[FOM] 686:Refuting the Continuum Hypothesis?/4
Harvey Friedman
hmflogic at gmail.com
Mon Jun 6 21:29:13 EDT 2016
We give a clear statement of the approach to the continuum hypothesis
being offered up here for investigation, with a pointer also to the
approach applied to PD.
REFUTING THE CONTINUUM HYPOTHESIS?
What follows here is another, perhaps more digestible, general
discussion, along the lines of
http://www.cs.nyu.edu/pipermail/fom/2016-May/019846.html
We have identified a number of successively richer languages L for
expressing properties of functions f:R into R, that appropriately
reflect the logical structure of the bulk of celebrated theorems
within the realm of continuous f:R into R. For these languages L, we
have investigated the following statements:
CONSISTENT TRUTH FOR L. CT(L). Every sentence in L that is consistent
with ZFC is true.
BOREL TRUTH FOR L. BT(L). Every sentence in L that is true for Borel
functions is true.
For these languages, we have found hat CT(L) and BT(L) are consistent
with ZFC, and refute the continuum hypothesis over ZFC. In fact, we
have found that CT(L) and BT(L) are equivalent to the negation of the
continuum hypothesis, over ZFC.
For each such language L, we can of course consider its dual L*.
CT(L*) and BT(L*) behave very differently. We have found that CT(L*)
is also consistent with ZFC, and proves the continuum hypothesis over
ZFC. In fact, we have found that CT(L*) is equivalent to the continuum
hypothesis. BT(L*) is a triviality.
Should we advocate the CT(L) or should we advocate the CT(L*)?
It makes sense to advocate the CT(L) since the L's are motivated by
the logical structure of celebrated classical mathematics. Obviously
the L*'s are in sharp conflict with the logical structure of
celebrated classical mathematics.
Since the CT(L)'s refute the continuum hypothesis, we have in a sense
refuted the continuum hypothesis.
Of course, as we consider richer and richer such L's, we may in fact
find that CT(L) at some point refutable within ZFC.
But we believe that such inconsistencies with ZFC are going to arise
only by using instances of CT(L) of very high complexity. There are
always core transparent low complexity instances of CT(L) that are
equivalent to the negation of the continuum hypothesis. So this
approach to refuting the continuum hypothesis survives via crude
complexity considerations.
Another way of saying this is that the L's can be based only on low
complexity sentences. Then we conjecture that we always have
consistent CT(L) and they are all equivalent to the negation of the
continuum hypothesis. In fact, most of the L's for which we have
worked out CT(L), BT(L), consist of a finite number of low complexity
statements.
Obviously by duality, the same can be said for the CT(L*) and proving
the continuum hypothesis. But we have a principled way of choosing
between the L's and the L*'s, namely: alignment with the actual
logical structure of fundamental classical mathematics. And we
envision the likelihood that they may more striking ways to choose
between the L's and the L*'s.
The core statement that is equivalent to the negation of the continuum
hypothesis is as follows.
$. For all f:R into R there exists x,y in R such that x (y) is not f
at any integer shift of y (x)
We work with $, but expect to work with the following as well.
$'. For all f:R into R there exists x,y in R such that x (y) is not f
at any multiple of y (x).
See http://www.cs.nyu.edu/pipermail/fom/2016-May/019805.html for a
calculation of a CT(L), BT(L). There I use the term list
ax+bn+c
ay+bn+c
f(ax+bn+c)
f(ay+bn+c)
where the formulas are finite conjunctions of equations and
inequations. I am now working this up with the larger term list
ax+by+cn+d
f(ax+by+cn+d)
I have also started to apply this general methodology, with different
details, to "prove" PD. See
http://www.cs.nyu.edu/pipermail/fom/2016-April/019775.html
***********************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 686th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html
600: Removing Deep Pathology 1 8/15/15 10:37PM
601: Finite Emulation Theory 1/perfect? 8/22/15 1:17AM
602: Removing Deep Pathology 2 8/23/15 6:35PM
603: Removing Deep Pathology 3 8/25/15 10:24AM
604: Finite Emulation Theory 2 8/26/15 2:54PM
605: Integer and Real Functions 8/27/15 1:50PM
606: Simple Theory of Types 8/29/15 6:30PM
607: Hindman's Theorem 8/30/15 3:58PM
608: Integer and Real Functions 2 9/1/15 6:40AM
609. Finite Continuation Theory 17 9/315 1:17PM
610: Function Continuation Theory 1 9/4/15 3:40PM
611: Function Emulation/Continuation Theory 2 9/8/15 12:58AM
612: Binary Operation Emulation and Continuation 1 9/7/15 4:35PM
613: Optimal Function Theory 1 9/13/15 11:30AM
614: Adventures in Formalization 1 9/14/15 1:43PM
615: Adventures in Formalization 2 9/14/15 1:44PM
616: Adventures in Formalization 3 9/14/15 1:45PM
617: Removing Connectives 1 9/115/15 7:47AM
618: Adventures in Formalization 4 9/15/15 3:07PM
619: Nonstandardism 1 9/17/15 9:57AM
620: Nonstandardism 2 9/18/15 2:12AM
621: Adventures in Formalization 5 9/18/15 12:54PM
622: Adventures in Formalization 6 9/29/15 3:33AM
623: Optimal Function Theory 2 9/22/15 12:02AM
624: Optimal Function Theory 3 9/22/15 11:18AM
625: Optimal Function Theory 4 9/23/15 10:16PM
626: Optimal Function Theory 5 9/2515 10:26PM
627: Optimal Function Theory 6 9/29/15 2:21AM
628: Optimal Function Theory 7 10/2/15 6:23PM
629: Boolean Algebra/Simplicity 10/3/15 9:41AM
630: Optimal Function Theory 8 10/3/15 6PM
631: Order Theoretic Optimization 1 10/1215 12:16AM
632: Rigorous Formalization of Mathematics 1 10/13/15 8:12PM
633: Constrained Function Theory 1 10/18/15 1AM
634: Fixed Point Minimization 1 10/20/15 11:47PM
635: Fixed Point Minimization 2 10/21/15 11:52PM
636: Fixed Point Minimization 3 10/22/15 5:49PM
637: Progress in Pi01 Incompleteness 1 10/25/15 8:45PM
638: Rigorous Formalization of Mathematics 2 10/25/15 10:47PM
639: Progress in Pi01 Incompleteness 2 10/27/15 10:38PM
640: Progress in Pi01 Incompleteness 3 10/30/15 2:30PM
641: Progress in Pi01 Incompleteness 4 10/31/15 8:12PM
642: Rigorous Formalization of Mathematics 3
643: Constrained Subsets of N, #1 11/3/15 11:57PM
644: Fixed Point Selectors 1 11/16/15 8:38AM
645: Fixed Point Minimizers #1 11/22/15 7:46PM
646: Philosophy of Incompleteness 1 Nov 24 17:19:46 EST 2015
647: General Incompleteness almost everywhere 1 11/30/15 6:52PM
648: Necessary Irrelevance 1 12/21/15 4:01AM
649: Necessary Irrelevance 2 12/21/15 8:53PM
650: Necessary Irrelevance 3 12/24/15 2:42AM
651: Pi01 Incompleteness Update 2/2/16 7:58AM
652: Pi01 Incompleteness Update/2 2/7/16 10:06PM
653: Pi01 Incompleteness/SRP,HUGE 2/8/16 3:20PM
654: Theory Inspired by Automated Proving 1 2/11/16 2:55AM
655: Pi01 Incompleteness/SRP,HUGE/2 2/12/16 11:40PM
656: Pi01 Incompleteness/SRP,HUGE/3 2/13/16 1:21PM
657: Definitional Complexity Theory 1 2/15/16 12:39AM
658: Definitional Complexity Theory 2 2/15/16 5:28AM
659: Pi01 Incompleteness/SRP,HUGE/4 2/22/16 4:26PM
660: Pi01 Incompleteness/SRP,HUGE/5 2/22/16 11:57PM
661: Pi01 Incompleteness/SRP,HUGE/6 2/24/16 1:12PM
662: Pi01 Incompleteness/SRP,HUGE/7 2/25/16 1:04AM
663: Pi01 Incompleteness/SRP,HUGE/8 2/25/16 3:59PM
664: Unsolvability in Number Theory 3/1/16 8:04AM
665: Pi01 Incompleteness/SRP,HUGE/9 3/1/16 9:07PM
666: Pi01 Incompleteness/SRP,HUGE/10 13/18/16 10:43AM
667: Pi01 Incompleteness/SRP,HUGE/11 3/24/16 9:56PM
668: Pi01 Incompleteness/SRP,HUGE/12 4/7/16 6:33PM
669: Pi01 Incompleteness/SRP,HUGE/13 4/17/16 2:51PM
670: Pi01 Incompleteness/SRP,HUGE/14 4/28/16 1:40AM
671: Pi01 Incompleteness/SRP,HUGE/15 4/30/16 12:03AM
672: Refuting the Continuum Hypothesis? 5/1/16 1:11AM
673: Pi01 Incompleteness/SRP,HUGE/16 5/1/16 11:27PM
674: Refuting the Continuum Hypothesis?/2 5/4/16 2:36AM
675: Embedded Maximality and Pi01 Incompleteness/1 5/7/16 12:45AM
676: Refuting the Continuum Hypothesis?/3 5/10/16 3:30AM
677: Embedded Maximality and Pi01 Incompleteness/2 5/17/16 7:50PM
678: Symmetric Optimality and Pi01 Incompleteness/1 5/19/16 1:22AM
679: Symmetric Maximality and Pi01 Incompleteness/1 5/23/16 9:21PM
680: Large Cardinals and Continuations/1 5/29/16 10:58PM
681: Large Cardinals and Continuations/2 6/1/16 4:01AM
682: Large Cardinals and Continuations/3 6/2/16 8:05AM
683: Large Cardinals and Continuations/4 6/2/16 11:21PM
684: Large Cardinals and Continuations/5 6/3/16 3:56AM
685: Large Cardinals and Continuations/6 6/4/16 8:39PM
Harvey Friedman
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