[math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
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
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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I think there is no really true or really false in mathematics; there's only provable, disprovable, and undecidable relative to the axioms and inference rules; and there's true or false in a model. Brent On 4/29/2018 4:29 PM, 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.
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Sun, Apr 29, 2018 at 5:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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.
Well, obviously CH is independent of ZF; there are models in which it holds, and models in which it doesn't. By your use of the word "really", you bring up the question of whether we can model ZF with physics somehow. It's conceivable that the universe itself is infinite, even if the observable universe isn't; we can think of the observable universe as something like our Moore neighborhood. However, the universe is expanding and accelerating, so there are parts of the universe that are now causally separated. The Bekenstein bound says there's a limit on how much information one can pack into a volume before it collapses into a black hole, so it looks like reality's limited to at best countable models if you include the whole universe and only finite models if you include only the observable universe. Since there aren't finite models of ZF, I don't think ZF describes reality. (That said, nobody's quite sure what to make of the cardinality of the set of probability amplitudes yet; there aren't any models I know of where probability amplitude is quantized, but neither is it something you can directly measure.) -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (5)
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Andres Valloud -
Brent Meeker -
Cris Moore -
Dan Asimov -
Mike Stay