932: Tangible Indiscernibles/3
Harvey Friedman
hmflogic at gmail.com
Sat May 14 13:34:40 EDT 2022
CONTINUING FROM 931: Tangible Indiscernibles/2
7. SECTIONS
The term "section" probably has no completely standard meaning, but we use
one very natural one.
Let S be a subset of X^k. The sections of S are certain subsets of the X^i,
1 <= i <= k, derived from S as follows.
Fix i coordinates of S with any chosen elements of X. Here 0 <= i < k.
These i coordinates do not have to be in front and can even have lots of
gaps.
This results in the obvious subset S' of X^k-i, where we read the remaining
k-i coordinates from left to right.
A section below c in X is a section where the fixed elements are <c. A
section at or below c is a section where the fixed elements are <=c.
Let E containedin X. E above c is the set of all elements of E that are >c.
E at or above c is the set of all elements of E that are >=c.
8. SECTIONAL INDISCERNIBLES AND RELATED NOTIONS
Let S containedin X^k. E is a sectional SOI for S and only if the following
holds for all c in E. E at or above c is an SOI for any section of S below
c.
E is a gap sectional SOI for S if and only if the following holds for all c
in E. E above c is an SOI for any section of S at or below c.
Note that every sectional SOI is a gap sectional SOI. In core mathematics,
gap sectional SOI are more common than sectional SOI.
Gap sectional SOI have been commonplace, at least intuitively, in core
mathematics for a very long time. However, they arose, using different
language, in the discrete setting of N = nonnegative integers, with its
usual linear ordering, in the work of Paris and Harrington. Paris, J.;
Harrington,
L. <https://en.wikipedia.org/wiki/Leo_Harrington> (1977). "A Mathematical
Incompleteness in Peano Arithmetic". In Barwise, J.
<https://en.wikipedia.org/wiki/Jon_Barwise> (ed.). *Handbook of
Mathematical Logic*. Amsterdam, Netherlands: North-Holland.
THEOREM 8.1. Every S containedin N^k has an infinite gap sectional SOI. It
can be taken to be a subset of any given infinite subset of N.
Theorem 8.1 is proved from Theorem 4.1 by an infinite recursive
construction.
Note the great generality of Theorem 8.1. No condition whatsoever on S
containedin N^k. This very much exploits being in N. Nothing like this can
be done for R even in two dimensions.
THEOREM 8.2. There exists (Borel measurable) S containedin R^2 with no gap
sectional SOI of cardinality 3.
Proof: Let f:R into R be such that the range of f on any nonempty open
interval is all of R. Use the graph of f as the S. QED
We also define the universal sectional indiscernibles, the universal gap
sectional indiscernibles, and the almost universal sectional
indiscernibles, the almost universal gap sectional indiscernibles.
9. GAP SECTIONAL INDISCERNIBLES IN R
Every piecewise linear subset of R^k has an infinite gap sectional SOI. In
fact,
THEOREM 9.1.. The family of all piecewise linear subsets of R^k has an
almost universal gap sectional SOI. In fact, the range of any sequence of
positive real numbers whose xn+1/xn goes to infinity is an almost universal
SOI.
THEOREM 9.2. There is a semi algebraic S containedin R^3 with no gap
sectional SOI of cardinality 3.
For let S be the set of all triples (x,y,1/|x-y| + max(x,y)). Then S has no
gap sectional SOI of cardinality 3.
THEOREM 9.3. semi algebraic S containedin R^2
10. SECTIONAL INDISCERNIBLES IN MATHEMATICS
Sectional SOI are very strong. Consider the set of all (x,x+1) from R^2.
There cannot be a sectional SOI of cardinality 2.
However, let us consider the ordered Abelian semigroup G in infinitely many
generators g1 < g2 < ... . The notion of piecewise linear is clear here,
namely Boolean combinations of inequalities between sums, with
constant terms allowed..
THEOREM 10.1. Every piecewise linear subset of G^k has the sectional SOI
{g1,g2,...}. In fact, {g1,g2,...} is a universal sectional SOI.
Thus we have the strongest possible situation -- a universal sectional SOI.
But if we use the Abelian group H in infinitely many generators g1 < g2 <
..., then we only get
THEOREM 10.2. Every piecewise linear subset of H^k has the gap sectional
SOI {g1,g2,...}.
##########################################
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 92nd 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 #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
900: Ultra Convergence/2 10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6 10/4/21 5:55AM
902: Mathematical L and OD/RM 10/7/21 5:13AM
903: Foundations of Large Cardinals/1 10/12/21 12:58AM
904: Foundations of Large Cardinals/2 10/13/21 3:17PM
905: Foundations of Large Cardinals/3 10/13/21 3:17PM
906: Foundations of Large Cardinals/4 10/13/21 3:17PM
907: Base Theory Proposals for Third Order RM/1 10/13/21 10:22PM
908: Base Theory Proposals for Third Order RM/2 10/17/21 3:15PM
909: Base Theory Proposals for Third Order RM/3 10/17/21 3:25PM
910: Base Theory Proposals for Third Order RM/4 10/17/21 3:36PM
911: Ultra Convergence/3 1017/21 4:33PM
912: Base Theory Proposals for Third Order RM/5 10/18/21 7:22PM
913: Base Theory Proposals for Third Order RM/6 10/18/21 7:23PM
914: Base Theory Proposals for Third Order RM/7 10/20/21 12:39PM
915: Base Theory Proposals for Third Order RM/8 10/20/21 7:48PM
916: Tangible Incompleteness and Clique Construction/1 12/8/21 7:25PM
917: Proof Theory of Arithmetic/1 12/8/21 7:43PM
918: Tangible Incompleteness and Clique Construction/1 12/11/21 10:15PM
919: Proof Theory of Arithmetic/2 12/11/21 10:17PM
920: Polynomials and PA 1/7/22 4:35PM
921: Polynomials and PA/2 1/9/22 6:57 PM
922: WQO Games 1/10/22 5:32AM
923: Polynomials and PA/3 1/11/22 10:30 PM
924: Polynomials and PA/4 1/13/22 2:02 AM
925: Polynomials and PA/5 2/1/22 9::04PM
926: Polynomials and PA/6 2/1/22 11:20AM
927: Order Invariant Games/1 03/04/22 9:11AM
928: Order Invariant Games/2 03/7/22 4:22AM
929: Physical Infinity/randomness *3/21/22 02:14AM *
930: Tangible Indiscernibles/1
931: Tangible Indiscernibles/2
Harvey Friedman
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