On Saturday, August 30, 2025 at 4:17:08 AM UTC-6 Alan Grayson wrote:
On Saturday, August 30, 2025 at 4:04:00 AM UTC-6 John Clark wrote: On Fri, Aug 29, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> 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. 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". * *It's only important if we want to understand the reality of transfinite cardinal numbers, and the conclusion that there is no infinite set strictly between the integers and the real numbers. AG* *But of course, that's what we don't know! AG * *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.* *John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis>* aoc -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/6b3d2ec6-3849-4fcb-a571-f6e725f9a1d6n%40googlegroups.com.
