[FOM] 689:Large Cardinals and Continuations/9
Harvey Friedman
hmflogic at gmail.com
Sat Jun 11 14:51:56 EDT 2016
THIS POSTING IS SELF CONTAINED
We continue to fine tune the headline statements. We restate
http://www.cs.nyu.edu/pipermail/fom/2016-June/019898.html . Our
headline statement now reads:
MAXIMAL CONTINUATION. MC. For all finite subsets of Q^2k|<0, some
maximal k-continuation within Q^2k|<=k has a translation from Q^k|<0 x
{(0,...,k-1)} onto Q^k|<0 x {(1,...,k)}.
followed by
MAXIMAL CONTINUATION (parametric). MCp. For all finite subsets of
Q^k|<0, some maximal n-continuation within Q^k|<=r has translations
from each Q^n|<m x {m,...,r-1}^k-n onto Q^n|<m x {m+1,...,r}^k-n.
*GLIMPSE OF THE FUTURE*
We are obviously making the symmetries involved more vivid. We are
looking toward a series of results that make these symmetries ever
more vivid, to the point of where we can show simple honest pictures
immediately digestible by the general public. At the advanced stage of
this series, we anticipate being in dimension 3 (a very nice dimension
to be in for seeds growing into plants), working in the unit cube with
a grid, using the shortest path from any grid point to the surface of
the unit cube (made of 6 two dimensional walls, of course). The idea
is that the patterns of the maximal continuation S on these shortest
paths have massive duplications. We see essentially the same thing
every time we look at S on these shortest paths. The other aspect is
obviously "maximal k-continuations within the unit cabe", So according
to this plan, there are going to be two numerical parameters used in
the unit cube in dimension 3. One is the fineness of the grid points -
with n^3 equally spaced grid points.. The other is the k in
k-continuations. After verifying that we get beyond ZFC using n,k as
variables, we anticipate a brain frying adventure to get both n and k
down to tiny numbers.
The plan is to speculate a bit on this glimpse of the future, but get
a writeup of SMC, SMC* below.
SYMMETRIC MAXIMAL CONTINUATIONS
by
Harvey M. Friedman
June 9, 2016
June 11, 2016
1. Maximal Continuation.
2. Regional Symmetry.
3. Symmetric Maximal Continuation.
4. Finite Symmetric Rich Continuation.
5. Smoothly Maximal Continuation.
6. Templates.
7. Subsets of N
8. Formal Systems Used.
1. MAXIMAL CONTINUATION
DEFINITION 1.1. Q,Z+,N is the set of all rationals, positive integers,
nonnegative integers, respectively. We use p,q for rationals and
n,m,r,s,t for positive integers, with and without subscripts, unless
otherwise indicated. A U. B is A U B if A,B are disjoint; undefined
otherwise (disjoint union). Let S containedin Q^k. S|<p, S|<=p, S|>p,
S|>=p is the set of all elements of S all of whose coordinate are <p,
<=p, >p, >=p, respectively.
DEFINITION 1.2. Let A containedin Q^k. h is an isomorphism from A onto
B if and only if B containedin Q^k, and h:Q into Q is an (everywhere
defined) order preserving bijection sending A onto B by the coordinate
action. (There are some variants of this, but not for finite A, the
only case we use). B is an r-continuation of A within C if and only if
A containedin B containedin C containedin Q^k, and every <=r element
subset of B is isomorphic to some subset of A. B is a maximal
r-continuation of A within C if and only if B is an r-continuation of A
within C, and no B U. {x} is an r-continuation of A within C.
We think of A containedin Q^k|<0 as a seed. We let the seed "grow"
within Q^k|<=s. We think of the r-continuations of A within Q^k|<=s as
plants. Maximal continuations of A within Q^k|<=s are thought of as
plants growing from A as much as possible, within the space constraint
Q^k|<=s.
THEOREM 1.1. Every A containedin Q^k|<0 has a maximal r-continuation
within Q^k|<s. Furthermore, there is an effective process that starts
with finite A and k,r,s, and produces a (normally infinite) maximal
r-continuation of A within Q^k|<=s.
The obvious constructions for Theorem 1.1 generally do not result in
maximal continuations with any symmetry.
DEFINITION 1.3. A sentence J is implicitly Pi01 if and only if for
some algorithm alpha processing finite strings, ZFC proves "J holds if
and only if alpha runs forever with the empty string input". J is
implicitly Pi01 over a formal system T if and only if t proves "J
holds if and only if alpha runs forever with the empty string input".
2. REGIONAL SYMMETRY
Regional symmetry refers to S containedin Q^k. We use the usual
Euclidean distance between elements of Q^k. Every isometry of Q^k
extends uniquely to an isometry of R^k. But the isometries of R^k that
extend an isometry of Q^k are very special. However, they include far
more than the translations by elements of Q^k.
DEFINITION 2.1. Let A,B containedin Q^k. A,B are isometric if and only
if there is a bijection f from A onto B such that for all x,y in A,
d(x,y) = d(fx,fy). Here d is the Euclidean distance.
DEFINITION 2.2. Let S,A,B containedin Q^k. S has an isometry from A
onto B if and only if S intersect A and S intersect B are isometric.
There are a number of obvious alternative definitions for these
notions. Our results hold for these variants.
There is a very special case of Definition 2.2 that occurs in our theory.
DEFINITION 2.3. Let S,A,B containedin Q^k. S has a translation from A
onto B if and only if there exists x in Q^k such that (S intersect A)
+ x = S intersect B.
In section 4, we will use the following weaker property.
DEFINITION 2.4. Let S,A,B containedin Q^k. S has a translation from A
into B if and only if there exists x in Q^k such that (S intersect A)
+ x containedin S intersect B.
We will be applying these definitions to nice A,B only.
DEFINITION 2.5. A rational interval is an interval in Q whose
endpoints are from Q union {-infinity,infinity}. A rational rectangle
is a finite product of rational intervals.
We will be focusing on rational rectangles, and finite unions of
rational rectangles.
3. SYMMETRIC MAXIMAL CONTINUATION
We first present our headline version of Symmetric Maximal
Continuation that uses just the numerical parameter k and two special
rational rectangles.
MAXIMAL CONTINUATION. MC. For all finite subsets of Q^2k|<0, some
maximal k-continuation within Q^2k|<=k has a translation from Q^k|<0 x
{(0,...,k-1)} onto Q^k|<0 x {(1,...,k)}.
THEOREM 2.1. MC is implicitly Pi01 via the Goedel Completeness
Theorem. In fact, MC is implicitly Pi01 over WKL_0.
Proof: First prove in RCA_0 that MC is equivalent to the refinement
that requires that the translation be by (0,...,0,1,...,1), using
dummy variables. Then show that this refinement asserts that each of
an
effectively given list of sentences in first order predicate calculus
with equality has a countable model. Now apply Goedel's completeness
Theorem. QED
MAXIMAL CONTINUATION (parametric). MCp. For all finite subsets of
Q^k|<0, some maximal s-continuation within Q^k|<=r has translations
from each Q^n|<m x {m,...,r-1}^k-n onto Q^n|<m x {m+1,...,r}^k-n.
Note that in MCp, we are using special finite unions of rational
rectangles. Obviously MCp implies MC over RCA_0.
THEOREM 2.2. MCp is provable in SRP+. If SRP is consistent, MC is not
provable in SRP. For each fixed k, MCp is provable in SRP. There is no
finite fragment of SRP which, for each fixed k, proves MC (assuming
SRP is consistent). Assuming ZFC is consistent, there is a fixed k
for which MC is not provable in ZFC. MC, MCp, and also MC, MCp for
some fixed k are independent of ZFC (assuming SRP is consistent). MC,
MCp are provably equivalent to Con(SRP) over WKL_0.
4. FINITE SYMMETRIC RICH CONTINUATION
Note that even though MC is implicitly Pi01, it is not explicitly
Pi01, because in general, maximal continuations are infinite.
We now give an explicitly Pi01 form of MC by weakening the maximality
to "rich", using the notion of height from number theory, so that
finite rich continuations can always be found.
DEFINITION 4.1. The height of a rational, rational vector, or finite
set of rational vectors, is the least integer i such that every
rational number involved is the ratio of two integers of magnitude at
most i. The height of an infinite set of rational vectors is infinity.
We use hgt for height.
DEFINITION 4.2. B is a rich k-continuation of A within C if and only
if B is a k-continuation of A within C, and no B U. {x}, hgt(x) <=
hgt(A), is an k-continuation of A within C.
THEOREM 4.1. For all finite subsets of Q^2k|<0, some finite rich
k-continuation within Q^2k|<=k has a translation from Q^k|<0 x
{(0,...,k-1)} onto
Q^k|<0 x {(1,...,k)}.
FINITE RICH CONTINUATION. FRC. For all finite subsets of Q^2k|<0,
some two successive finite rich k-continuations within Q^2k|<=k have
translations from Q^k|<0 x {(0,...,k-1)} onto Q^k|<0 x {(1,...,k)}.
FINITE RICH CONTINUATION (parametric). FRCp. For all finite subsets of
Q^k|<0, some t successive finite rich s-continuations within Q^k|<=r
have translations from each Q^n|<m x {m,...,r-1}^k-n onto Q^n|<m x
{m+1,...,r}^k-n.
FRC, FRCp are explicitly Pi02. There is an obvious a priori
exponential upper bound on the height of the successive finite rich
continuations. This results in explicitly Pi01 statements.
THEOREM 4.2. FRC, FRCp are provably equivalent to Con(SRP) over EFA.
5. SMOOTHLY MAXIMAL CONTINUATIONS
Here instead of continuing subsets of Q^k by subsets of Q^k, we
continue Q,k systems by Q,k systems.
DEFINITION 5.1. A Q,k system is a pair (A,R), where A containedin Q
and R containedin A^k. (A,R) is finite if and only if A is finite.
(A,R) is negative if and only if A containedin Q|<0.
DEFINITION 5.2. Let (A,R) be a Q,k system. (B,S) is a k-continuation
of (A,R) if and only if
i. (B,S) is a Q,k system.
ii. A U {0,1} containedin B.
iii. Every subset of S with <=k elements is isomorphic to some subset of R.
DEFINITION 5.3. Let (A,R) be a Q,k system. (B,S) is a smoothly maximal
k-continuation of (A,R) if and only if (B,S) is a k-continuation of
(A,R), and no (B|<=max(x),S U. {x}), min(x) < 0 < max(x), is a
k-continuation of (A,R).
DEFINITION 5.4. Let (A,R) be a Q,k system, B,C containedin Q^k. (A,R)
has a translation from B onto (into) C if and only if R has a translation from
B onto (into) C.
SMOOTHLY MAXIMAL CONTINUATION. SMC. For all finite negative Q,2k
systems, some smoothly maximal k-continuation has a translation from
Q^k|<1 x Q^k|>=1 into Q^k|<1 x Q^k|>=2.
DEFINITION 5.5. Let (A,R) be a Q,k system. (B,S) is a smoothly rich
k-continuation of (A,R) if and only if (B,S) is a k-continuation of
(A,R), and no (B|<=max(x),S U. {x}), min(x) < 0 < max(x), hgt(x) <=
hgt(A), is a k-continuation of (A,R).
STRONG SMOOTHLY MAXIMAL CONTINUATION. SSMC. For all finite negative
Q,2k systems, some smoothly maximal k-continuation has translations
from Q^k|<1 x Q^k|>=1 into Q^k|<1 x Q^k|>=2 and from Q[1,k]^k x
{k+(3/2)}^k onto Q[2,k+1]^k x {k+(3/2)}^k.
THEOREM 5.1. SMC is provably equivalent to Con(SRP) over WKL_0. SSMC
is provably equivalent to Con(HUGE) over WKL_0. This holds even for a
fixed k.
FINITE SMOOTHLY MAXIMAL CONTINUATION. FSMC. For all finite negative
Q,2k systems, some two successive finite smoothly rich k-continuations
have translations from Q^k|<1 x Q^k|>=1 into Q^k|<1 x Q^k|>=2.
STRONG FINITE SMOOTHLY MAXIMAL CONTINUATION. SFSMC. For all finite
negative Q,2k systems, some two successive finite smoothly rich
k-continuations have translations from Q^k|<1 x Q^k|>=1 into Q^k|<1 x Q^k|>=2
and from Q[1,k]^k x {k+(3/2)}^k onto Q[2,k+1]^k x {k+(3/2)}^k.
The above two statements are explicitly Pi02. A priori upper bounds
can be given on the heights of the rich k-continuations, resulting in
explicitly Pi01 forms.
THEOREM 5.2. FSMC is provably equivalent to Con(SRP) over EFA. SFSMC
is provably equivalent to Con(HUGE) over WKL_0.
The same results apply to obvious parametric forms of SMC, SSMC, FSMC,
SFSMC. (Among the parameters to be introduced is the number t of
successive continuations, rather than just t = 2).
6. TEMPLATES
MAXIMAL CONTINUATION TEMPLATE. MCT. Let A,B,C_1,...,C_i,D_1,...,D_i be
finite unions of rational rectangles in a common dimension, and t >=
1. For all finite subsets of A, some maximal t-continuation within B
has translations from C_1,...,C_i onto D_1,...,D_i, respectively.
CONJECTURE. Every instance of MCT is refutable from RCA_0 or provable in SRP.
This Conjecture looks quite accessible if all of the rational
rectangles involved are a product of rational intervals, each of which
are singletons or have endpoint -infinity.
SMOOTHLY MAXIMAL CONTINUATION TEMPLATE. SMCT. Same as MCT but with
maximal replaced by smoothly maximal.
CONJECTURE. Every instance of MCT is refutable from RCA_0 or provable in SRP.
This Conjecture looks quite accessible without restricting the
rational rectangles involved.
STRONG SMOOTHLY MAXIMAL CONTINUATION TEMPLATE. SSMCT. Let
A,B,C_1,...,C_i,D_1,...,D_i,E_1,...,E_j,F_1,...,F_j be finite unions
of rational rectangles in a common dimension, and t >= 1. For all
finite subsets of A, some maximal t-continuation within B has
translations from C_1,...,C_i onto D_1,...,D_i, and from E_1,...,E_j
into F_1,...,F_j, respectively.
Obviously each instance of MC, MCp is an instance of MCT. Each
instance of SMC is an instance of SMCT. Each instance of SSMC is an
instance of SSMCT.
7. SUBSETS OF N
We can go down to one dimension if we involve addition. In fact, we
can work within N. See
http://www.cs.nyu.edu/pipermail/fom/2016-June/019892.html.
8. FORMAL SYSTEMS USED
EFA
RCA_0
WKL_0
ACA
ZFC
SRP
SRP+
MAH
MAH+
HUGE
HUGE+
***********************************************
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 689th 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
684: Large Cardinals and Continuations/5 6/3/16 3:56AM
685: Large Cardinals and Continuations/6 6/4/16 8:39PM
686: Refuting the Continuum Hypothesis?/4 6/616 9:29PM
687: Large Cardinals and Continuations/7 6/7/16 10:28PM
688: Large Cardinals and Continuations/8 6/9/16 11:41PM
***********************************************
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 689th 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
684: Large Cardinals and Continuations/5 6/3/16 3:56AM
685: Large Cardinals and Continuations/6 6/4/16 8:39PM
686: Refuting the Continuum Hypothesis?/4 6/616 9:29PM
687: Large Cardinals and Continuations/7 6/7/16 10:28PM
688: Large Cardinals and Continuations/8 Jun 9 23:41:05 EDT 2016
Harvey Friedman
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