# [FOM] 404: Set Equation Tower Theory 2

Harvey Friedman friedman at math.ohio-state.edu
Wed Mar 24 00:13:26 EDT 2010

```THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.

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Here we only improve on the NOTATION used in the statements. HOWEVER,
notation is important for this project!

MOTIVATING THEME: Do certain set equations emanating out of an
important area in graph theory going back to von Neumann (game theory)
have nontrivially self embeddable solutions? And if so, what are the
first integers moved by the self embeddings?

IMAGE EQUATION TOWER THEORY 1
by
Harvey M. Friedman
March 23, 2010

1. SET EQUATIONS AND TOWERS.
2. SET EQUATION TOWER THEOREMS - infinite.
3. SET EQUATION TOWER THEOREMS - finite.

1. SET EQUATIONS AND TOWERS.

Identical to section 1 of http://www.cs.nyu.edu/pipermail/fom/2010-March/014475.html

KEY DEFINITIONS: For A contained in N^k, fld(A) is the set of integers
appearing as coordinates in A, and A# = fld(A)^k.

For R contained in N^k x N^k, R<[A] = {y: there exists x in A with
max(x) < max(y) and R(x,y)}.

Let A,B contained in N^k. h embeds A into B iff h:fld(A) into fld(B)
is one-one and order preserving, and for all x_1,...,x_k in fld(A),
(x_1,...,x_k) in A iff (h(x_1),...,h(x_k)) in B.

2. SET EQUATION TOWER THEOREMS - infinite.

We think of A = A#\R<[B] as an approximation to A = A#\R<[A], and we
have seen that this behaves differently: compare Theorems 1.7 and 1.8.

A tower is a sequence of sets, each one a subset of the next.

We say that h is a self embedding of a tower {A_i}, 1 <= i <= r, of
infinite subsets of N^k, if and only if h is a self embedding of A_r
that maps each fld(A_i) into fld(A_i).

An even better approximation to A = A#\R<[A] are the towers {A_i = A_i#
\R<[A_i+1]}, 1 <= i <= r, of infinite subsets of N^k.

Note that according to our terminology, we must have A_1 contained
in ... contained in A_r, but not necessarily A_r contained in A_r+1.
This means that we need a tower of length 4 to have the effect of the
3 equations in http://www.cs.nyu.edu/pipermail/fom/2010-March/014475.html
.

PROPOSITION 2.1. For all R contained in N^+k x N^+k, there is a tower
{A_i = A_i#\R<[A_i+1]}, 1 <= i <= 4, of infinite subsets of N^k, with
a self embedding with critical point min(fld(A_1)).

PROPOSITION 2.2. For all k,p >= 1 and R contained in N^+k x N^+k,
there is a tower {A_i = A_i#\R<[A_i+1]}, 1 <= i <= p, of infinite
subsets of N^k, with a self embedding with critical point min(fld(A_1)).

THEOREM 2.3. Propositions 2.1, 2.2 are provable in SRP+, but not from
any consequence of SRP consistent with RCA_0. Propositions 2.1, 2.2
are provably equivalent to Con(SRP) over ACA'.

Here SRP+ = ZFC + "for all k there exists a limit ordinal with the k-
SRP". SRP = ZFC + {there exists a limit ordinal with the k-SRP}_k. We
say that lambda has the k-SRP if and only if lambda is a limit
ordinal, where every partition of the unordered k tuples from lambda
into two pieces has a homogenous set which is a stationary subset of
lambda. ACA' is ACA_0 + "for all n and x contained in omega, the n-th
Turing jump of x exists".

3. SET EQUATION TOWER THEOREMS - finite

We use |A| for the cardinality of A. We work with R contained in
{1,...,t}^k x {1,...,t}^k.

Since we are working with finite sets, we must relax the notion of
embedding. An embedding from A_1,...,A_p into B_1,...,B_p is as in
section 2, except that it only maps fld(A_p) intersect [0,a] into
fld(B_p) intersect [0,b], where a is the second largest element of
fld(A_1) and b is the second largest element of fld(B_1).

PROPOSITION 3.1. Let t >> k,r >= 1. For all R contained in
{1,...,t}^k, there is a tower {A_i = A_i#\R<[A_i+1]}, 1 <= i <= r, of
subsets of {0,...,t}^k with at least r+1 elements, with a self
embedding with critical point min(fld(A_1)). Furthermore, it suffices
to assume that t is greater than an exponential stack of 8kr 2's.

PROPOSITION 3.2. Let t >> k,r,p >= 1. For all R contained in
{1,...,t}^k, there is a tower {A_i = A_i#\R<[A_i+1]}, 1 <= i <= p, of
subsets of {0,...,t}^k with at least r+1 elements, with a self
embedding with critical point min(fld(A_1)). Furthermore, it suffices
to assume that t is greater than an exponential stack of 8krp 2's.

THEOREM 3.3. Propositions 3.1, 3.2 are each provable in SRP+, but not
from any consequence of SRP consistent with SEFA. Propositions 3.1,
3.2 are each provably equivalent to Con(SRP) over SEFA.

**********************

manuscripts. This is the 404th 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 #1-349 can be found at http://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.

350: one dimensional set series  7/23/09  12:11AM
351: Mapping Theorems/Mahlo/Subtle  8/6/09  10:59PM
352: Mapping Theorems/simpler  8/7/09  10:06PM
353: Function Generation 1  8/9/09  12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1  8/9/09  6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2  8/10/09  6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem  8/14/09  9:31AM
357: HIGH SCHOOL Games/Update  8/20/09  10:42AM
358: clearer statements of HIGH SCHOOL Games  8/23/09  2:42AM
359: finite two person HIGH SCHOOL games  8/24/09  1:28PM
360: Finite Linear/Limited Memory Games  8/31/09  5:43PM
361: Finite Promise Games  9/2/09  7:04AM
362: Simplest Order Invariant Game  9/7/09  11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1  9/24/09  1:06PM
366: Free Reductions and Large Cardinals/polished  9/28/09  2:19PM
367: Upper Shift Fixed Points and Large Cardinals  10/4/09  2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction  10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement  10/29/09  2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09  12:14PM
371: Vector Reduction and Large Cardinals  11/21/09  1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09  5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals  12/7/09  9:17AM
374: Upper Shift Greedy Chain Games  12/12/09  5:56AM
375: Upper Shift Clique Games and Large Cardinals 1
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09  2:23PM
377: The Polynomial Shift Theorem  12/25/09  2:39PM
378: Upper Shift Clique Sequences and Large Cardinals  12/25/09  2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems  12/28/09  7:06AM
381: Trigonometric Shift Theorem  12/29/09  11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals  12/30/09  2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09  3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION  12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences  1/1/10  7:35PM
386: Terrifically and Extremely Long Finite Sequences  1/1/10  7:35PM
387: Better Polynomial Shift Translation/typos  1/6/10  10:41PM
388: Goedel's Second Again/definitive?  1/7/10  11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1  2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2  2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3  2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4  2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5  2/22/10
3:50AM
394: Free Reduction Theory 1  3/2/10  7:30PM
395: Free Reduction Theory 2  3/7/10  5:41PM
396: Free Reduction Theory 3  3/7/10  11:30PMs
397: Free Reduction Theory 4  3/8/10  9:05AM
398: New Free Reduction Theory 1  3/10/10  5:26AM
399: New Free Reduction Theory 2  3/12/10  9:36AM
400: New Free Reduction Theory 3  3/14/10  11:55AM
401: New Free Reduction Theory 4  3/15/10  4:12PM
402: New Free Reduction Theory 5  3/19/10  12:59PM
403: Set Equation Tower Theory 1  3/22/10  2:45PM

Harvey Friedman

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