Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
And < e is equivalent to saying that M isn’t universal.
Post’s problem was solved in the affirmative by Richard Friedberg and Albert
Muchnik, working independently in 1956. And further work by mathematical logicians such
as Gerald Sacks has shown that the degrees of unsolvability represent about as messy and
unruly an ordering as one can imagine. 10
There are infinitely many distinct degrees of unsovability between the recursive and
the universal degrees.
The degrees are dense in the sense that if P < e Q, there is an S such that P < e S and P
< e S.
And to make things gnarlier, the degrees of unsolvability don’t fall into a linear
ordering, that is, we can find P and Q such neither P ≤ e Q nor Q ≤ e P is true, that is, P can’t
emulate Q, and Q can’t emulate P,
Historical Analogy Regarding Intermediate Degrees
History. One might say that the Greeks worked primarily with real numbers that can
be expressed either as the fraction of two whole numbers, or which can be obtained by the
process of taking square roots. By the time of the Renaissance, mathematicians had learned
to work with roots of all kinds, that is, with the full class of algebraic numbers ⎯ where an
algebraic number can be expressed as the solution to some polynomial algebraic equation
formulated in terms of whole numbers. The non-algebraic numbers were dubbed the
transcendental numbers. And, for a time, nobody was sure if any transcendental numbers
existed.
Analogy. Until the mid-1950s, it seemed possible that, whatever precise notion of
degrees of unsolvability one uses, there might be only two degrees of unsolvability among
the computations: the recursive degree and the universal degree . Nobody was sure if
intermediate degrees existed, or if there were non-universal computations that had unsolvable
halting problems.
History. The first constructions of transcendental real numbers were carried out by
Joseph Liouville, starting in 1884. Liouville’s numbers were, however, quite artificial, such
as the so-called Liouvillian number 0.1100010000000000000000010000... which has a 1 in
10 See the books on recursion theory referenced at the beginning of our Technical Appendix, and see
Richard Shore, “Conjectures and Questions from Gerald Sack’s Degrees of Unsolvability”, Archive for
Mathematical Logic 36 (1997), 233-253. The paper is available online at,
http://www.math.cornell.edu/~shore/papers/pdf/sackstp3.pdf.
In point of fact, the Friedberg, Muchnik and Sacks results regarding degrees of unsolvability were
proved for an ordering ≤ T known as Turing reducibility rather than the emulation degree comparison ≤ e that I’m
using here. My ≤ e is in fact equivalent to what recursion theorists call one-one reducibility ≤ 1 . But for ≤ 1 it’s
also the case that pairs of incomparable degrees exist, that the degrees are dense, and that there are infinitely
many degrees. This follows as a corollary to the Friedberg, Muchnik and Sacks results for ≤ T because ≤ 1 is a
weaker notion than ≤ T . And independent proofs of some of the same facts about ≤ 1 result from work by J. C. E.
Dekker, also in the 1950s. Dekker was my professor in a class on Recursion Theory at Rutgers, years ago. I
used to think I had an Erdös number thanks to Dekker, who wrote a paper with my thesis advisor Erik
Ellentuck, whom I also wrote a paper with, but I was mistaken, Dekker didn’t write a paper with Paul Erdös, so
far as I know. But Ellentuck also collaborated with Richard T. Bumby on a paper, whose Erdos number is 2,
by way of R. Silverman who wrote papers with both Bumby and Erdos, so my Erdos number is (at most) 4.
Thanks to Jerry Grossman, http://personalwebs.oakland.edu/~grossman/, for explaing this to me.
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