I think the typical mathematician these days believes that 1) first-order statements about the integers, like the Twin Prime Conjecture, are either true or false: that there is a truth of the matter. But they also believe that 2) where transfinite cardinals are concerned, things like CH are theological questions on which reasonable people can disagree, so it’s fine that different axioms give different answers (or no answer) about them. But I know dissidents who 1) doubt that statements about the integers have definite truth values, and others who 2) feel that questions about transfinite numbers do have definite truth values. - Cris
On Apr 29, 2018, at 5:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The twin prime conjecture (TPC) says there are infinitely many pairs of prime numbers separated only by 2, like 3 and 5, 5 an 7, 11 and 13, etc.
No one knows if it's true or false. But whichever is the case, it might not be possible to *prove* this fact.
A similar situation is the Continuum Hypothesis (CH), which states that there is no infinity strictly larger than the size of the integers and strictly smaller than the size of the real numbers.
Kurt Goedel proved about 1940 that CH was consistent with the axioms of set theory (ZF, for Zermelo-Frankel), and Paul Cohen proved in 1963 that the negation of CH was *also* consistent with ZF.
So in this case there is *no doubt*: There exists neither a proof of CH, nor a proof of its negation ~CH, purely based on the axioms ZF+AC of set theory.
Question: --------- So, does that mean that CH is neither really true nor really false, but some third option? Or what?
I'm curious what people think about this.
—Dan
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