observation in the last comment of the above conversation illustrates. It appears that following George Boole's codification of logical operations, and some early studies by Charles Babbage and Lady Ada, the study of logic didn't progress between 1850s and Frege's and Russell's efforts around 1910. While UK mathematics stagnated till the late 1800s due to their dogmas of Newton-worship, mainland Europe saw a revolution in the development of mathematical foundations. Rigor was emphasized over the intuitive, application-oriented British (English, UK, whatever :P ) approach to mathematics pioneered by Isaac Newton. Mainland pioneers included, Euler, Gauss, Weierstrass, Riemann, Hilbert, Cantor, Bolzano, etc. By the 1900s, with the foundations firmly established, people started worrying seriously about the philosophy of mathematics. (Mandatory nod to Euclid and Pythagoras for being the original dada's of the method of proof in mathematics.)

This anxiety was highlighted by David Hilbert in his famous description of 23 open problems in mathematics. His second problem was about the foundations of mathematical foundations, "Is (Paeno) arithmetic consistent?" (The question was definitively answered by Gentzen in 1938, though Godel's work dealt a fatal blow to Hilbert's second question.) Hilbert was a so-called formalist, i.e., he believed that mathematics is like a chess game played on a chess board with its own set of rules, etc.- an intellectual masturbation of sorts. But, Godel proved otherwise, showing that trying to play within such formal systems is impossible as completeness and consistency cannot be simultaneously achieved.

... (hopefully I will ramble a little longer on this thread, albeit in a day or two.)
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#19
(21-Nov-2011, 01:47 PM)Kanad Kanhere Wrote:
(21-Nov-2011, 01:48 AM)somnath mazumder Wrote:
(01-Oct-2010, 09:00 PM)Lije Wrote: Godel's theorem states that "In any sufficiently powerful logical system statements can be formulated which can neither be proved nor disproved within the system, unless possibly the system itself is inconsistent"

The first time I read about Godel's theorem, I immediately knew that it is a ripe target for god abuse. But I did not know how to counter such an use as I didn't (and still don't) really understand the theorem.

Somebody has posted a proof of god using Godel's Theorem and Good Math, Bad Math has a scathing critique of the proof. Makes for a very interesting read.
Godel's theorems are a demonstration of the failure of first order logic to account for inconsistency in sufficiently complete structures like Peano's axioms. Based upon Godel's measure. To cut it short, to eliminate the related difficulties lets put it this way... 1st theorem goes like this: If atheism satisfy basic algebraic truths (that is if an atheist believes 2+2 = 4, 2X3=6 etc) then according to Godel's 1st theorem a statement can be made using the postulates of atheistic paradigm that cannot be proved with atheism. 2nd theorem says: if an atheist claim that he has got complete explanation of the universe using his "postulates" of atheism, then his system will be inconsistent with some other conclusion of his claim. Peace! Big Grin

Don't think that last statement is correct. Actually depends on what you define by "universe", but in general the 2nd theorem just says a sufficiently complicated axiomatic systems cannot prove the consistency of its axioms. So the atheist won't be able to prove his "postulates" to be consistent. But that doesn't mean conclusions of his claim will be inconsistent.
By the way why the stress on "atheism" here.. Same is equally true for "theism".

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. 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.
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.


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#20
(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.
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#21
Hi guys. Ironically - I was introduced to Godel's incompleteness theorem through a lecture on nature of belief and God by a guy named Gary Miller. It was fascinating, because, he himself as theist, claimed that the common 3 arguments used to prove the existence of God (ontological, teleological and cosmological) are actually all faulty, and have been debunked atleast 200 years, if not 800 years, ago. As a glancing observation on the 3 arguments, he states that their very foundation is faulty. You believe in an absolute which is God - that means God depends on nothing. But when you go to prove the existence of God, God becomes dependent on the assumptions you make, which defeats the idea of God being absolute.

However, using Godel's incompleteness theorems, he confirms the internal belief in God, the absolute. Here is his postulation: applying Godel's theorem on the universe - if one believes that the universe makes sense, then he must also admit that no one can then know all the assumptions or truths of the universe. And that there has to be an absolute outside the universe, which also we can never understand or know. And since the universe does make sense, it is necessary for this absolute outside of the universe for the universe to be consistent. It is impossible for us within the system of the universe to know this absolute.

He also gives an explanation of why it has to be an absolute and not something relative that has to be outside the universe - but I can't remember exactly how he laid that down. In the end, we can never know the nature of God or his existence - just that it is necessary for a God (an absolute) to be outside the universe, because the universe makes sense and is consistent.

Now, as this guy is a muslim, and I am a muslim too - this argument is very appealing to me as it fits very well with our fundamental belief in God. I don't want to go into why right now - as it may come off as me trying to propagate a theistic, if not Islamic viewpoint.

But the reason I am posting this here is I want to know if there is any flaw in this logic. Or in using Godel's theorem like this. I myself am no logician, so as per my simple logic - this argument by Gary Miller is very solid.

Thanks for reading guys. And thanks in advance for all responses to this argument.
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#22
Aktheka, there are multiple resources in this thread that debunk such claims. The very first post in this thread has a link that uses same logic, and later posts show how that is flawed.

(28-Mar-2013, 04:10 PM)AKtheKA Wrote: However, using Godel's incompleteness theorems, he confirms the internal belief in God, the absolute. Here is his postulation: applying Godel's theorem on the universe - if one believes that the universe makes sense, then he must also admit that no one can then know all the assumptions or truths of the universe. And that there has to be an absolute outside the universe, which also we can never understand or know. And since the universe does make sense, it is necessary for this absolute outside of the universe for the universe to be consistent. It is impossible for us within the system of the universe to know this absolute.

Unfortunately Godel's theorem is nothing like what is given above. There is no outside part in Godel's incompleteness theorem, and most certainly not in any physical sense (outside the universe). And Godel's theorem is about proof for logical consistency of axioms and again nothing to do with "makes sense". The rest of the part about absolutes etc. again has no place in Godel's theorem.
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#23
(28-Mar-2013, 04:10 PM)AKtheKA Wrote: Now, as this guy is a muslim, and I am a muslim too - this argument is very appealing to me as it fits very well with our fundamental belief in God. I don't want to go into why right now - as it may come off as me trying to propagate a theistic, if not Islamic viewpoint.

Can you please answer a few questions.

1) Were you a Muslim before you read the argument made by this Muslim guy?

2) If you are shown the flaw(s) in the argument made by this guy, will you give up on Islam and become an apostate?

3) Had this guy been a Catholic will you have converted to Catholicism?

Just curious.
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#24
The only commonality between Godel and God is that the God is a substring in the string Godel.
But only such as that o is umlaut o, not an ordinary O.
As such, probably people should not get so hyper about Godels involvement with God.
Anyone believes in God, in God sense that prime mover -- needs to get their heads checked, or give up the notion
of Good/Evil. The argument of Evil is really really dangerous for the existence of a Benevolent God.
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