FOM: Harvey and the CH story for non-mathematicians
martind at cs.berkeley.edu
Tue Jan 13 01:00:26 EST 1998
At 05:02 AM 1/6/98 +0100, Harvey Friedman wrote:
>Let me mention a primary example. Let us consider
>Godel/Cohen to be the complex of results about the consistency and
>independence of the axiom of choice and of the continuum hypothesis (in the
>presence of the axiom of choice) from ZF (ZFC). One can on one extreme
>a) state the axioms of ZFC, the axiom of choice, the continuum hypothesis,
>the notion of formal proof; also give background about the notion of
>cardinality in connection with the statement of the continuum hypothesis.
>Then state Godel/Cohen.
>Or at another extreme
>b) assert that there is a commonly accepted system of axioms and rules that
>is considered more than adequate to formalize all currently accepted
>mathematical proofs. Assert that this commonly accepted system is based on
>the single concept of set with membership, and is called standard axiomatic
>set theory. That they form the usual axioms and rules of inference for
>mathematics. Assert that the two most important problems in set theory -
>indeed the two main problems emphasized by the founder of set theory - were
>shown to be neither provable nor refutable within standard axiomatic set
>theory. More specifically, the first was shown to be neither provable nor
>refutable within standard axiomatic set theory, but has been regarded as
>having the flavor of an additional axiom. And that when added as an
>additional axiom - as is common today - the resulting system is not
>sufficient to prove or refute the second of the most important problems in
>set theory. This second of the most important problems in set theory is, in
>contrast, not of the flavor of an axiom.
I agree with Harvey that the CH story can be profitably told to a general
audience in a way that brings out its general intellectual content. (I would
prefer to regard AC as a quite different story, and to keep them separate.)
Here (in brief outline) is how I'd tell it:
Begin with suspician of mathematicians about infinity. Gauss's warning that
in mathematics it's always just a figure of speech (facon de parler).
Galileo and the one-one correspondence between poisitive integers and
squares. Then, Cantor's amazing discovery that infinity comes in different
sizes. Given any infinite collection of things, Cantor showed how to get
even bigger collections. But there was an embarassment of riches: he found
two entirely different ways to do this, and struggled to see how they fit
together. The obvious guess was that the size of the next biggest collection
after a given collection obtained either way was the same. Cantor believed
this was true and at various times tried to prove it. But he never
succeeded. Even the simplest case, where the collection we start out with is
just the familar collection of positive integers, remained unresolved. It is
this particular case that came to be known as Cantor's continuum hypothesis,
CH. The general case is called GCH. Hilbert's 1900 speech. Lo! Cantor's CH
led the rest. Real progress wasn't obtained until work by G\"odel in 1938.
Now between 1900 and the 1930s, mathematics had passed through a period of
real doubt and reexamination largely related to the difficulties of
incorporating Cantor's work into the body of mathematics. Although there are
still skeptics, practicing mathematicians had pretty much come to accept
the general theory of sets or collections as an appropriate foundation for
mathematics. All mathematical concepts could be understood as sets of some
kind. In addition a system of axioms was developed (called ZFC) from which
every theorem of ordinary mathemaics could be proved using just the basic
rules of logic. What G\"odel proved in 1938 was that CH (and even GCH) is
consistent with the ZFC axioms so CH could not be refuted from these axioms.
Considerably later, in the 1960s, Paul Cohen showed that CH couldn't be
proved from the ZFC axioms either.
A remarkable situation: all of ordinary mathematics, in all of its branches,
is a logical consequence of the ZFC axioms, but these axioms do not suffice
to decide CH. No wonder Cantor didn't succeed!
Some mathematicians who have given the matter thought believe that the
G\"odel-Cohen results settle the matter, that no final decision as to the
truth or falsity of CH is to be expected, if indeed the question has any
meaning. Others believe that all that has been established is that the ZFC
axioms are too weak to deal with CH and look forward to the discovery of new
appropriate axioms which, finally, will provide the answer to Cantor's question.
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