On 8/30/2025 3:03 AM, John Clark wrote:
On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <[email protected]>
wrote:
/>>> Actually, sometimes even in pure mathematics we can't
always reach absolute conclusions, a good example of which
is the CONTINUUM HYPOTHESIS. AG /
*>> But it has been proven you can assume that the continuum
hypothesis is true or you can assume that the continuum
hypothesis is not true, but neither assumption will produce a
contradiction to existing mathematics. It doesn't matter, so
to my mind that indicates that the continuum hypothesis is
just not very important. *
/> What's "important" here is in the mind of mathematicans. And
IMO you've misstated the result. AG/
*In 1940 Kurt Gödel proved that the truth of the Continuum Hypothesis
is consistent with existing mathematics, that is to say if it's true
then it would not change anything.*
* In 1963 Paul Cohen proved that the NEGATION of the Continuum
Hypothesis is ALSO consistent with existing mathematics. As a result
of these developments I don't think the Continuum Hypothesis is
meaningless but I do think it's unimportant. *
It may be unimportant, I don't know of anything it's led to, but you're
wrong to say that adding it as an axiom to would not change anything.
It would not change anything already proven, but it might make some
theorems, yet unformulated, decidable which were not before.
*I say that because, if neither the truth nor the falsehood of a
conjecture would change anything and if the word has any meaning then
that conjecture is "unimportant". *
*
*
*And we can't just add the Continuum Hypothesis as an axiom because an
axiom needs to be simple and self-evidently true, and the Continuum
Hypothesis is neither of those things. And the same thing could be
said about its negation.*
Where did you get that from, and who decides what is simple and
self-evident?
Brent
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