I Am a Strange Loop

Page 137 suppose you wanted to know if statement X is true or false… The dream of a mechanical method for reliably placing statements into two bins — ‘true’ and ‘false’ — is known as the quest for a decision procedure. The absolute nonexistence of a decision procedure for truth (or for provability) is discussed in [DeLong], [Boolos and Jeffrey], [Jeffrey], [Hennie], [Davis 1965], [Wolf], and [Hofstadter 1979].

Page 139 No formula can literally contain… [Nagel and Newman] presents this idea very clearly, as does [Smullyan 1961]. See also [Hofstadter 1982].

Page 139 an elegant linguistic analogy… See [Quine] for the original idea (which is actually a variation of Gödel’s idea (which is itself a variation of an idea of Jules Richard (which is a variation of an idea of Georg Cantor (which is a variation of an idea of Euclid (with help from Epimenides))))), and [Hofstadter 1979] for a variation on Quine’s theme.

Page 147 “…and Related Systems (I)”… Gödel put a roman numeral at the end of the title of his article because he feared he had not spelled out sufficiently clearly some of his ideas, and expected he would have to produce a sequel. However, his paper quickly received high praise from John von Neumann and other respected figures, catapulting the unknown Gödel to a position of great fame in a short time, even though it took most of the mathematical community decades to absorb the meaning of his results.

Page 150 respect for …the most mundane of analogies… See [Hofstadter 2001] and [Sander], as well as Chapter 24 in [Hofstadter 1985] and [Hofstadter and FARG].

Page 159 X’s play is so mega-inconsistent… This should be heard as “X’s play is omega-inconsistent”, which makes a phonetic hat-tip to the metamathematical concepts of omega-inconsistency and omega-incompleteness, discussed in many books in the Bibliography, such as [DeLong], [Nagel and Newman], [Hofstadter 1979], [Smullyan 1992], [Boolos and Jeffrey], and others. For our more modest purposes here, however, it suffices to know that this “o”-containing quip, plus the one two lines below it, is a play on words.

Page 160 Indeed, some years after Gödel, such self-affirming formulas were concocted… See [Smullyan 1992], [Boolos and Jeffrey], and [Wolf].

Page 164 Why would logicians …give such good odds… See [Kneebone], [Wilder], and [Nagel and Newman], for reasons to believe strongly in the consistency of PM-like systems.

Page 165 not only although…but worse, because… For another treatment of the perverse theme of “although” turning into “because”, see Chapter 13 of [Hofstadter 1985].

Page 166 the same Gödelian trap would succeed in catching it… For an amusing interpretation of the infinite repeatability of Gödel’s construction as demonstrating the impossibility of artificial intelligence, see the chapter by J. R. Lucas in [Anderson], which is carefully analyzed (and hopefully refuted) in [DeLong], [Webb], and [Hofstadter 1979].

Page 167 called “the Hilbert Program”… See [DeLong], [Wolf ], [Kneebone], and [Wilder].

Page 170 In that most delightful though most unlikely of scenarios… [DeLong], [Goodstein], and [Chaitin] discuss non-Gödelian formulas that are undecidable for Gödelian reasons.

Page 172 No reliable prim/saucy distinguisher can exist… See [DeLong], [Boolos and Jeffrey] , [Jeffrey], [Goodstein], [Hennie], [Wolf ], and [Hofstadter 1979] for discussions of many limitative results such as this one (which is Church’s theorem).

Page 172 It was logician Alfred Tarski who put one of the last nails… See [Smullyan 1992] and [Hofstadter 1979] for discussions of Tarski’s deep result. In the latter, there is a novel approach to the classical liar paradox (“This sentence is not true”) using Tarski’s ideas, with the substrate taken to be the human brain instead of an axiomatic system.

Page 172 what appears to be a kind of upside-down causality… See [Andersen] for a detailed technical discussion of downward causality. Less technical discussions are found in [Pattee] and [Simon]. See also Chapters 11 and 20 in [Hofstadter and Dennett], and especially the Reflections. [Laughlin] gives fascinating arguments for the thesis that in physics, the macroscopic arena is more fundamental or “deeper” than the microscopic.