[FOM] 695: Refuting the Continuum Hypothesis?/6

[Harvey Friedman]

Here are some reflective remarks about this Consistent Truth program:

[1] H. Friedman,
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#89.

The traditional Platonist point of view, that there is a matter of
fact absolute truth for sentences of set theory, even sentences about
(V(omega + 2),epsilon), is in a weakening position from most points of
view. Of course, it still has its strong adherents.

An interesting way of arguing against matter of fact, absolute truth
for set theory is to give examples of concepts for which this matter
of fact, absolute truth, is easy to reject - and then inquire as to
why set theory should be any different. Of course, when pointing to a
concept for this purpose, the inevitable response from many is simply
to say that "this is different - your notion is subjective, whereas
arbitrary sets are objective". So there remains the basic
methodological question of assigning burdens to opposing parties in
such an argument.

Perhaps relevant is that in any kind of physical science, attempts to
maintain matter of fact absolute truth have historically run into
grave difficulties. Consider the *actual length of this steel rod*, or
to eliminate some obvious difficulties connected with units of
measurement, that "these two steel rods have the same length". There
is practically no end to the difficulties that arise with this notion
when analyzed carefully. Or for that matter, just about any notion
whatsoever that is not considered to have been faithfully reduced to
the purely mathematical. And are such considerations from the realm of
physical science and just about everywhere, really relevant to the
issues of matter of act absolute truth in set theory? Even this can be
argued for and against without decisive outcome.

So unless you really want to adhere to the matter of fact absolute
truth of set theory (say at the level of V(omega + 2) if not V), you
should be open to the idea of finding some sort of basic principle for
assigning truth values to statements in set theory that are not
decided by ZFC.

MY TENTATIVE BELIEF: ZFC appears to be the rough limit or maybe exact
limit of what can be reasonably argued to be "obvious upon reflection
on the notion of set". It appears that adopting axioms or assigning
truth values to statements, beyond what is provided by ZFC, have to be
argued for on very different grounds that "obvious upon reflection on
the notion of set". The Consistent Truth approach is being offered up
as a new tool for arguing on such "very different grounds".

HOWEVER: I still do try to come up with some sort of intrinsically
fundamental considerations or clarifications or elaborations of what
sets are that give axioms going beyond ZFC - with some promising
lines. But I have to admit that what I look at along these lines is
much more problematic than ZFC. Much more promising seems to be a
related enterprise. Namely, coming up with fundamental ideas that are
argued to be more fundamental than set theory, leading to theories of
great strength in which one builds models of ZFC extended with large
cardinals. I have called this general enterprise Concept Calculus,
where I use first order predicate calculus as the mode of
presentation. This provide models of and consistency proofs for ZFC
extended by large cardinal hypotheses. I want to revisit Concept
Calculus taking second and higher order logic as the basic mode of
presentation, where one is always fundamentally using the standard
corresponding first order formulations, viewed as critical important
fundamental approximations. Of these ideas, I think that the Exploding
Universe idea looks most attractive at this point.

FINITE EXTRAPOLATION. There is a possible reason why ZFC is so
fundamental. It appears to be exactly what you get when you want to
lift what is true from the finite to the infinite. I posted theorems
to this effect on the FOM some time ago, and should revisit this
matter in the near future. One technical problem I ran into is that
for the formal results, there was a problem in incorporating the
axioms of choice and foundation, and so I only get to ZF with
foundation replaced by the very weak form of foundation,asserting that
there is no finite epsilon chain. But I think the stuff should be
reworked so that foundation and choice are fully incorporated. After
all, both foundation and choice are obvious in the finite. So I need
to refine my extension principle from the finite to the infinite to
accommodate them.

Back to the Principle of Consistent Truth. So this is being offered,
not as being based on any kind of fundamental reflection on or
clarification of or elaboration of the notion of set. But rather as a
new principle for assigning truth values to certain statements of set
theory that are not proved or refuted by the usual ZFC axioms for set
theory.

PRINCIPLE OF CONSISTENT TRUTH. Any "conforming" sentence of set theory
formally consistent with the very powerful but limited subsystem ZC of
ZFC, is true.

In [1], I stated it this way:

CONSISTENT TRUTH FOR L. CT(L). Any sentence of the language L
consistent with ZC is true.

and proved that CT(L) is provably equivalent to notCH over ZFC for a
particular limited language L surrounding my $. I conjectured that
this is the case for considerably stronger such languages L, but that
it breaks at some point. I conjectured that even after it breaks,
where we have Inconsistent Truth!,

$. For every f:R into R there exist (distinct) x,y in R, neither being
the value of f at an integral shift of the other.

NOTE: I have added the option "distinct", as the proof in [1] and
http://www.cs.nyu.edu/pipermail/fom/2016-June/019905.html of the
equivalence of $ goes through with or without "distinct".

So there is the matter of which languages L should be chosen for
analysis of CT(L) with an eye to the adoption of CT(L). I.e., what is
a "conforming language L"?

The idea is for L to *conform* to the normal grammar of normal
mathematics. In [1] I discussed the normal grammar of fundamental
classical real analysis. There is a vivid asymmetry here between
sentences and their negations, so that conforming languages L will
certainly not be closed under negation. In fact, generally throughout
mathematics, we see

*For all big objects, something simple holds among its parts*

In [1], I mention a bunch of examples of this that are icons of the
mathematics curriculum where the big objects are the continuous
functions from R into R. Of course, when aiming at notCH, we want to
consider all functions from R into R.

Now what would the dual class look like?

*There exist a big object for which something simple holds among its parts*

And why don't we simply use languages L* based on this and then CT(L*)
is provably equivalent to CH?

Well, L is the language confirming to fundamental theorems, and L* is
the language conforming to fundamental counterexamples. And
fundamental theorems are more important and more numerous and more
dominating in normal mathematics than are counterexamples.

NOW, there is a related category of mathematical theorems that do
start with an existential quantifier, but have an additional crucial
feature. Whereas these theorems are less numerous, they can also be
fundamentally very imports. I am thinking of

**There exist a big object, which is unique up to isomorphism, for
which something simple holds among its parts**

For instance, in the treatment of the fundamental number systems:
positive integers, integers, rationals, reals, complexes, real
algebraics, complex algebraics, and also Hilbert space, etcetera.

But here it would appear that for natural languages L' conforming to
** above, CT(L) may be vacuous.And it appears that when we get L' for
which CT(L') is not vacuous, it is consistent and does not imply CH.

Concerning the mainline languages L envisioned in [1]. we also
envisioned, after a long series, of ever more expressive languages,
that we run into CT(L) being inconsistent. But here we expect that the
inconsistencies must involve extremely complicated sentences in the L
that arise. Unconscionably more complicated that the motivating $. So
the Consistent Truth program gets restarted by cutting down the
languages to sentences of reasonable complexity.

***********************************************
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 695th 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
686: Refuting the Continuum Hypothesis?/4  6/616  9:29PM
687: Large Cardinals and Continuations/7  6/7/16  10:28PM
688: Large Cardinals and Continuations/8  6/9/16  11::41PM
689: Large Cardinals and Continuations/9  6/11/16  2:51PM
690: Two Brief Sketches  6/13/16  1:18AM
691: Large Cardinals and Continuations/10  6/13/16  9:09PM
692: Large Cardinals and Continuations/11  6/15/16  10:22PM
693: Refuting the Continuum Hypothesis?/5  6/21/16  10:44AM
694: Large Cardinals and Continuations/12  6/29/16  11:46PM

Harvey Friedman