Re: [asa] Goedel's theorem and religion

This is a long note, but I want to call attention to the bottom where I try to answer Dave's original request from my own personal opinions.
 
First, I want to back up what Dave Siemens says in response to Iain, but I can't speak as authoritatively as Dave can. It's been a couple years since I struggled to try to understand Godel's proof, and I must admit that there was one step in the proof that I did not grasp. I intend to return to it one day and understand it fully before I die! :-)
 
There is a whole class of these self-referential arguments, and some are indeed trivial but others have deep consequences. As an example of how these "proofs" may be very consequential, consider this question. Can God's infinite mind think of a list containing all the real numbers that lie between 0 and 1? Well, if He could do so, then He could write them in base-2 (e.g., 0.001011010010...). He could write them in the form of a table, each row being one of the real numbers, with each row being infinitely long and there being an infinite number of such rows. Then He could take the numerals lying on the diagonal of that table and then take the binary complement of that diagonal (i.e., replace all its 1's with 0's and vice versa). This new number will also be a real number between 0 and 1. So had it already been recorded somewhere in the infinite list? Well Yes, by definition, if God recorded all of them. But No it cannot be in the list, because if it was in the list th
 en it would cross itself on the diagonal somewhere, and at that crossing point it would need to be both 0 and 1 simultaneously. This proves that there exists a single number that cannot be on the list. Therefore, God cannot list all the real numbers.
 
Getting to Iain's question, is this trivial because it only deals with one single, self-referential number? No, indeed. It proves that there exists no infinite list of real numbers that contains all of them. It's not as simple as just adding that one self-referential number onto the list in order to complete the list, since that emended list, too, cannot contain its diagonal. You can never emend the list to completeness. Thus it has been said that "Not even God in his infinite mind can imagine a list of all the real numbers!" And that is true. God cannot construct a list of the real numbers, because "being listable" is a meaningless concept in regard to real numbers. It is like saying "Not even God can make a three-sided square," which is a very true statement. This is not a limitation in God; to the contrary it is a limitation in man, in that it is not more obvious to us how meaninglessness it is to talk about three-sided squares or the listing of real numbers! Thu
 s, the proof has laid bare to our limited minds a very important concept that God himself already knew about real numbers, that they cannot be listed, and thus to us mortals it is a very useful proof! And yet we obtained it by appealing to one single, self-referential number, the one that happens to be the complement of the diagonal.
 
Because of this, such proofs are sometimes called "diagonalizations".
 
So in response to Iain I'd say that Godel's use of a single, self-referential statement does not necessarily impose a limitated application of the proof. Godel constructed a diagonalization of a meta-mathematical statement, which proves that you cannot decide all of the meaningful statements that exist in an axiomatic system containing number theory. No matter how you emend the system by adding more axioms, it can never be complete. Even after you add an infinite number of axioms, it still contains meaningful concepts that are a true part of the system and yet cannot be decided by that system. Further, Godel showed that you can never be sure that your axioms are not contradictory with one another. You cannot perform less than an infinite amount of reasoning and be sure that no contradictions exist in your system.
 
What I find interesing about this (as a physicist) is that, to my thinking, it may point to a similar diagonalization that may exist in the program of scientific reductionism. The program of physics is to discover the physical "axioms" of the universe. These are not mathematical axioms, but physical entities or laws. But they operate like axioms in that nature "derives" all subsequent events from this simple set of basic laws. In doing this derivation, nature follows its own axioms, and no information is injected into the system from outside, if naturalism is true. Well, if the universe truly is really a closed, "axiomatic" system like this, then can we appeal to a Godel-esque argument to say that the universe itself cannot know everything that must occur in the universe? That additional axioms must be added to make up for what is not derivable, so that events may occur that the universe could not decide from its own finite set of initial axioms? And that (therefore) n
 ature's process of deriving everything that must occur in nature is not a property of what will occur in nature? That an infinite number of axioms must be added to nature in order to make up for the lack of completeness in reductionistic physics, and yet the system will always be incomplete? I have a hunch that this is true. I suspect that this is related to the existence of free moral agency. An alternative position might be to appeal to multi-verses, saying that nature adds its own new "axioms" in opposing pairs in each quantum mechanical bifurcation of the universe (the Everett view). So this extension of Godel may not produce an absolute proof of God, but it might point to the impossibility of reductionism, which is compatible with Christian theism. (And "multi-verse" may simply be a fancy way of saying "Mind of God" for all we know.)
 
Likewise (since Godel not only proved incompleteness, but also the unprovability on non-contradictions in the axioms), there is the question how physics could know that its own axiomatic laws are not self-contradictory? If the universe performed an infinite amount of processing of its own axioms, still it seems the universe could never "know" if it were fundamentally self-contradictory at its heart. Since we usually assume physics is self-consistent, then this raises the question how it could ever arise at self-consistency if it is indeed an axiomatic system as the program of reductionism assumes. Again, it may be possible to appeal to an extension of quantum indeterminacy, that nature does not actually decide any of its axioms a priori, so this does not necessarily produce an air-tight proof of God to my mind (at least not until someone far beyond my abilities takes it up!). But again, it certainly smacks of Christian theism, the idea that an infinite mind who eternally
 knows all deductions a priori has created everything. It makes one wonder if reductionism is inherently impossible, and that only a being described as fully God can self-exist. It all is very appealing to me. It fits with what I already know about God in that He came to us on the cross and in my heart, and spoke to us in the Bible and through prophets, and what He has communicated agrees with this extension of Godel. So it seems very appealing and I can't help but wonder if this is all true.
 
I also understand why many people are very hostile to these possible extensions of Godel. It is because some less-careful souls latch onto Godel's proof the same way that young earth creationists latch onto "evidences" for a young earth. The skeptical scientist inside us notices this tendency, and we want to spurn the error, and then the social dynamics of us as a group reinforces our desire to reject all attempts to relate Godel to other things. I appreciate the importance of this. Notice also that I tried to be careful to say "I suspect" these things are true because I don't think it is a trivial matter to simply quote Godel as if his proof directly applies to any ontological questions. I think there are some subtle difficulties in applying Godel to physics or religion. For example, does physics contain number theory (a necessary part of Godel's proof)? It would seem so (as in quantum numbers), but is this appearance of mathematical "numbers" just a construct of huma
 n minds projected into our description of physics? There are probably many such problems that arise when you try to go from pure reason to the emprics of physics or faith.
 
Phil Metzger
 
 
 
 
-----Original Message-----
From: dfsiemensjr@juno.com
To: igd.strachan@gmail.com
Cc: d.nield@auckland.ac.nz; asa@calvin.edu
Sent: Fri, 13 Oct 2006 2:24 PM
Subject: Re: [asa] Goedel's theorem and religion

Sorry, Iain, but the application is broader, not to a single form of sentence. With Alonzo Church's extension, it is proved that there are true sentences that can be formulated within the vocabulary of the system but that cannot be proved within the system. In application, there are theological truths that cannot be proved by theology.
Dave
 
On Fri, 13 Oct 2006 06:59:08 +0100 "Iain Strachan" <igd.strachan@gmail.com> writes:

 
On 10/13/06, D. F. Siemens, Jr. <dfsiemensjr@juno.com> wrote:
As a logician, I have to say that any study that involves logic beyond
the level of the sentential, is affected by Goedel's theorem. It does not
interfere with derivations within limits, as it also does not restrict a
proof in geometry or number theory. But it presents a strong claim: there
are true statements in all these areas that cannot be proved true.

Yes, but do you not agree that the proof of Goedel's theorem is simply an existence proof by provision of a counter-example, and furthermore that the counter-example belongs to a particular class of self-referential statements of the form "This statement is formally undecidable"? It cannot imply the existence of formally undecidable propositions that don't have this explicit form of self-reference.

I'm struggling to see how any theological statement could fit into this category.

Iain

 
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Received on Sat Oct 14 00:23:16 2006