[FOM] 684:Large Cardinals and Continuations/5
Harvey Friedman
hmflogic at gmail.com
Fri Jun 3 03:56:06 EDT 2016
THIS POSTING IS SELF CONTAINED
This posting corrects the postings in the Large Cardinals and
Continuations series, and also presents, for the first time,
explicitly Pi01 sentences corresponding to HUGE.
***IMPLICITLY AND EXPLICITLY Pi01 FOR SRP***
in the rationals
This is sections 1 and 2 of
http://www.cs.nyu.edu/pipermail/fom/2016-June/019886.html
*no change here*
***IMPLICITLY Pi01 FOR HUGE***
Here we make a small change in the definition of smoothly maximal
r-continuation and smoothly(<=) maximal r-continuation.
DEFINITION 1. A Q,k system is a pair (A,R), where A containedin Q and R
containedin A^k. A negative Q,k system is a Q,k system with its domain
A containedin Q|<0.
We say that (A,R) is a part of (B,S) if and only if A containedin B and R
containedin S. The cardinality of a Q,k system is the cardinality of its
domain. Isomorphisms from (A,R) onto (B,S) are order preserving
bijections h:A into B that send R onto S.
DEFINITION 2. (B,S) is an r-continuation of the Q,k system (A,R) if and
only if (A,R) is a part of the Q,k system (B,S), 0 in B, and every part
of (B,S) of cardinality r is isomorphic to some part of (A,R). If r =
2 then we omit r.
DEFINITION 3. (B,S) is a smoothly maximal r-continuation of the
Q,k system (A,R) if and only if (B,S) is an r-continuation of (A,R),
where no (B|<=max(x),S|<max(x) U. {x}), max(x) >= 0, is an
r-continuation of (A,R).
DEFINITION 4. Let (A,R) be a Q,k system. A self embedding h of (A,R) is
a one-one h:Q into Q such that for all p_1,...,p_k in A,
R(p_1,...,p_k) if and only if R(h(p_1),...,h(p_k)).
SMOOTHLY EMBEDDED MAXIMAL CONTINUATIONS. SEMC. For finite negative Q,k
systems, some smoothly maximal continuation is self embedded by the
function p if p < 0; p+1 otherwise.
We now present a weakened form of smoothly maximal r-continuations.
DEFINITION 5. (B,S) is a smoothly/<= maximal r-continuation of (A,R)
if and only if (B,S) is an r-continuation of (A,R), where no
(B|<=max(x),S|<max(x) U. {x}), max(x) >= 0, x increasing, is an
r-continuation of
(A,R). Here x in Q^k is increasing if and only if x_1 <= ... <= x_k.
DEFINITION 6. Let x in Q^k. x is positive if and only if min(x) > 0. x
is increasing if and only if x_1 <= ... <= x_k. E containedin Q^k is
positive if and only if all elements of E are positive. The sections
of Q,k systems (A,R) are the sets of one lower dimension obtained from R
by fixing one or more arguments from A in any positions.
SMOOTHLY/<= EMBEDDED MAXIMAL CONTINUATIONS (<=). SEMC(<=). For finite
negative Q,k systems, the positive tuples and bounded positive
sections of some smoothly/<= maximal continuation are closed under +1.
THEOREM 1. SEMC is equivalent to Con(SRP) over WKL_0. SEMC(<=) is equivalent to
Con(HUGE) over WKL_0.
***EXPLICITLY Pi01 FOR HUGE***
this is new
We now give a finite form of SEMC(<=) using number theoretic heights
as we did when we presented the explicitly Pi01 sentences
corresponding to SRP.
DEFINITION 7. We use hgt(A) for the height of A containedin Q^k. For R
containedin Q^k, R[p] is the section resulting from fixing all but the
last coordinate to be p.
DEFINITION 8. (B,S) is a smoothly/<= rich r-continuation of (A,R)
if and only if (B,S) is an r-continuation of (A,R), where no
(B|<=max(x),S|<max(x) U. {x}), max(x) >= 0, x increasing, hgt(x) <=
hgt(A), is an r-continuation of (A,R).
Note that we have merely inserted the height inequality.
FINITE SMOOTHLY/<= EMBEDDED MAXIMAL CONTINUATIONS. FSEMC(<=). Each
finite negative Q,k system (A1,R1) starts k successive finite
smoothly/<= rich continuations, (A1,R1),(A2,R2),...,(Ak,Rk), where for
all i,j < k, Ri + 1 containedin Ri+1, and Ri[j+.5] = Ai + 1|<=j.
FSEMC(<=) is explicitly Pi02. An obvious bound can be placed on the
height of Ak, putting it in explicitly Pi01 form.
THEOREM 2. FSEMC(<=) is provably equivalent to Con(HUGE) over EFA.
***Pi01 FOR THE MAHLO HIERARCHY***
In the nonnegative integers
DEFINITION 1. Let A,B containedin N. We say that A,B are k-isomorphic
if and only if A,B are of the same (possibly infinite) cardinality,
and for any signed sum of at most k elements of A, the corresponding
signed sum from elements of B (under the unique order preserving
bijection from A onto B) has the same sign - positive, negative, or 0.
DEFINITION 2. Let A containedin N. B is a k-continuation of A if and
only if A containedin B containedin N, and A,B have the same k element
subsets up to isomorphism. B is a maximal k-continuation of A if and
only if B is a k-continuation of A, where no B U. {n} is a
k-continuation of A.
Here U. indicates disjoint union.
THEOREM 1. Every A contianedin N has a unique maximal k-continuation.
We now use a weaker form of maximal k-continuation. .
DEFINITION 3. Let X containedin N. A proper k sum from X is an
unsigned sum of k elements of X greater than all of its summands.
DEFINITION 4. B is a +,X optimal k-continuation of A if and only if B
is a k-continuation of A, where no B U. {n}|<=n U A, n > k, is a
k-continuation of A, n a proper k sum from A U X.
OPTIMAL CONTINUATIONS IN N. OC(N). Every finite subset of [k] has r
successive +,N! maximal k-continuations omitting almost all of N!.
FINITE OPTIMAL CONTINUATIONS IN N . FOC(N). Every finite subset of [k]
has two successive finite +,[n]! maximal k-continuations omitting (8k)!!.
Here [n] = {0,...,n}.
Note that FOC(N) is explicitly Pi02. Using an obvious upper bound, it
becomes explicitly Pi01.
THEOREM 2. OC(N) is provably equivalent to Con(MAH) over RCA_0. FOC(N)
is provably equivalent to Con(MAH) over EFA.
***********************************************
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 684th 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 #1-599 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2015-August/018887.html
600: Removing Deep Pathology 1 8/15/15 10:37PM
601: Finite Emulation Theory 1/perfect? 8/22/15 1:17AM
602: Removing Deep Pathology 2 8/23/15 6:35PM
603: Removing Deep Pathology 3 8/25/15 10:24AM
604: Finite Emulation Theory 2 8/26/15 2:54PM
605: Integer and Real Functions 8/27/15 1:50PM
606: Simple Theory of Types 8/29/15 6:30PM
607: Hindman's Theorem 8/30/15 3:58PM
608: Integer and Real Functions 2 9/1/15 6:40AM
609. Finite Continuation Theory 17 9/315 1:17PM
610: Function Continuation Theory 1 9/4/15 3:40PM
611: Function Emulation/Continuation Theory 2 9/8/15 12:58AM
612: Binary Operation Emulation and Continuation 1 9/7/15 4:35PM
613: Optimal Function Theory 1 9/13/15 11:30AM
614: Adventures in Formalization 1 9/14/15 1:43PM
615: Adventures in Formalization 2 9/14/15 1:44PM
616: Adventures in Formalization 3 9/14/15 1:45PM
617: Removing Connectives 1 9/115/15 7:47AM
618: Adventures in Formalization 4 9/15/15 3:07PM
619: Nonstandardism 1 9/17/15 9:57AM
620: Nonstandardism 2 9/18/15 2:12AM
621: Adventures in Formalization 5 9/18/15 12:54PM
622: Adventures in Formalization 6 9/29/15 3:33AM
623: Optimal Function Theory 2 9/22/15 12:02AM
624: Optimal Function Theory 3 9/22/15 11:18AM
625: Optimal Function Theory 4 9/23/15 10:16PM
626: Optimal Function Theory 5 9/2515 10:26PM
627: Optimal Function Theory 6 9/29/15 2:21AM
628: Optimal Function Theory 7 10/2/15 6:23PM
629: Boolean Algebra/Simplicity 10/3/15 9:41AM
630: Optimal Function Theory 8 10/3/15 6PM
631: Order Theoretic Optimization 1 10/1215 12:16AM
632: Rigorous Formalization of Mathematics 1 10/13/15 8:12PM
633: Constrained Function Theory 1 10/18/15 1AM
634: Fixed Point Minimization 1 10/20/15 11:47PM
635: Fixed Point Minimization 2 10/21/15 11:52PM
636: Fixed Point Minimization 3 10/22/15 5:49PM
637: Progress in Pi01 Incompleteness 1 10/25/15 8:45PM
638: Rigorous Formalization of Mathematics 2 10/25/15 10:47PM
639: Progress in Pi01 Incompleteness 2 10/27/15 10:38PM
640: Progress in Pi01 Incompleteness 3 10/30/15 2:30PM
641: Progress in Pi01 Incompleteness 4 10/31/15 8:12PM
642: Rigorous Formalization of Mathematics 3
643: Constrained Subsets of N, #1 11/3/15 11:57PM
644: Fixed Point Selectors 1 11/16/15 8:38AM
645: Fixed Point Minimizers #1 11/22/15 7:46PM
646: Philosophy of Incompleteness 1 Nov 24 17:19:46 EST 2015
647: General Incompleteness almost everywhere 1 11/30/15 6:52PM
648: Necessary Irrelevance 1 12/21/15 4:01AM
649: Necessary Irrelevance 2 12/21/15 8:53PM
650: Necessary Irrelevance 3 12/24/15 2:42AM
651: Pi01 Incompleteness Update 2/2/16 7:58AM
652: Pi01 Incompleteness Update/2 2/7/16 10:06PM
653: Pi01 Incompleteness/SRP,HUGE 2/8/16 3:20PM
654: Theory Inspired by Automated Proving 1 2/11/16 2:55AM
655: Pi01 Incompleteness/SRP,HUGE/2 2/12/16 11:40PM
656: Pi01 Incompleteness/SRP,HUGE/3 2/13/16 1:21PM
657: Definitional Complexity Theory 1 2/15/16 12:39AM
658: Definitional Complexity Theory 2 2/15/16 5:28AM
659: Pi01 Incompleteness/SRP,HUGE/4 2/22/16 4:26PM
660: Pi01 Incompleteness/SRP,HUGE/5 2/22/16 11:57PM
661: Pi01 Incompleteness/SRP,HUGE/6 2/24/16 1:12PM
662: Pi01 Incompleteness/SRP,HUGE/7 2/25/16 1:04AM
663: Pi01 Incompleteness/SRP,HUGE/8 2/25/16 3:59PM
664: Unsolvability in Number Theory 3/1/16 8:04AM
665: Pi01 Incompleteness/SRP,HUGE/9 3/1/16 9:07PM
666: Pi01 Incompleteness/SRP,HUGE/10 13/18/16 10:43AM
667: Pi01 Incompleteness/SRP,HUGE/11 3/24/16 9:56PM
668: Pi01 Incompleteness/SRP,HUGE/12 4/7/16 6:33PM
669: Pi01 Incompleteness/SRP,HUGE/13 4/17/16 2:51PM
670: Pi01 Incompleteness/SRP,HUGE/14 4/28/16 1:40AM
671: Pi01 Incompleteness/SRP,HUGE/15 4/30/16 12:03AM
672: Refuting the Continuum Hypothesis? 5/1/16 1:11AM
673: Pi01 Incompleteness/SRP,HUGE/16 5/1/16 11:27PM
674: Refuting the Continuum Hypothesis?/2 5/4/16 2:36AM
675: Embedded Maximality and Pi01 Incompleteness/1 5/7/16 12:45AM
676: Refuting the Continuum Hypothesis?/3 5/10/16 3:30AM
677: Embedded Maximality and Pi01 Incompleteness/2 5/17/16 7:50PM
678: Symmetric Optimality and Pi01 Incompleteness/1 5/19/16 1:22AM
679: Symmetric Maximality and Pi01 Incompleteness/1 5/23/16 9:21PM
680: Large Cardinals and Continuations/1 5/29/16 10:58PM
681 :Large Cardinals and Continuations/2 6/1/16 4:01AM
682:Large Cardinals and Continuations/3 6/2/16 8:05AM
683: Large Cardinals and Continuations/4 6/2/16 11:21PM
Harvey Friedman
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