[some big basic results, with sketch of proofs]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
*Computability*
Entropy, randomness (d'apres Carnot, Clausius, Boltzmann and Gibbs,
Shannon, AIT: Solomonoff/Kolmogorov/Chaitin/Martin-L\"of/Levin)
thermodynamics, information theory, probability
(The resolution of the paradox of Maxwell's demon.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The most celebrated theorems in logic (to this non-logician) are
widely
- Cantor's uncountability
- Goedel's incompleteness/inconsistency theorem
(the result, not the proof
this has a definitive write-up
(see ???)
sketch:
* number all statements
(defined *syntactically* as "well formed formulae" or wff, pronounced "wiff")
* hence you can have self-referential statements
* encode Russell's paradox
the "incomplete \emph{or} inconsistent" duality is little-appreciated,
since inconsistent logic tends to revolt people
added note:
any inconsistencies don't affect arithmetic (proved in 70s/80s?)
- Cohen's proof of the independence of Axiom of Choice and (Generalized) Continuum
Hypothesis
(via set forcing; again, the result, but not the proof)
Contrast this with proofs of non-Euclidean geometries:
those were constructive: they produced a Euclidean geometry,
and a non-Euclidean, hence parallel postulate was independent.
Cohen's proof is non-constructive, for a very good reason.
First, you produce the constructible universe (where AC and GCH hold).
Now, *this is all that you can construct!* (as the name indicates).
You're not going to have much luck constructing a model of set theory
where they fail.
Instead, Cohen showed that given a contradiction in ZF+~AC or ZF+~GCH
(and so forth) that you could force it down into a contradition just
in ZF.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- independence of other axioms
Note that other axioms in ZF (which had at some point in time been
considered controversial) had been proven to be independent of the
other axioms.
[is my chronology right? This was before forcing, no?
I think the method was different.]
Along the lines of GCH, cardinal arithmetic ($2^{\aleph_x}$) has been extensively
studied, and with a few exceptions (limit cardinals), arithmetic can
be whatever it feels like.
[BTW, you can do combinatorics on cardinals,
so long as you've categorified them properly (as numbers of maps
between spaces), and they all reduce to 2^\aleph stuff, I think;
see old, old emails]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions of independence}
One conclusion of the independence of these (and other) axioms is
by analogy with non-Euclidean geometry, we can and should study
all of these possible logics.
I'm told that in ~2005, a very respected and elder logician
wrote a well-regarded article
[I think printed in some form in the Notices]
that said (among other things?)
that we should *not* think of these other logics as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Notes on incompleteness
Some theories *are* complete, including:
- Euclidean geometry
- real closed field
- algebraically closed, characteristic zero
That is, there are first order axiomatizations of these,
but you can't encode the natural numbers with addition and
multiplication:
you can't tell if a given real number is an integer *in a first order way*
(this follows from the fact that the theory is complete -- otherwise
Godel would apply)