Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
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
I have read (iirc) that Gödel thought CH was either true or false (I think he favored false), and that its independence of ZFC revealed the inadequacy of ZFC. In contrast, Cohen felt the independence of CH ended the discussion. There is lots more (and more than I want to look at) in the Stanford Encyclopedia of Philosophy, at this link: https://plato.stanford.edu/entries/continuum-hypothesis/ On Sun, Apr 29, 2018 at 7:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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.
—Dan
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Having looked a bit at this article: I think the typical modern mathematician is a pluralist (we can have it both ways) with respect to transfinite numbers, but a non-pluralist (i.e., there is a truth of the matter, and if we can’t prove it either way then our axioms are at fault) about first-order sentences regarding the integers. However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> - Cris
On Apr 29, 2018, at 8:03 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
I have read (iirc) that Gödel thought CH was either true or false (I think he favored false), and that its independence of ZFC revealed the inadequacy of ZFC. In contrast, Cohen felt the independence of CH ended the discussion. There is lots more (and more than I want to look at) in the Stanford Encyclopedia of Philosophy, at this link: https://plato.stanford.edu/entries/continuum-hypothesis/
On Sun, Apr 29, 2018 at 7:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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.
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Apr 30, 2018, at 3:21 PM, Cris Moore <moore@santafe.edu> wrote:
However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>>
Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult. But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here. - Cris
On Mon, Apr 30, 2018 at 3:28 PM, Cris Moore <moore@santafe.edu> wrote:
On Apr 30, 2018, at 3:21 PM, Cris Moore <moore@santafe.edu> wrote:
However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>>
Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult.
But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here.
Well, physically we can't build a Turing machine, so it depends on your idealization. Once you're idealizing, "really" stops having much meaning. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
On Sun, Apr 29, 2018 at 8:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
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.
I don't think it's that simple. ZFC is an attempt to codify our intuitions about what a set is. There may be other things about sets that people agree are "true" in the intended model---that is, there could be new axioms that become generally accepted. While there are lots of non-isomorphic models of ZFC, it's pretty clear to me that some of them are other systems that happen to satisfy the axioms that are not the intended object that ZFC is an attempt to axiomatize. For example ZFC has countable models, but I think people generally agree that there are an uncountable number of sets---it's just an unfortunate property of first-order logic that any system with a model has a countable model. And you can't prove Consis(ZFC) in ZFC, so there are models where ZFC is true, but Consis(ZFC) is false, These are pretty clearly not the models we are interested in. I don't think anyone is interested at all in what you can prove from ZFC + not(Consis(ZFC)), while if you could prove CH from ZFC + Consis(ZFC) (unfortunately, you provably can't), then I think almost all mathematicians would accept CH as true. .
Now consider physics. I find CH completely antithetical to quantum theory -- e.g., "ultraviolet catastrophe" and all that.
What does the ultraviolet catastrophe have to do with CH?
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.
I've never seen any QM math that made use of CH; can you point me to a reference to one of these uses? Andy
participants (5)
-
Andy Latto -
Cris Moore -
Henry Baker -
James Buddenhagen -
Mike Stay