It's pretty clear: if there are consistent models with CH and consistent models w/o CH, then that makes CH *independent* -- to choose whichever way you please. Just like the Euclid's parallel postulate: if you like it, you get Euclid's plane geometry, whereas if you don't like it, you get one of the non-flat geometries -- e.g., Einstein. That's mathematics/mathematical logic. Now consider physics. I find CH completely antithetical to quantum theory -- e.g., "ultraviolet catastrophe" and all that. So while CH is really nice for a lot of math, I don't think that it describes post-19th C. physics very well. Curiously, much of the math of QM seems to make liberal use of CH, so something's wrong with the math at the basis of QM. I believe that this was the essence of Fredkin's, Bennett's, Feynman's (?) etc., arguments. BTW, I seriously doubt that TPC is independent, but I don't have any proofs. At 04:29 PM 4/29/2018, Dan Asimov 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.
ÂDan