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

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:
[ ... ] 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.

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. 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.

That was a long time ago with most of the interest being in the structure
of how mathematical theories are developed and extended rather than in
the precise technical content. It certainly was a good choice for a
travelling sponsored lecturer.

To say that
it could be either asserted or denied and that both possibilities could lead
to consistent systems has a rather different sound than to say that it could
not be decided even if both seem to have the same meaning.

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.
The axiom of choice permits choosing an element out of every set of
a collection of sets. Precise statements are longer of course!)

Unless you have misread a "this" or a "that" there is no suggestion that
the two are the same as the whole discussion was based on the fact that
they were quite distinct. There is evidently a progression of further
axioms that I know nothing about and I am sure matter to logicians. The
often (mis)quoted disassembly of a sphere into two equal parts follows
from one those. Makes for good stories about "mad mathematicians" who
are disconnected from the real world!

(We could take this over to sci.logic, but there are a lot of quacks
over there...)

Perhaps Lane Staatman might find someone there to help in his quest.





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Relevant Pages

  • Re: How do We Know that ZF is the Axiomatization that Proves everything Provable?
    ... As to determinacy, large cardinals, descriptive set theory, and the ... no claim has been made that all of the mathematics dealing with ... continuum hypothesis, i.e. that there are no infinite sets of real ...
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  • Re: Why float is called as float, not real?
    ... establishment by proving that the Continuum Hypothesis could not be decided. ... continuum hypothesis to be independent of the axiom of choice. ... rest of set theory; probably a wise choice. ... cardinality of the reals then the continuum hypotheis is that C = X_1. ...
    (comp.lang.fortran)
  • Re: Cantors Donut Paradox
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