https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714

"In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history.

"Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths.

"But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency.

"His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

"In the 89 years since Gödel’s discovery, mathematicians have stumbled upon just the kinds of unanswerable questions his theorems foretold. For example, Gödel himself helped establish that the continuum hypothesis, which concerns the sizes of infinity, is undecidable, as is the halting problem, which asks whether a computer program fed with a random input will run forever or eventually halt. Undecidable questions have even arisen in physics, suggesting that Gödelian incompleteness afflicts not just math, but — in some ill-understood way — reality.

***

"Gödel’s main maneuver was to map statements about a system of axioms onto statements within the system — that is, onto statements about numbers. This mapping allows a system of axioms to talk cogently about itself.

"The first step in this process is to map any possible mathematical statement, or series of statements, to a unique number called a Gödel number.

[a long clear discussion of Godel's proof follows]

"However, although G is undecidable, it’s clearly true. G says, “The formula with Gödel number sub(n, n, 17) cannot be proved,” and that’s exactly what we’ve found to be the case! Since G is true yet undecidable within the axiomatic system used to construct it, that system is incomplete.

"You might think you could just posit some extra axiom, use it to prove G, and resolve the paradox. But you can’t. Gödel showed that the augmented axiomatic system will allow the construction of a new, true formula Gʹ (according to a similar blueprint as before) that can’t be proved within the new, augmented system. In striving for a complete mathematical system, you can never catch your own tail.

"We’ve learned that if a set of axioms is consistent, then it is incomplete. That’s Gödel’s first incompleteness theorem. The second — that no set of axioms can prove its own consistency — easily follows.

"What would it mean if a set of axioms could prove it will never yield a contradiction? It would mean that there exists a sequence of formulas built from these axioms that proves the formula that means, metamathematically, “This set of axioms is consistent.” By the first theorem, this set of axioms would then necessarily be incomplete.

"But “The set of axioms is incomplete” is the same as saying, “There is a true formula that cannot be proved.” This statement is equivalent to our formula G. And we know the axioms can’t prove G.

"So Gödel has created a proof by contradiction: If a set of axioms could prove its own consistency, then we would be able to prove G. But we can’t. Therefore, no set of axioms can prove its own consistency.

"Gödel’s proof killed the search for a consistent, complete mathematical system. The meaning of incompleteness “has not been fully fathomed,” Nagel and Newman wrote in 1958. It remains true today."

Comment: this applies to using math to absolutely prove any part of reality. It means we have to accept certain concepts about reality on faith.

Studying the advancing cosmology and particle physics of the 20th Century, convinced me, a la' Paul Davies (I've followed his metamorphosis for years), that something was afoot behind the Big Bang. I don't believe string theory and multiverses are reasonable, more like mental masturbation as Smolin and Woit have described in their recent books. And I find John Leslie's conclusion that 'either there are multiverses and/or God' (rough quote) most convincing, since I don't buy multiverses, unproveable and anti-Occam, despite Andrei Linde and others.-Paul Davies asks the darndest questions. He wants to know where the laws of the universe come from! He thinks scientists have too much faith in what they expect from their studies and their results. At the same time he rejects any supernatural source. He feels that the laws of the universe have a source that must be explained from within the universe, not multiverses.-http://www.nytimes.com/2007/11/24/opinion/24davies.html?pagewanted=1

That is some amazing-sounding chemistry. It very likely is a step in the right direction. - Yes, a tiny step. 
 
> I have to ask you again however, how do you determine what theological claims are valid?
> Science operates under the assumption that we cannot differentiate from natural and supernatural events, and science cannot work without that assumption. - But scientific discoveries can be used to argue there is a God. - 
> Thus, we cannot use science when we discuss theology. - Again I agree.
 
> So again, what do you use to determine what kind of theological claims are valid and invalid? - The validity of God is not ABSOLUTELY proven and never will be. That is where faith comes in. But proof can be accepted beyond a reasonable doubt by those who wish to: Read Mortimer J. Adler's "How to Think About God",1980. Perhaps you know him. He was a leading 20th Century philosopher/theologan. Antony Flew, the famous atheist philosopher, now accepts that approach in his book,"There is a God", 2007.

The biggest issue to our disagreement is that you assume life must have the exact same rules and function that it has now... - Correct. There is no evidence that the laws ofthe universe change, escept for some recent debate about the speed of light.
 
> My argument is that before biochemistry would begin, standard organic and inorganic chemistry would have to apply. - I agree. 
 
> I still have yet to read Shapiro's book (I will do that in my august vacation), - Be sure to read his article in the June 2007 Quarterly Review of Biology. It is a great current followup.

After reading the primer, I take issue with the seeming argument that seems to be made about the concept of 'created by chance.' The author seems ignorant concerning the fundamentals of chemistry and what the words 'chance' and 'spontaneity' mean in this context.
 
> spontaneity refers to the fact that if the ambient conditions are right for a chemical reaction, it will happen spontaneously, any time the right components are present. - These statements from Matt are his basis for thinking origin of life can be spontaneous. Read the folowing link of a recent lab discovery about artifical DNA, formed by intelligent design, with comments by Robert Shapiro. Note that no enzyme was used after the formation of the artifical DNA-like molecule. - http://www.nature.com/news/2009/090612/full/news.2009.563.html

specifically "Supernatural phenomenon cannot be discerned from physical phenomenon." This assumption is accepted by the broadest range of scientists (as you are certainly aware most scientists are theistic on some level, contrary to Dawkins' assertion.) - Your second thought first: Surveys in the US find that 90+% of academic physical scientists (National Academy of Science) are atheist or agnostic;while in biologic sciences 40% are believers. I think this is because we work so closely with the amazing complexities of life, which partially answers your words below. I agree that the supernatural cannot be directly detected, but Mortimer J. Adler accepts 'proof beyond a reasonable doubt', and so do I.
 
 
> Would you agree at large that god appears to take no great interest in the day-to-day dealings with people here on earth? - Adler and I both agree with you. 
 
> I ask because I feel that the god of deism is best suited as the potential creator, in which case, abiogenesis is a physical problem and the only invocation of a god is in getting the ball rolling in the first place. (Big bang.) But how could you detect the deist God? - I don't buy deism, on the logical grounds that a supernatural intelligence in creating the universe must have some purpose other than creation and watching. I believe my intelligence is a small part of the universal intelligence, and thus 'we are made in the image of god'.
 
> Am I also correct in that your leaning towards design is primarily due to your incredulity of our complexities? - Yes, as above. - > If I am, how is this reasoning different than the "God of the Gaps?" How can we detect the creator, if we accept a supernatural being? - When the 'gaps' are closed by science we will have 'proof' as Adler above. My expectation is that we will find that RNA is the master control, not DNA, and that RNA has managed evolution since the beginning, that is, the increasing complexity in evolution is coded from the beginning. 
The DNA of an amoeba is longer than human DNA. There is little if any 'junk' DNA, most of it is turning out to be interference-RNA of various functions, currently about six different functions found so far. This my expectd 'proof' and covers all three forms of deism/theism. - > Have you investigated Process theology? - Yes, a little and I'm not impressed. I consider myself a panentheist, a bit more theist than Einstein or Spinoza. As a child my Mother told me there was a God, and I believed her. After medical school I was agnostic, despite the complexities, especially of human biochemistry. I bought into the idea of materialistc reductionism solving all problems. That is now an intellectual mirage in my view. - Studying the advancing cosmology and particle physics of the 20th Century, convinced me, a la' Paul Davies (I've followed his metamorphosis for years), that something was afoot behind the Big Bang. I don't believe string theory and multiverses are reasonable, more like mental masturbation as Smolin and Woit have described in their recent books. And I find John Leslie's conclusion that 'either there are multiverses and/or God' (rough quote) most convincing, since I don't buy multiverses, unproveable and anti-Occam, despite Andrei Linde and others. - My objections to Darwin, I'll cover at another time.

Matt: I haven't had time to offer you a full response describing my reasoning, and I apologize. I'll take the time when I can. What you are presenting in the link is microevolution with mutations changing information. I accept every bit of that. - Read the following link. It confirms Reznik's studies from over 20 years ago. - http://www.sciencedaily.com/releases/2009/06/090602133551.htm Both studies indicated that the DNA/RNA system can in time of danger from environmentdal change mediate rapid change in phenotype.

At the risk of beating a dead horse, I'm going to give you another link that deals with the issue of claims that random evolutionary processes do not increase information.
 
> Again, I recognize your non-dismissal of evolution but that line of argumentation is typically used by creationists so you'll have to ignore that part of the material. http://www.talkorigins.org/indexcc/CB/CB102.html - 
Matt: I haven't had time to offer you a full response describing my reasoning, and I apologize. I'll take the time when I can. What you are presenting in the link is microevolution with mutations changing information. I accept every bit of that.

Could I bug you again for that link to the Shapiro paper you mentioned yesterday? I should be doing homework but its on a subject I have a hard time making myself study. - It is in my June 4, 19:23 entry, chance and design thread, I think

I've read Kuhn, his observations don't apply here. Kuhn's great criticism is in how a grand idea appears and then creates a new paradigm that is filled by research within the paradigm, - Sorry I was implying the other aspect of Kuhn, overturning an existing paradigm. The fight to get rid of a paradigm. I saw that when I was practicing. The GI ulcer problem was thought to be excess gastic acid. The original work on Helicobacter pyelori took 15-20 years for the US GI community to accept. I tried it on hard-to-treat patients when it first appeared and it worked beautifully. Be skeptical of all you are taught and think for yourself, is my personal rule, and you appear to be doing that.

All the feedback loops are still to be worked out.I'll find the reference I just mentioned and post it later. http://www.sciencedaily.com/releases/2009/06/090604144326.htm

If you want refutations of Behe's arguments, talkorigins is probably the best place to start. It's run by scientists--people who actually do the research into evolution. 
 
> http://www.talkorigins.org/faqs/behe.html - Thanks for the reference. I've looked at Dorit's and Orr's reviews of Black Box. 
> Their refutations of Behe are incredibly thorough and they cite every source and theory to a fault, his arguments are defeated on his science alone. - They have refuted Behe in the same way Ernst Mayr refuted mathematicians (who were claiming Darwin theory did not work)at the Wistar Institute Symposium 1966 by saying: "We are comforted by knowing that evolution has occurred". Of course evolution has occurred, which many IDer's deny. I pick and choose from the ID playbook, what I think is appropriate. Dorit and Orr and Mayr are the same breed of cat. There are alternate evolutionary explanations to complex biochemistry and complex organs. Chance hunt-and-peck, step-by-step did not occur in the Cambrian Explosion. CE is saltation. Examples of exaptation (Gould) appear later in complex organisms, to explain jumps like CE.
 
> And as for Behe's scientific output:
> http://rationalwiki.com/wiki/Michael_Behe - Have not looked at this just yet. Anything WIKI is often biased from individual input and therefore bias.
 
> Look at the section "What kind of scientist is Michael Behe." There's two excerpts from a paper published by another researcher comparing Behe's scientific output to those of a scientist universally considered as "top-notch." - I'm not noted as a 'top-notch' anything, but I can think and I'll match my IQ with anyone. This is spurious stuff. It doesn't help your point of view. Everyone has the right to interpret scientific results of other top-notch folks. Being top-notch doesn't mean they have the right conclusion. Read Thomas Kuhn. - Kenneth Miller's discussion of the human blood clotting cascade is filled with misstatements and errors, in my opinion. By the way, the feedback mechanism that stops clotting at the right moment for the type of injury has just been found. When I was in medical school there were ten clotting factors, now over 17+. All the feedback loops are still to be worked out.I'll find the reference I just mentioned and post it later.