Notes and References for the 10/31/11 Post on Kurt Gödel

The notes are (only) on things that might not be easy to find via Google, or that reflect some unstated intentions. They go in order down the post.


1.9 degrees Kelvin is the expected temperature of the yet-undetected Cosmic Neutrino Background Radiation. OK, 1.95K.

I doubt that the Penrose-Hameroff consciousness model is in any shape to countenance neutrinos, but hey, this is fiction and the truth of 95% of stuff in the universe being currently unknown is stranger.

Of course we are riffing on the OPERA claim of super-c neutrinos; if your reaction to their possibly conducting information to/from the past is "In your dreams!", we'll say: Precisely. As for the time of year, I originally had All Souls' Day (Nov. 2) in mind, but that would go better with Gödelian ideas on consciousness which we put off to later. Each of the three interview sections was begun by Dick and finished by me.

Incompleteness Theorems

1971 was chosen because it is a clean 40 years prior, for Cook's Theorem, and for being after Paul Cohen's independence results---which are on tap. That it was right after KG's work on his "Ontological Argument" was not noticed until mid-post.

Bio details on eating, raising his discovered flaw in the Constitution on his 1947 citizenship day, having Tarski's Theorem, and never meeting Turing are readily looked up.

Sources seem agreed that Gödel never met Turing. In fact the closest he came was visiting the IAS (Institute for Advanced Study, in Princeton) in the fall of 1938, Turing having left in summer.

My source for KG's doubt on Turing's argument is Oron Shagrir's essay Gödel on Turing on Computability. It focuses on a "short note [1972a]" by Gödel titled "A philosophical error in Turing's work", which KG would have been thinking about in late 1971. Shagrir's essay makes clear that KG regarded Turing's paper as correct for mechanical computability but queried Turing's assumption that there are only finitely many distinguishable states of a human mind. (Also note per Robert Soare's 2009 essay on Turing that Gödel initially rejected Church's argument.)

The accompanying disparagement of Turing machines is my invention, though consistent with Gödel's own approach to recursion and the related ideas as stated. To be fair to Steve Cook, he has led a great program joined by others at Toronto to re-base polynomial time and related classes on systems of logic and recursion that are more computationally structured.

The reference at the end is to Janna Levin's 2006 semi-biographical novella, A Madman Dreams of Turing Machines.

The "Lost Letter"

The Einstein photo is here, KG chiding Einstein is imagined.

The text of the "Lost Letter" itself is on the blog site itself; see also Juris Hartmanis 1989 Bulletin of the EATCS column.

The interpretation of the time bounds including "N" is mine, but consistent with the text of the letter and attitudes of the time, IMHO.

Time Travel

The paper by the "two other people" is here; in it is a personal communication from Gödel saying he had one of their "temporally benign" models. See also here (page 84).

We recently blogged on Lambda > 0. The consequence for observability of the Big Bang is via this paper and Scientific American article by Lawrence Krauss and Robert Scherrer.

Here is a paper with beautiful diagrams visualizing Gödel's solution to Einstein's equations. My own version of Gödel describing his paper is most strongly influenced by Franz Embacher's short paper on the history, see also his slides. The importance of Mach's principle is emphasized by many sources, including notably Brian Greene in The Fabric of the Cosmos.

Walter Isaacson's 2007 biography Einstein: His Life and Universe discusses efforts to "convert" Einstein to quantum mechanics on page 515, and for "...he kept writing equations" here is page 511: "In his quest for a unified field theory, he [Einstein] still had no compelling physical guide his way, so his endeavors remained a groping through clouds of abstract mathematical equations with no ground lights to orient him." The "12 years" referred to are 1905 to 1917; one could also say 1907 or 1912 to late 1915. Brian Greene on p70 of The Fabric of the Cosmos called it a "ten-year struggle", and Einstein famously got the field equations wrong the first time, in 1915.

The last paragraphs refer to Gödel's rendition in modal temporal logic of (Leibniz' version of) St. Anselm's "Ontological Proof" of the existence of God. Modal temporal logic was introduced by Arthur Prior, who was a centerpiece of the second of three lectures by Moshe Vardi which we covered here, though I did not mention Prior in that post. Gödel's notes on his proof date to 1970 with the final version listed as "[1972a]", so he would have been thinking about it in late 1971. Since Gödel was "baptized Lutheran" and Anselm is on the Lutheran Calendar of Saints, Gödel could have used the "St."

Christopher Small of the University of Waterloo has a huge treatment of Kurt Gödel's form of the argument in several artwork-studded long webpages beginning here. About Anselm's original form, my old Oxford chess team captain put forth a novel objection here; see this for a rebuttal and Peter Millican's rejoinder here.

The analogy between Anselm's argument and collection in set theory is my doing, but I think I'm just filling in reasoning that I'd probably find in Gödel's record if I took full trouble to look. Set theory has the "V = L? Question" of equality between John von Neumann's universe and Gödel's Constructible Universe, and the debate about whether it's true "in reality" rages on. The leap from there to thoughts about God was already made by Georg Cantor with his "Ultimate Aleph".

I have Gödel scaling back his claim from a proof of God's existence G to claiming only that G is a proposition such that ◊G ⇒ ☐G. That is the limit of what is consistent with my own partial fideism expressed here. From a standard Christian perspective my putting into his mouth the words "Well, you don't have to believe in it!" is chiding, but I tend toward David Stern's understanding of pstisis as trust in the New Testament, which for me is a different matter from intellectual assent. There is also joy in finding a really good formulation of what someone else purposed to say, one good enough to provoke the artful response by Small, and to spur some insights about monotone Boolean functions, for instance.

Open Questions

Reference is to the Harry Potter "Pensieve".