[FOM] 696: Refuting the Continuum Hypothesis?/7

[Harvey Friedman]

This is about a MULTIDIMENSIONAL version of $.

Recall

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

$. For all f:R into R, there exist two (distinct) reals, neither being
f at an integral shift of the other.

I now want to look at

$[k]. For all f:R into R, there exist k (distinct) reals, none being f
at an integral shift of the sum of the others.

Thus $ = $[2]. $[1] makes sense, and is equivalent to: for all f:R
into R, there exist a real which is not 0. Hence $[1] is provable.

(I always follow the convention that letters like "k" denote positive
integers, unless indicated otherwise).

I was expecting that for k >= 2, $[k] is equivalent to |R| >= omega_k.
However,, my proofs were completely blocked, which I found
frustrating. Here is the reason.

THEOREM A. $[k] s provable in ZC + notCH.

LEMMA 1. Assume notCH.  Let A,B be subsets of R, where A has
cardinality omega_1, the complement of B has cardinality at most
omega_1, and k in A. There is a real number u such that for all x in A
there exists y in B with kx+y = u.

Proof: Let A,B,k be as given. For each x in A, {kx+y: y in B} has
complement of cardinality at most omega_1. Hence the intersection over
x in A of these sets has complement of cardinality at most omega_1.
Hence by notCH, the intersection over x in A of these sets meets B.
Let u in B lie in this intersection over A. Then for all x in A, u
lies in {kx+y: y in B}. QED

Proof of A: Assume not CH and $[k] is false, for fixed k >= 3. Let f:R
into R be a counterexample. I.e., for any k distinct reals, one is f
at an integral shift of the sum of the others. Below n ranges over Z.

Fix A containedin R with cardinality omega_1. For all x in R, look at
x,x+1,...,x+k-2,y, for varying y outside {x,x+1,...,x+k-2}. For all
but countably many y in R, y is not an integral shift of the sum of
the x's. Hence for all but countably many y, some of x,x+1,...,x+k-2
is some f((k-2)x+y+n). I.e., for each x in A, the exceptional set of
y's is countable. Therefore, the union of the exceptional sets of y's
for the various x in A has cardinality at most omega_1.

Hence we have a set B of reals, with complement of cardinality at most
omega_1, such that for all x in A and y in B, some of x,x+1,...,x+k-2
is some f((k-2)x+y+n). By Lemma 1, fix u such that for all x in A,
there exists y in B such that (k-2)x+ y = u.. Then for all x in A,
some of x,x+1,...,x+k-2 is some f(u+n). Let A' be an uncountable
subset of A no two elements of which differ by an integer. Then we see
that uncountably many reals must be some f(u+n), which is impossible.
QED

LEMMA 2. Assume CH and k >= 2.There exists f:R into R such that for
all reals x_1,...,x_k, one is f at an integral shift of the sum of the
others.

Proof: Assume all hypotheses. Let G_alpha, alpha < omega_1, have the
following properties.
i. G_0 is the set of all rationals.
ii. G_alpha is the divisible additive subgroup of R generated by the
union of the G_beta, beta < alpha, and a single real u_alpha not in
any G_beta, beta < alpha.
iii. R is the union of the G_alpha.

Note that each G_alpha is countably infinite and they form a tower
under proper inclusion.

Fix alpha < omega_1, and suppose we have defined f:G_<alpha into
G_<alpha so that for all reals x_1,...,x_k+1 in G_<alpha, one is f at
an integral shift of the sum of the others. We now extend f to be from
G_alpha into G_alpha, maintaining this property, as follows

Let x_1,...,x_k+1 be from G_alpha, but not all from G_<alpha. We claim
that we can remove one of the terms so that the sum of the others is
not in G_alpha. To see this, first suppose that the sum of all of the
terms lies in G_<alpha. Then we can remove any term not lying in
G_<alpha. Now suppose that the sum of all of the terms does not lie in
G_<alpha. If we remove each successive term and sum the others, and
then we obtain k+1 sums-with removal. The sum of these sums-with
removal is sum of all these sums-with-removal, then this overall sum
is k times the sum of all terms, which is not in G_<alpha. Thus one of
the sums-with-removal is not in G_<alpha.

Now enumerate the x_1,...,x_k+1 from G_alpha, but not all from
G_<alpha, without repetition. By induction, we can define partially on
G_alpha\G_<alpha so that for each x_1,...,x_k+1, some x_i is f at an
integral shift of the sum of the others. We can do this because we
have infinitely many possible integral shifts available at every
stage. QED

THEOREM B. Let k >= 2. The following are provably equivalent over ZC.
i. $ (both forms).
ii. $[k] (both forms).
iii. For all k >= 1, $[k] (both forms).
iv. notCH.

Proof: By Theorem A and Lemma 2. QED

***********************************************
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 696th 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
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622: Adventures in Formalization 6  9/29/15  3:33AM
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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
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630: Optimal Function Theory 8  10/3/15  6PM
631: Order Theoretic Optimization 1  10/1215  12:16AM
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633: Constrained Function Theory 1  10/18/15 1AM
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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
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653: Pi01 Incompleteness/SRP,HUGE  2/8/16  3:20PM
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655: Pi01 Incompleteness/SRP,HUGE/2  2/12/16  11:40PM
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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
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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
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676: Refuting the Continuum Hypothesis?/3  5/10/16  3:30AM
677: Embedded Maximality and Pi01 Incompleteness/2  5/17/16  7:50PM
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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
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686: Refuting the Continuum Hypothesis?/4  6/616  9:29PM
687: Large Cardinals and Continuations/7  6/7/16  10:28PM
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694: Large Cardinals and Continuations/12  6/29/16  11:46PM
695: Refuting the Continuum Hypothesis?/6  ul 1 02:28:09 EDT 2016

Harvey Friedman