[FOM] 449: Maximal Sets and Large Cardinals I
Harvey Friedman
friedman at math.ohio-state.edu
Sat Dec 4 18:00:57 EST 2010
THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION
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THIS POSTING IS SELF CONTAINED.
In the series Kernels and Large Cardinals I-IV, kernels represent a
strong kind of maximal set. We have isolated this strong maximality,
which we call LOCAL MAXIMALITY. It applies well to arbitrary binary
relations, without any condition like "downward" which is needed in
order to use kernels.
Even more basic is simply the notion of MAXIMAL CLIQUE for a binary
relation on a set. We will use this as well as the notion of LOCALLY
MAXIMAL CLIQUE.
We also present a new view of the finite form. This takes the form of
a finite sequential construction of vectors, more straightforward than
previous versions of this some time ago.
We postpone any reconsideration of the Exotic statements that
correspond to HUGE to a later posting.
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1. A-RELATIONS, CLIQUES, UPPER SHIFT.
Fix a subset A of the set Q of all rationals. The A-relations on A^k
are the order invariant subsets R of A^k x A^k = A^2k. I.e., binary
relations R, where if x,y in A^2k are order equivalent, then x in R if
and only if y in R.
The A-relations are the A-relations on the various A^k.
Let R be contained in A^k x A^k. B is an R clique if and only if B x B
is contained in R.
B is a maximal R clique if and only if B is an R clique which is not
properly contained in any R clique.
B is a locally maximal R clique if and only if for all x in A, B|<=x
is a maximal clique in R|<=x. Here T|<=x is T restricted to the
vectors whose coordinates are <= x.
Note that local maximality implies maximality. However, the converse
fails.
The upper shift of a vector from Q is obtained by adding 1 to all
nonnegative coordinates. The upper shift of a set of vectors from Q is
the set of upper shifts of its elements.
2. THE UPPER SHIFT CLIQUE THEOREMS.
THE UPPER SHIFT MAXIMAL CLIQUE THEOREM. There exists 0 in A contained
in Q such that every A-relation has a maximal clique that contains its
upper shift.
THE UPPER SHIFT LOCALLY MAXIMAL CLIQUE THEOREM. There exists 0 in A
contained in Q such that every A-relation has a locally maximal clique
that contains its upper shift.
The only proof we know of The Upper Shift Maximal Clique Theorem uses
large cardinals. We don't know if it can be proved in ZFC, or even in
RCA_0.
We do know that it is necessary and sufficient to use large cardinals
in order to prove The Upper Shift Locally Maximal Clique Theorem.
The same situation obtains even if we restrict ourselves to symmetric
A-relations.
Specifically, let SRP+ = ZFC + "for all k, there is a limit ordinal
with the k-SRP". SRP = ZFC + {there is a limit ordinal with the k-
SRP}_k. The k-SRP asserts that every 2 coloring of the unordered k-
tuples has a stationary monochromatic set.
THEOREM 2.1. SRP+ proves The Upper Shift Locally Maximal Clique
Theorem. In fact, it is provably equivalent to Con(SRP) over WKL_0.
3. SEQUENTIAL CLIQUE CONSTRUCTION THEOREMS.
Fix a reflexive order invariant R contained in Q^k x Q^k. We present a
nondeterministic construction of a "rich" R clique.
The R clique constructions take the following form:
INITIALIZATION. Form the sequence of length 1 consisting of (0,...,0)
in Q^k. This is obviously an R clique.
CONTINUATION. Make successive R clique continuations, as prescribed
below.
INFINITE SEQUENTIAL CLIQUE CONSTRUCTION THEOREM. For each reflexive
order invariant R contained in Q^k x Q^k, there is a R clique
construction with infinitely many continuations.
FINITE SEQUENTIAL CLIQUE CONSTRUCTION THEOREM. For each reflexive
order invariant R contained in Q^k x Q^k, there are R clique
constructions with any given finite number of continuations.
Let x_1,...,x_p, p >= 1, be an R clique. An R clique continuation of
x_1,...,x_p take the form of an R clique
x_1,...,x_p,y_1',...,y_q',ush(y_1'),...,ush(y_q')
where ush is the upper shift.
The R clique continuations are constructed in three steps.
STEP 1. Choose an enumeration y_1,...,y_q without repetition, of all k-
tuples whose coordinates are among the coordinates of x_1,...,x_p. Of
course, we cannot expect x_1,...,x_p,y_1,...,y_q to be an R clique.
STEP 2. Replace none, some, or all of the y_i by a vector from Q^k of
lower maximum coordinate, which is not related to y_i by R (not a
predecessor and not a successor). Write the resulting sequence as
y_1',...,y_q'.
STEP 3. Return x_1,...,x_p,y_1',...,y_q',ush(y_1'),...,ush(y_q').
Note that because of the enumeration without repetition in STEP 1, we
have obvious bounds on the lengths of successive continuations. Also,
because clique constructions are entirely order theoretic, we can put
obvious bounds on the numerators and denominators that are used in the
successive continuations. This results in an explicitly Pi01 form of
the Finite Sequential Construction Theorem.
THEOREM 3.1. If we omit the ush terms, then the Infinite Sequential
Clique Construction Theorem is provable in RCA_0.
THEOREM 3.2. The Infinite Sequential Clique Construction Theorem is
provably equivalent to Con(SRP) over WKL_0. The Finite Sequential
Clique Construction Theorem is provably equivalent to Con(SRP) over EFA.
An alternative is to modify STEP 1 by using only the k-tuples that are
a subsequence of the concatenated sequence x_1,...,x_p. The same
results apply using this alternative.
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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 449th 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
446: Kernels and Large Cardinals III 11/22/10 2:50PM
447: Kernels and Large Cardinals IV 11/23/10 3:51PM
448: Naturalness/PA Independence 12/3/10 12:19AM
Harvey Friedman
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