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

Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker
the string n is found on the Turing machine tape.
Definition. The integer n is Chaitin-Kolmogorov incompressible if for no e < n do we
have T e (0) = n.
Rather than working with arbitrary kinds of computations, the proof of Chaitin’s
theorem focuses on Turing machines with two symbols. The key notion behind the proof is
that both the programs and the outputs of these Turing machines can both be thought of as
strings of 0s and 1s. Because of this, a program can itself be regarded as possibly being the
output of a smaller program. For any P, let P* be the program such that if P(0) halts with
output Q, Q is treated as a program and Q(0) is run. There will be some fixed number
overhead such that size(P*) = size(P) + overhead.
Now Chaitin argues that if of, once again, the unsovability of the halting problem.
know that a string is a name for another string, you need to know that, when fed into the
universal computer, the name produces a computation that writes the target string and halts.
So proving that a number is Chaitin complex involves proving that certain computations
don’t halt and therefore fail to name the string. And, as we know by now, some
computations never halt even though you can’t prove this to be so.
A more transparent way to get at the proof of Chaitin’s theorem is to use the
following line of thought. If that if Chaitin’s theorem weren’t true, then T could prove there
is a number called BerryT which is definable as “the first number that doesn’t have a
definition shorter than itself.” And then T would be in trouble as this proof provides a short
name for BerryT.
People Who Email Me About Gödel’s Proof
I’ve written about Gödel’s proof before, like in Infinity and the Mind and in Mind
Tools, and I get email about it every few months, usually from people who think the proof is
wrong. Guys wanting to tell me they’re smarter than Kurt Gödel. “Oooooh kay.”
Adding Quantifiers
The simple sentences G I’ve been discussing can be characterized as having a single
kind of quantifier ⎯ where a quantifier is a phrase like “for all n” or “for some n”.
Although we’ve speaking of G as having the form “For all n, g[n] is false,” we could also
express it as the negation of a sentence of the form “There is an n such that g[n].”
Adding more quantifiers of the same kind doesn’t change much. But if we allow
alternating quantifiers we get into a richer zone of undecidability.
Course-of-values Simulation
A slightly stronger form of simulation is what one might call course-of-values
simulation. Say that Big with translation function tr is a course-of-values-simulation of
Small iff, for any s0, if the computation Small(s0) produces, at some increasing sequence of
times, the output states s1, s2, s3, etc., then the Big(tr(s0)) will also produce, for some
increasing sequence of times, the states tr(s1), tr(s2), tr(s3), etc.
References
David Deutsch, The Fabric of Reality: The Science of Parallel Universes — and Its
p. 106