# [FOM] 446: Kernels and Large Cardinals III

Harvey Friedman friedman at math.ohio-state.edu
Mon Nov 22 02:50:42 EST 2010

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

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

THIS POSTING IS ENTIRELY SELF CONTAINED

Here we present better formulations of the Exotic Kernel Theorems,
which involve extremely large cardinals.

We also take a different approach to finite forms.

We present several general approaches to finite forms which are quite
different than the approach used in 443: Kernels and large cardinals
I, http://www.cs.nyu.edu/pipermail/fom/2010-October/015099.html

We also give improved formulations of the Infinite Exotic Kernel
Theorem. We now see that the latter, the Infinite Extended Exotic
Kernel Theorem, of http://www.cs.nyu.edu/pipermail/fom/2010-October/015099.html
, was stated too strongly, and is inconsistent.

With these new approaches to finite forms, where we no longer try to
give a compelling specific finite statement, we no longer will use the
words "Infinite" and "Finite" in the title of the theorems.

Thus we refer to

The Upper Shift Kernel Theorem
The Exotic Kernel Theorem

KERNELS AND LARGE CARDINALS
by
Harvey M. Friedman

1. Upper Shift Kernel Theorem.
2. Exotic Kernel Theorem.
3. Concrete Aspects.

1. 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 that
are order invariant. I.e., membership depends only on the order type.
It is clear that the A-sets in A^k are also the subsets of A^k that
are given by a Boolean combination of inequalities in variables
x_1,...,x_k ranging over A, with no constants allowed.

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

The upper shift of a tuple from Q is obtained by adding 1 to all
nonnegative coordinates. The upper shift of a set of tuples from Q is
the set of upper shifts of its elements.

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. ACA_0 proves that USKT is equivalent to Con(SRP).

Here SRP = ZFC + {there exists a cardinal kappa with the stationary
Ramsey property of order k}_k.

2. EXOTIC KERNEL THEOREM.

We fix f:A^2 into A, where A is a subset of Q.

The f-sets in A^k are the subsets of A^k that are given by a Boolean
combination of inequalities in f and variables x_1,...,x_k ranging
over A, with no constants allowed.

Note that the f-sets are analogous to the multidimensional
semialgebraic sets in the reals, using +,dot,< only, with no
constants. This is the same as the multidimensional semialgebraic sets
in the reals, over the field of rationals.

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

For x in A, we write f//x for the cross section f_x restricted to (-
infinity,x). Here f//x is defined if and only if x is in A.

For partial g:Q into Q and S contained in Q^k, the diagonal image of S
under g is {(g(x_1),...,g(x_k)): x_1,...,x_k in S}.

EXOTIC KERNEL THEOREM. There exists f:A^2 into A, A contained in Q,
such that every downward f-digraph has a kernel containing its
diagonal image under the non identity function f//1 union f//2
union ... .

THEOREM 2.1. WKL_0 proves that EXKT is equivalent to Con(HUGE).

Here HUGE = ZFC + {there exists a k-huge cardinal}_k.

3. CONCRETE ASPECTS.

THEOREM 3.1. ACA_0 proves that if USKT holds then USKT holds in some
countable model of RCA_0. ACA_0 proves that if EXKT holds then EXKT
holds in some countable model of RCA_0.

THEOREM 3.2. ACA_0 proves that if USKT holds then USKT holds where the
A and the kernels are recursive in the Turing jump of 0. ACA_0 proves
that if EXKT holds then EXKT holds where the f and the kernels are
recursive in the Turing jump of 0.

COROLLARY 3.3. There are algorithms alpha, beta such that the
following is provable in ACA_0. USKT if and only if alpha does not
terminate. EXKT if and only if beta does not terminate.

With a more careful argument, we can show the following.

THEOREM 3.4. There is algorithm beta such that the following is
provable in WKL_0. EXKT if and only if beta does not terminate.

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

manuscripts. This is the 446th 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
445: Kernels and Large Cardinals II  11/17/10  10:13PM

Harvey Friedman
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