[FOM] 693:Refuting the Continuum Hypothesis?/5

Tue Jun 21 10:44:35 EDT 2016

FIRST PART CONCERNS CH, SECOND PART IS COMMENTS ON
http://www.cs.nyu.edu/pipermail/fom/2016-June/019911.html

I have placed a draft paper here:

http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#89. On the Principle of Consistent Truth, June 20, 2016, 13 pages,
draft.

concerning the Consistent Truth Program for refuting(?) the continuum
hypothesis and other(?) set theoretic problems.

There are some new items there. $ is of course old. PPNU is much
older, due to Sierpinski.

$. For all f:R into R,  there exist two reals neither of which is the
value of f at an integral shift of the other.

$'. For all f:R into R, there exist two reals neither of which is an
integral shift of the value of f at the other.

PNUF. The plane is not the union of countably many functions y = f(x)
and x = f(y).

PNII. The plane is not the union of the integral isometries of any
function y = f(x).

$,PNII, and also PNUF(Sierpinski?) are equivalent to not CH. $' is
provable in Z.

This only treats a rather baby language, and the next few steps
promise to be considerably more difficult.

HIGHER DIMENSIONS

I think the following is either well known or essentially due to
Kuratowski with his free set theorem:

THEOREM 1. R^k is the union of countably many functions from R^k-1
into R (all k types of them) if and only if |R| <= omega_k-1.

I am very PRESSED FOR TIME right now so I will throw out some
questions here. Presumably it is well known or follows at least from
Kuratowski's free set theorem, 1950, that

THEOREM?. A set is the countable function of k-ary functions (all k
types of them) if and only if its cardinality is at most omega_k.

So in analogy to what we have been discussing regarding refuting(?)
CH, we should have

PNII^k. R^k is not the  union of the integral isometries of any
function of k-1 variables (all k types of them).

THEOREM(?). PNI^k is equivalent to |R| >= omega_k-1.

But it is particularly interesting to see what can be done along these
lines in ONE dimension.

$k. For all f:R into R, there exist k reals none of which is the value
of f at an integral shift of the sum of the others.
$k'. For all f:R into R, there exist k reals where no value at an
integral shift of any one is the sum of the remaining.

This of course immediately follows from PNII^k and so presumably from
|R| >= omega_k. But how strong is $k? I played a little with $k and
some rethinking of Jonssonism seemed to surface, and then I had to
quit.

Hope you enjoy this combinatorial set theory, if it hasn't been looked at.

COMMENTS ON http://www.cs.nyu.edu/pipermail/fom/2016-June/019911.html

The feedback from core math concerning this development is limited,
but so far has been at an entirely different level of tone than
previously.

"THEOREM 1.1. Every A containedin Q[-n,0)^k has a maximal r-continuation
within Q[-n,m]^k. Furthermore, there is an effective process that starts
with finite A and k,n,m and produces a (normally infinite) maximal
r-continuation of A within Q[-n,m]^k."

Revision:

THEOREM 1.1. Every A containedin Q[-n,m)^k has a maximal r-continuation
within Q[-n,m]^k. Furthermore, there is an effective process that starts
with finite A and k,n,m and produces a (normally infinite) maximal
r-continuation of A within Q[-n,m]^k."

"DEFINITION 2.1. Let S,A,B contained Q^k. An isometry from A onto B is
a bijection f:A onto B which is Euclidean distance preserving. S is
congruent from A onto B if and only if there is an isometry from S
intersect A onto S intersect B."

There are a number of variants here, none of which change any of the
results. We can define an isometry from A onto B as a distance
preserving bijection of Q^k that sends A onto B. We can't use
Q[-n,m]^k for this purpose, so we have to use the space Q^k in which
our real ambient space Q[-n,m]^k resides. I don't like this
alternative, but it should be mentioned.

We can define "S is congruent from A onto B" as "there is an isometry
of A onto B which sends S intersect A onto S intersect B. More wordy,
slightly different, but again it doesn't affect the results.

"THEOREM 2.1. (background) In Q^k, x||y and z|w are congruent if and
only if d(x,y) = d()z,w)."

Revision:

THEOREM 2.1. (background) In Q^k, x||y and z||w are congruent if and
only if d(x,y) = d(z,w).

>From Definition 1.2:

Let A containedin Q^k. h is an isomorphism from A onto
B if and only if B containedin Q^k, and h:Q into Q is an (everywhere
defined) order preserving bijection sending A onto B by the coordinate
action. (There are some variants of this, but not for finite A, the
only case we use).

This is essentially the ONLY nontrivial definition for CONGRUENT
MAXIMAL CONTINUATION (CMC), and it is obviously trivial.

There is the alternative definitions where h maps the field of A onto
the field of B, or h maps the field of A into Q, or h maps at least
the field of A, etcetera. All such definitions are the same for finite
A,B, and for finite A. Perhaps the most intuitively pleasing
definition is the one using fields (the field of a set of k-tuples is
the set of all coordinates of all of its elements). However, it should
be noted that it requires another definition, that of field, together
with the disclaimer that this has nothing to do with fields in
algebra.

>From Definition 1.2:

B is an r-continuation of A within C if and only if
A containedin B containedin C containedin Q^k, and every <=r element
subset of B is isomorphic to some subset of A.

We can remove <=. I.e., replace <=r by r. This cuts down a symbol. It
actually makes a literal difference, even in the maximal
r-continuation within Q[-n,m]^k concept. However, it in no way affects
the results. Given the preschool biological philosophy behind all of
this, I prefer to leave it the way it is, with <=r, and remark that it
can be changed to no serious effect.

"CONGRUENT MAXIMAL CONTINUATION (specific). CMCs. For finite subsets of
Q[-k,0)^k, some maximal 2-continuation within Q[-k,k]^k is congruent
from (0,1,...,k-1)||(1,1,...,k-1) onto open (0,2,...,k)||(1,2,...,k)."

"CONGRUENT MAXIMAL CONTINUATION (parametric). CMCp. For finite subsets
of Q[-n,0)^k, some maximal r-continuation within Q[-n,m]^k is
congruent from any (i+1,...,i+k)||(i+2,i+2,...,i+k) onto any
(j+1,...,j+k)||(j+2,j+2,...,j+k), -1 <= i,j <= m-k."

Both of these statements can be strengthened by changing Q[n,0)^k to
Q[n,1)^k, without affecting any of the results. For the first one, I
still like to keep it with Q[-j,0)^k because of the preschool
biological philosophy. 0 represents the ground, and so the given
finite set is below ground, just like seeds usually are. Well, maybe
seeds can lie on the ground, or partly, and then maybe I should use
Q[-n,0]^k instead! So leave it this way instead, For the second
statement, we can also replace Q]-n,0)^k with Q[-n,1)^k or if you
like, Q[-n,0]^k. I am warming up to using Q[-k,0]^k, with the
advantage that both of the displayed intervals (two of them) are
closed.

***********************************************
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 693rd 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

Harvey Friedman