# [FOM] 445: Kernels and Large Cardinals II

Harvey Friedman friedman at math.ohio-state.edu
Wed Nov 17 15:36:39 EST 2010

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

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THIS POSTING IS ENTIRELY SELF CONTAINED.

This is a follow up to http://www.cs.nyu.edu/pipermail/fom/2010-October/015099.html

Here we concentrate only on small simplificattions in the statements
of the Finite and Infinite Upper Shift Kernel Theorem.

We will state these without regard to any statements of the much
stronger Finite and Infinite Exotic Kernel Theorems from http://www.cs.nyu.edu/pipermail/fom/2010-October/015099.html

1. INFINITE UPPER SHIFT KERNEL THEOREM.

A directed graph, or digraph, is a pair (V,E), where V is a nonempty
set of vertices, and E contained in V^2 is a set of edges. We say that
x connects to y if and only if (x,y) in E.

A kernel in (V,E) is an S contained in V such that

i. No element of S connects to any element of S.
ii. Every element of V\S connects to some element of S.

We let Q be the set of all rational numbers with the usual ordering.
We will focus on digraphs (A^k,E) where A is contained in Q.

We say that (A^k,E) is downward if and only if x E y implies max(x) >
max(y).

We fix A contained in Q. The A-sets in A^k are the subsets of A^k
which are order invariant in the following sense. If x lies in the
subset, then every y in A^k that is order equivalent to x also lies in
the subset. Note that the A-sets in A^k form a finite Boolean algebra
of subsets of A^k.

The A-digraphs are the digraphs (A^k,E), where E is an A-set in
A^2k.

The A-sets are the A-sets in A^k, k >= 1.

The upper shift of x in Q^k is obtained from x by adding 1 to all
nonnegative coordinates of x.

The upper shift of S contained in Q^k is the set of all upper shifts
of the elements of S.

INFINITE UPPER SHIFT KERNEL THEOREM. There exists 0 in A contained in
Q such that every downward A-digraph has a kernel containing its upper
shift.

THEOREM 1.1. The Infinite Upper Shift Kernel Theorem is provably
equivalent to Con(SRP) over ACA_0. ACA_0 + Con(SRP) proves that A and
the kernels can be taken to be recursive in the Turing jump of 0.

Here SRP = ZFC + {there exists lambda with the k-SRP}_k. Lambda has
the k-SRP if and only if lambda is a cardinal such that every f:
[lambda]^k into 2 is constant on some [E]^k, E a stationary subset of
lambda. Here [E]^k is the set of all unordered k-tuples from E.

2. FINITE UPPER SHIFT KERNEL THEOREM.

There is a close correspondence between the Infinite Upper Shift
Kernel Theorem and the Finite Upper Shift Kernel Theorem:

INFINITE UPPER SHIFT KERNEL THEOREM. There exists 0 in A contained in
Q such that every downward A-digraph has a kernel containing its upper
shift.

FINITE UPPER SHIFT KERNEL THEOREM. Let n >= 1. There exists finite 0
in A contained in Q such that every downward A-digraph has an n-kernel
that contains its bounded upper shift. We can require the norms of
elements of A to be at most 8n^2.

The norm of x in Q^k is the sum of the magnitudes of the coordinates
of x when put in reduced form.

Let (A^k,E) be an A-digraph, A contained in Q. An n-kernel of (A^k,E)
is a set S contained in A^k such that

i. No element of S connects to any element of S.
ii. Every x in Q^k\S of norm p <= n connects to some element of S of
norm at most 8p^2.

The bounded upper shift of S contained in Q^k is the set of elements
of the upper shift whose max is at most the max of some element of S.

Note that the Finite Upper Shift Kernel Theorem is explicitly Pi01.

THEOREM 2.1. The Finite Upper Shift Kernel Theorem (even without the
last clause) is provably equivalent to Con(SRP) over EFA.

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manuscripts. This is the 444th 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 athttp://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 1graham
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  11:30PM
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
404: Set Equation Tower Theory 2  3/24/10  11:18PM
405: Some Countable Model Theory 1  3/24/10  11:20PM
406: Set Equation Tower Theory 3  3/25/10  6:24PM
407: Kernel Tower Theory 1  3/31/10  12:02PM
408: Kernel tower Theory 2  4/1/10  6:46PM
409: Kernel Tower Theory 3  4/5/10  4:04PM
410: Kernel Function Theory 1  4/8/10  7:39PM
411: Free Generation Theory 1  4/13/10  2:55PM
412: Local Basis Construction Theory 1  4/17/10  11:23PM
413: Local Basis Construction Theory 2  4/20/10  1:51PM
414: Integer Decomposition Theory  4/23/10  12:45PM
415: Integer Decomposition Theory 2  4/24/10  3:49PM
416: Integer Decomposition Theory 3  4/26/10  7:04PM
417: Integer Decomposition Theory 4  4/28/10  6:25PM
418: Integer Decomposition Theory 5  4/29/10  4:08PM
419: Integer Decomposition Theory 6  5/4/10   10:39PM
420: Reduction Function Theory 1  5/17/10   2:53AM
421: Reduction Function Theory 2  5/19/10   12:00PM
422: Well Behaved Reduction Functions 1  5/23/10  4:12PM
423: Well Behaved Reduction Functions 2  5/27/10  3:01PM
424: Well Behaved Reduction Functions 3  5/29/10  8:06PM
425: Well Behaved Reduction Functions 4  5/31/10  5:05PM
426: Well Behaved Reduction Functions 5  6/2/10  12:43PM
427: Finite Games and Incompleteness 1  6/10/10  4:08PM
428: Typo Correction in #427  6/11/10  12:11AM
429: Finite Games and Incompleteness 2  6/16/10  7:26PM
430: Finite Games and Incompleteness 3  6/18/10  6:14PM
431: Finite Incompleteness/Combinatorially Simplest  6/20/10  11:22PM
432: Finite Games and Incompleteness 4  6/26/10  8:39PM
433: Finite Games and Incompleteness 5  6/27/10  3:33PM
434: Digraph Kernel Structure Theory 1  7/4/10  3:17PM
435: Kernel Structure Theory 1  7/5/10  5:55PM
436: Kernel Structure Theory 2  7/9/10  5:21PM
437: Twin Prime Polynomial  7/15/10  2:01PM
438: Twin Prime Polynomial/error  9/17/10  1:22PM
439: Twin Prime Polynomial/corrected 9/19/10  2:16PM
440: Finite Phase Transitions  9/26/10  1:28PM
441: Equational Representations  9/27/10  4:59PM
442: Kernel Structure Theory Restated  10/11/10  9:01PM
443: Kernels and Large Cardinals 1  10/21/10  12:16AM
444: The Exploding Universe 1  11/1/10  1:46AMs

Harvey Friedman

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