[FOM] 681:Large Cardinals and Continuations/2
Harvey Friedman
hmflogic at gmail.com
Wed Jun 1 04:01:20 EDT 2016
THIS POSTING IS SELF CONTAINED
In http://www.cs.nyu.edu/pipermail/fom/2016-May/019880.html I started
with a lot of background information about the kind of symmetry being
considered.
Here I want to headline the clear connection with the most elemental
intuitive features of a very basic biological system known to
preschoolers - specifically, seeds growing into plants.
1. PRESCHOOL BIOLOGY
1. Seeds lie under the ground. (Seeds are finite subsets V of Q^k|<0 =
the set of all elements of Q^k whose coordinates are all < 0).
2. Plants grow out of seeds. They are subject to a height restriction.
(Plants are subsets of Q^k|<=n = the set of all elements of Q^k whose
coordinates are all <= n).
3. Basic features of a plant growing from a seed are already present
in the seed. Like modern DNA theory. (A plant continues a seed, or
more generally, a plant continues a plant, if and only if the former
contains the latter, and every two element subset of the former is
isomorphic to some subset of the latter. More generally, a plant
r-continues a plant if and only if the former contains the latter, and
every r element subset of the former is isomorphic to some subset of
the latter.)
NOTE: In http://www.cs.nyu.edu/pipermail/fom/2016-May/019880.html we
used perpetuation for 3-continuation. I now want the greater
flexibility with the notation r-continuation, where continuations are
2-continuations.
4. Plants grow as much as they can subject to a height restriction (at
least assuming that the physical conditions are good). (We consider
maximal plants continuing a given seed in the sense of inclusion).
5. Plants exhibit lots of symmetry. (Maximal continuations of seeds
can always be found with some explicitly given partial self
embeddings, and also global self embeddings satisfying very simple
conditions).
In http://www.cs.nyu.edu/pipermail/fom/2016-May/019880.html, I only
presented the explicitly given partial self embeddings, and didn't use
global self embeddings. A partial self embedding is a 1-1 partial h:Q
into Q that works for all p's from dom(h). A global self embedding is
a partial self embedding whose domain is Q.
The reason I didn't use global self embeddings in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019880.html was because
it is not yet clear that I get independence from ZFC this way. There
are some open issues.
NOTE: In a future revision, we will take spatial features more
seriously. We should seek dimension k = 3, which is the dimension of
your garden. Also, you garden isn't simply represented spatially by
Q^3|<=3, but a product of three more reasonable intervals, maybe not
all three the same interval.If we use k = 3, we will probably have to
use r-continuations, where r is at least somewhat greater than 2.
EMBEDDED MAXIMAL CONTINUATIONS (global). EMCg. For finite subsets of
Q^k|<0, some maximal continuation within Q^k|<=n has a self embedding
which pointwise fixes exactly the negative rationals.
EMBEDDED MAXIMAL CONTINUATIONS (partial). EMCp. For finite subsets of
Q^k|<0, some maximal continuation within Q^k|<=n is partially self
embedded by the function p if p < 0; p+1 if p = 0,...,n-1.
FINITE EMBEDDED MAXIMAL CONTINUATIONS. FEMC. For finite subsets of
Q^k|<0, some two successive finite rich 3-continuations within Q^k|<=n
are partially self embedded by the function p if p < 0; p+1 if p =
0,...,n-1.
Here rich is weaker than maximal, and only requires maximality with
respect to rational tuples of number theoretic height at most that of
the set being continued. Maximal continuations are generally infinite,
whereas rich continuations can be finite. But they need to be iterated
at least once to gain their tremendous logical power.
NOTE: The latter two statements appear in
http://www.cs.nyu.edu/pipermail/fom/2016-May/019880.html with n = k in
order to save a parameter. I changed my mind about using an additional
parameter here. I want the greater flexibility, e.g., for growth
spurts. Also I intend to show that k,n can both be taken to be tiny
integers (for incompleteness), but not necessarily the same tiny
integer.
NOTE: All three statements are implicitly Pi01 via the Goedel
Completeness Theorem. The latter two statements are equivalent to
Con(SRP) over WKL_0. However, we don't know if the first statement is
provable in ZFC.
NOTE: If we replace Q^k|<=n with Q^k in these three statements, then
we we not know if they become provable in ZFC.
2. GROWTH SPURTS - STEP MAXIMALITY
First we can grow from seed V to S maximally within Q^k|<=0. Then we
can grow from S to S' maximally within Q^k|<=1. Then we can grow from
S' to S'' maximally within Q^k|<=2. And so forth. Let S* be the union.
What symmetry are we expected to have for S*?
It will be useful to be more formal about this and make the following
definition. S is a step maximal continuation of finite V containedin
Q^k|<0 if and only if
*) Each S|<=n-1 is a maximal continuation of V within Q^k|<=n-1.
(Recall that by convention, n is a positive integer).
In terms of global symmetry, we want a global embedding of S* which
pointwise fixes exactly the negative rationals. We can ask for more:
EMBEDDED STEP MAXIMAL CONTINUATIONS (global). ESMCg. For finite
subsets of Q^k|<0, some step maximal continuation has a self embedding
which pointwise fixes the negative rationals and adds 1 to all
nonnegative integers.
EMBEDDED STEP MAXIMAL CONTINUATIONS (partial). ESMCp. For finite
subsets of Q^k|<0, some step maximal continuation is partially self
embedded by the function p if p < 0; p+1 if p is in N.
These statements are equivalent to Con(SRP) over WKL_0. They are
implicitly Pi01 via the Goedel Completeness Theorem.
3. SMOOTH GROWTH
The growth in section 2 is evidently much smoother than general growth
within Q^k.
We can try for the ultimate in smoothness:
**) For all p >= 0, S|<=p is a maximal continuation of V within Q^k|<=p.
Unfortunately, there are V for which we cannot attain **). But we can
get reasonably close to this in the following sense.
Define fld(S) to be the set of all coordinates of elements of S.
***) For all p >= 0, S|<=p is a maximal continuation of V within
(fld(S) union N)^k|<=p.
Thus we say that S is a smoothly maximal continuation of V if ***) holds.
EMBEDDED SMOOTHLY MAXIMAL CONTiNUATIONS. ESMC. For finite subsets of
Q^k|<0, some smoothly maximal continuation is embedded by the function
p if p < 0; p+1 otherwise.
ESMC is provably equivalent to Con(SRP) over WKL_0.
***********************************************
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 681st 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
Harvey Friedman
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