Re: Why float is called as 'float', not 'real'?

Gordon Sande <g.sande@xxxxxxxxxxxxxxxx> wrote:
On 2007-03-30 23:37:53 -0300, pa@xxxxxxxxxxxxxxxxxxxxx (Pierre Asselin) said:
Gordon Sande <g.sande@xxxxxxxxxxxxxxxx> wrote:
On 2007-03-29 18:59:37 -0300, "Lane Straatman" <invalid@xxxxxxxxxxx> said:
[ Paul Cohen's obituary ]

[ ... ] There aren't any logicians sharper than Paul
Cohen, who died last week. From his obit: "Cohen shocked the math
establishment by proving that the Continuum Hypothesis could not be decided.
The notion that conventional mathematics couldn't prove or disprove concrete
and well known assertions caused an uproar among academics." May he rest in
peace.


I expect that a better technical description is that he showed that the
continuum hypothesis to be independent of the axiom of choice.

No, the obit was pretty accurate. Kurt Goedel had shown that the
axiom of choice and the continuum hypothesis are consistent with
the other axioms of set theory; Cohen showed that their negations
are consistent as well. The reporter decided not to mention the
rest of set theory; probably a wise choice.

The obit was accurrate if understood but appeared to be misused by
Lane Staatman so more context was supplied.

Yeah, it was a bit of a non-sequitur.

Not being an expert in set theory all I can do is rely on the report
that Cohen gave in seminar when he was a sponsored traveling lecturer.
His summary was that the axiom of choice had been inadequate to prove
the continuum hypothesis inspite of various attempts. The presumption
was this was mostly a lack of skill or insight. Cohen then developed
a model in which in which AC was true but CH could be false. His
summary was that this showed CH to be independent of AC.

Ah yes, that's correct. That is indeed one of his results.

The most
interesting aspect of the seminar was his observation that within six
months of the announcement of his work the set theory technicians had
so advanced the use of his notions that he could no longer read that
literature. His summary was that he was an outsider who had made a
detour into set theory and he was fully intending to get back to real
mathematics. He suggested that the Riemann Hypothesis looked interesting
as a problem. He was a good speaker so the last suggestion may have been
playing to (or with) the audience.

Fascinating. His methods revolutionized the field and are at least as
important as the theorems he proved. I always assumed that he kept up
with the field he opened up.

The axiom of choice and the continuum hypothesis mean different things.

(If X_0 (Aleph nought) is the cardinality of the integers and X_1 =
2 ^ X_0 (cardinality of the power set of the integers) and C is the
cardinality of the reals then the continuum hypotheis is that C = X_1.

No, no. The cardinal 2^X_0 is equal to C and is strictly greater
than X_0, that was known since Cantor. X_1 is the smallest cardinal
greater than X_0, so X_0 < X_1 <= 2^X_0 = C. The CH is that the
second inequality "<=" is actually an "=".

One place the axiom of choice comes in is that it establishes
trichotomy for cardinals, a<b or a=b or a>b ; without choice,
it is possible that two cardinals might be incomparable.

Ok, that's far enough off-topic. I stop.

--
pa at panix dot com
.

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