[FOM] 694: Large Cardinals and Continuations/12

[Harvey Friedman]

THIS POSTING IS SELF CONTAINED

This supersedes http://www.cs.nyu.edu/pipermail/fom/2016-June/019911.html

Now I really think I am done with this run of ideas so that I can get
back to writing detailed proofs.. I should have details available for
the core stuff this calendar year.

LARGE CARDINALS AND CONTINUATIONS
draft
by
Harvey M. Friedman
June 29, 2016

ABSTRACT. We consider continuations of finite subsets of Q[-1,0)^k in
Q[-1,1]^k that are inclusion maximal (Q is the rationals). Here a
continuation is a superset with the same <=2 element subsets, up to
order isomorphism. Such maximal continuations are generally infinite,
and are easily effectively constructed using a greedy procedure.
Maximal Continuation Translation asserts that we can always find such
maximal continuations with some  translation symmetry from some given
rational line segments. This is proved by necessarily using far more
than the usual ZFC axioms for mathematics. Some explicitly finite
forms of Maximal Continuation Translation (Rich Continuation
Translation in Q^k and in N) are also proved by necessarily going far
beyond ZFC.

1. Maximal Continuation Translation in Q[-1,1]^k.
  1.1. Line Translation.
  1.2. Box Translation.
  1.3. Embedding.
2. Rich Continuation Translation in Q[-1,1]^k.
3. # Continuation Translation  in Q^k.
4. ## Continuation Translation  in Q^k.
5. Rich Factorial Continuation in N.
6. Rich Continuation Translation in N.
7. Future Research
8. Logical Notions.

1. MAXIMAL CONTINUATION TRANSLATION N Q[-1,1]^k

DEFINITION 1.1. Q,Z+,N is the set of all rationals, positive integers,
nonnegative integers, respectively. We use p,q for rationals and
i,j,k,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). Q[p,q], Q[p,q) is Q intersect [p,q], Q
intersect [p,q), respectively.

DEFINITION 1.2..Let A containedin Q^k. A+p = {x+p: x in A}. A/p =
{x/p: x in A}.The field of A, fld(A), is the set of all coordinates of
elements of A. h is an order isomorphism from A onto B if and only if
B containedin Q^k and
h is an increasing bijection from fld(A) onto fld(B) which sends A
onto B.

DEFINITION 1.3. Let A containedin Q^k. 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 has an order isomorphism onto a 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, where no B U. {x} is an r-continuation
of A within C.

We think of finite A containedin Q[-1,0)^k as a seed. We let the seed
"grow" within Q[-1,1]^k. We think of the r-continuations of seed A
within Q[-1,1]^k as plants (seeds grow into plants). Every small
pattern in a plant must be already present in the seed. Maximal
continuations of seed A within Q[-1,1]^k are thought of as plants
growing from seed A as much as possible, under the space constraint
Q[-1,1]^k.

THEOREM 1.1. Every A containedin Q[-1,1]^k has a maximal r-continuation
within Q[-1,1]^k. Furthermore, there is an effective process that starts
with finite A and k and produces a (normally infinite) maximal
r-continuation of A within Q[-1,1]^k.

The obvious greedy constructions for Theorem 1.1 generally do not
result in maximal continuations with any significant kind of symmetry
or embedding. Obtaining such is the focus of sections 1.1, 1.2, 1.3.

      1.1. LINE TRANSLATiON

DEFINITION 1.1.1. Let S containedin Q^k. S translates A by c if and
only if c in Q^k, and for all x in A, x in S iff x+c in S.

This corresponds to visual translation symmetry if we think of S as a
set of black and white pixels, with black indicating membership.

We now present our headline statement that can only be proved by going
well beyond the usual ZFC axioms for mathematics. We have chosen to
use only one numerical parameter k, and some specific rational numbers
involving k. We follow the usual convention that the statement itself
quantifies over all choices of parameters.

MAXIMAL CONTINUATION TRANSLATION (special). MCTs. For finite subsets
of Q[-1,0)^k, some maximal 2-continuation within Q[-1,1]^k translates
{(0,...,k-2)}/k-1 x Q[-1,0) by (1,...,1,0).

For k = 1,  {(0,...,k-2)}/k-1 = {x/(k-1): x in {(0,...,k-2)}} =
emptyset =  {(0,...,k-2)}/k-1 x Q[-1,0), validating the statement.

NOTE CONCERNING GRAMMAR. We have avoided the following prima facie
preferable English grammar:

MCT*. Every finite subset of Q[-1,0)^k has a maximal 2-continuation
within Q[-1,1]^k translating {(0,...,k-2)}/k-1 x Q[-1,0) by
(1,...,1,0).

The reason we have avoided MCT* is that because of the way "maximal"
is used throughout mathematics, we might be looking for a maximal
object subject not only to the condition "within Q[-1,1]^k", but also
subject to the condition  "translating {(0,...,k-2)}/k-1 x Q[-1,0) by
(1,...,1,0)". Notice how this potential ambiguity is avoided by MCTs.
However, in sections 2-5, we do not use "maximal", and therefore there
will not be any ambiguity in following the grammatical construction in
MCT*.

DEFINITION 1.1.2. x,y in Q^k are order equivalent if and only if for
all 1 <= i,j <= k, x_i < x_j iff y_i < y_j. x,y in Q^k is order
equivalent over negatives if and only if for all p < 0, (x,p), (y,p)
are order equivalent.

DEFINITION 1.1.3. Let A containedin Q[-1,1]^k. A is order preserved by
c (over negatives) if and only if every x in A is order equivalent to
x+c in Q[-1,1]^k (over negatives).

MAXIMAL CONTINUATION TRANSLATION (lines). MCTL. Let A containedin
Q[-1,1]^k be a rational line segment, c in Q^k, r >= 1. The following
are equivalent.
i. A containedin Q[-1,1]^k is order preserved by c over negatives.
ii. For finite subsets of Q[-1,0)^k, some maximal r-continuation
within Q[-1,1]^k translates A by c.

THEOREM 1.1.1. MCTL for ii implies i only is provable in RCA_0. MCTs,
MCTL, and MCTL with i implies ii only, are implicitly Pi01 over WKL_0
via the Goedel Completeness Theorem.

We now consider multiple lines at once. We will not attempt to obtain
complete information here. We will just give a rather special
sufficient condition.

DEFINITION 1.1.4. A rational line segment in Q[-1,1]^k is negatively
based if and only if every projection on the k axes is either a
singleton or has a negative element.

MAXIMAL CONTINUATION TRANSLATION (multiple lines). MCTML Let
A_1,...,A_n and B_1,...,B_n be negatively based rational line segments
in Q[-1,1]^k which are order preserved by c_1,...,c_n over negatives,
respectively. For finite subsets of Q[-1,0)^k, some maximal
r-continuation within Q[-1,1]^k translates each A_i by c_i.

THEOREM 1.1.2. Let T be any of MCTs, MCTL, MCTL with i implies ii
only, MCTML. T is provably equivalent to Con(SRP) over WKL_0.

     1.2. BOX TRANSLATION

DEFINITION 1.2.1. A rational interval is a nonempty interval of
rationals with endpoints from Q U {-infinity,infinity). A rational
interval is trivial if and only if it has exactly one element. A is a
box if and only if A
is a product of nonzero finite length of rational intervals. The
dimension of of a box is the number of its intervals that are nontrivial,
counting repetitions.

MAXIMAL CONTINUATION TRANSLATION (boxes). MCTB. Let A containedin
Q[-1,1]^k be a box, c in Q^k, r >= 1. The following are equivalent.
i. A containedin Q[-1,1]^k is order preserved by c over negatives.
ii. For finite subsets of Q[-1,0)^k, some maximal r-continuation
within Q[-1,1]^k translates A by c.

THEOREM 1.2.1. MCTB for ii implies i only is provable in RCA_0. MCTB,
and MCTB with i implies ii only, are implicitly Pi01 over WKL_0 via
the Goedel Completeness Theorem.

We also consider multiple boxes. Here we only use a rather special
sufficient condition.

DEFINITION 1.2.2. A box in Q[-1,1]^k is negatively based if and only
if every interval of the box is either a singleton or has a negative
element.

iMAXIMAL CONTINUATION TRANSLATION (multiple boxes). MCTMB. Let
A_1,...,A_n and B_1,...,B_n be negatively based boxes in Q[-1,1]^k
which are order preserved by c_1,...,c_n over negatives, respectively.
For finite subsets of Q[-1,0)^k, some maximal r-continuation within
Q[-1,1]^k translates each A_i by c_i.

THEOREM 1.2.2. Let T be any of MCTB, MCTB with i implies ii only,
MCTMB. T is provably equivalent to Con(SRP) over WKL_0.

     1.3. EMBEDDING

DEFINITION 1.3.1. Let S containedin Q^k. A partial embedding of S is a
strictly increasing partial function h:Q into Q such that for all
p_1,...,p_k in dom(h), (p_1,...,p_k) in S iff (h(p_1),...,h(p_k)) in
S.

DEFINITION 1.3.2. h:Q[-1,1] into Q[-1,1] is a partial piecewise
translation if and only if dom(h) is a finite union of rational
intervals, and h is a translation on each of them.

MAXIMAL CONTINUATION EMBEDDING (special). MCEs. For finite subsets of
Q[-1,0)^k, some maximal 2-continuation within Q[-1,1]^k is partially
embedded by the function p if p < 0; p+(1/k) if p = 0,1/k,...,(k-1)/k.

MAXIMAL CONTINUATION EMBEDDING/1. MCE/1. For finite subsets of
Q[-1,0)^k, some maximal r-continuations within Q[-1,1]^k is partially
embedded by the function p if p < 0; p+(1/t) if p = 0,1/t,...,(t-1)/t.

MAXIMAL CONTINUATION EMBEDDING (general). MCEg. Let h:Q[-1,1] into
Q[-1,1] be a strictly increasing partial piecewise translation. For
finite subsets of Q[-1,0)^k, some maximal r-continuation within
Q[-1,1]^k is embedded by h.

THEOREM 1.3.4. Let T be any of MCEs, MCE/1, MCEg. T is provably
equivalent to Con(SRP) over WKL_0.

2. RICH CONTINUATION TRANSLATION IN Q[-1,1]^k

We now give an explicitly Pi01 form of MCT by weakening the maximality
to "rich", using the notion of height from number theory.

DEFINITION 2.1. The height of a rational, rational tuple, or finite
set of rational tuples, is the least integer i such that every
rational number involved is the ratio of two integers of magnitude at
most i. Every finite set of rational tuples has a (the) height.

DEFINITION 2.2.  Let A,B,C containedin Q^k be finite. B is a rich
r-continuation of A within C if and only
if B is a finite r-continuation of A within C, and no B U. {x}, hgt(x) <=
hgt(A), is an r-continuation of A within C.

RICH CONTINUATION TRANSLATION. RCT. Every finite subset of Q[-1,0)^k
has 2 successive rich 2-continuations within Q[-1,1]^k translating
{(0,...,k-2)}/k-1 x Q[-1,0) onto {(1,...,k-1)}/k-1 x Q[-1,0).

RICH CONTINUATION TRANSLATION/1. RCT/1. Every finite subset of
Q[-1,0)^k has n successive rich r-continuations within Q[-1,1]^k
translating {(0,...,k-2)}/k-1 x Q[-1,0) onto {(1,...,k-1)}/k-1 x
Q[-1,0).

RICH CONTINUATION EMBEDDING. RCE. Every finite subset of Q[-1,0)^k has
2 successive rich 2-continuations within Q[-1,1]^k partially embedded
by the function p if p < 0; p+(1/k) if p = 0,1/k,...,(k-1)/k.

RICH CONTINUATION EMBEDDING/1. RCE/1. Every finite subset of Q[-1,0)^k
has n successive rich r-continuations within Q[-1,1]^k partially
embedded by the function p if p < 0; p+(1/t) if p = 0,1/t,...,(t-1)/t.

All four of the above are explicitly Pi02. There is an obvious a
priori exponential upper bound on the height of the successive rich
continuations. This results in an explicitly Pi01 statement.

THEOREM 2.1. RCT, RCA/1, RCE, RCE/1 are provably equivalent to
Con(SRP) over EFA. They are provable in EFA using only 1 rich
r-continuation.

3. #  CONTINUATION TRANSLATION IN Q^k

For # continuation, we use the ambient space Q^k instead of Q[-1,1]^k.

# will be a specific natural operator from subsets of Q^k into subsets of Q^k.

DEFINITION 3.1. Let E contianedin Q^k. E|<p, E|<=p, E|>p, E|>=p is the
set of all elements of E all of whose coordinates are <p, <=p, >p, >=p
respectively. E# = (fld(E) U {0})^k|>=0. E injectively translates from
A containedin Q^k by c if and only if c in Q^k, and for all x in A, if
x in E then
x+c in E.

DEFINITION 3.2. B is a #r-continuation of A containedin Q^k if and
only if B is an r-continuation of A, and no B|<max(x) U {x}, x in
B#\B, is an r-continuation of A.

# CONTINUATION/1. #C1. Every finite subset of Q^k|<0 has a
#2--continuation injectively translating from Q|<0 x Q^k-1|>=0 by
(0,1,...,1).

# CONTINUATION/3. #C2. Every finite subset of Q^k|<0 has a
#2-continuation injectively translating from Q^k|>=0 by (1,1,...,1) and
translating from {-1} x {k+(3/2)} x Q[0,k]^k-2 by (0,0,1,...,1).

# CONTINUATION/4. #C3. Every finite subset of Q^k|<0 has a #
r-continuation injectively translating from Q|<0 x Q^k|>=0 by
(0,1,...,1) and translating from each {-1} x {s+(1/s)} x Q[0,s-1]^k-2
by (0,0,1,...,1), s >= 2.

THEOREM 3.1. #C1 is provably equivalent to Con(SRP) over WKL_0.
#C2,#C3 are provably equivalent to Con(HUGE) over WKL_0.

4. ## CONTINUATION TRANSLATION IN Q^k

DEFINITION 4.1. B is a ##r-continuation of A containedin Q^k if and
only if B is a finite r-continuation of A, and no B|<max(x) U {x}, x
in A#\B, is an r-continuation of A.

## CONTINUATION/1. ##C1. Every finite subset of Q^k|<0 has 2
successive ##2-continuations injectively translating from Q|<0 x
Q^k-1|>=0 by (0,1,...,1).

## CONTINUATION/3. ##C2. Every finite subset of Q^k|<0 has 2
successive ##2-continuations injectively translating from Q^k|>=0 by
(1,1,...,1) and translating from {-1} x {k+(3/2)} x Q[0,k]^k-2 by
(0,0,1,...,1).

## CONTINUATION/4. ##C3. Every finite subset of Q^k|<0,has n
successive ##r-continuations injectively translating from Q|<0 x
Q^k|>=0 by (0,1,...,1) and translating from each {-1} x {s+(1/s)} x
Q[0,s-1]^k-2 by (0,0,1,...,1), s >= 2.

All of the above are explicitly Pi02. There is an obvious a priori
exponential type upper bound on the height of the successive rich
continuations. This results in an explicitly Pi01 statements.

THEOREM 4.1. ##C1 is provably equivalent to Con(SRP) over EFA.
##C2,##C3 are provably equivalent to Con(HUGE) over EFA.

5. RICH FACTORIAL CONTINUATION IN N

To simplify the presentation, we use only the numerical parameter n in
Definition 5.1.

DEFINITION 5.1. Let A containedin N. B is an n+ continuation of A if
and only if A containedin B and every system of at most n linear
inequalities (<,<=) and n linear congruences in n unknowns, with
coefficients of magnitude at most n and moduli at most n, that has a
solution over B has a solution over A. SUM(A,n) is the set of all sums
from A of length n. B is a rich/k n+ continuation of A if and only if
B is an n+ continuation of A, where no B|<t U {t}, t in SUM(A,n)\B
above k is an n+ continuation of A. A! = {m!: m in A}.

INFINITE FACTORIAL CONTINUATION. IFC. Every A contianedin {0,...,k} U
N! has 2 successive rich/k n+ continuations avoiding (8kn)!-1.

INFINITE FACTORIAL CONTINUATION/1. IFC/1. Every A contianedin
{0,...,k} U N! has r successive rich/k n+ continuations avoiding
[8krn,infinity)!-1.

FINITE FACTORIAL CONTINUATION. FFC. Every A containedin {0,...,k} U
{0,...,r}! has 2 successive rich/k n+ continuations avoiding (8kn)!-1.

FINITE FACTORIAL CONTINUATION/1. FFC/1. Every A containedin {0,...,k}
U {0,...,t}! has r successive rich/k n+ continuations avoiding
[8rkn,t]!-1.

There are obvious exponential type upper bounds on the continuations
involved in FFC and FFC/1. This puts FFC and FFC/1 in explicitly Pi01
form.

THEOREM 5.1. IFC, IFC/1, FFC, FFC/1 are provably equivalent to
Con(SMAH) over ACA

6. RICH N CONTINUATION TRANSLATION

DEFINITION 6.1. Let A containedin N. N translates from n to m by r if
and only if for all 0 <= i <= r, n+i in A iff m+i in A.

Note that Definition 6.1 gives us a form of translation symmetry
whereby A looks the same on [n,n+r] and [m,m+r].

RICH N CONTINUATION TRANSLATION (specific). RNCTs. Every A containedin
{0,...,(6k)!,(7k)!,(8k)!,(9k)!} has k successive rich/(6k)! k+
continuations translating from  (7k)!+(8k)! to (8k)!+(9k)! by (7k)!-1.

RICH N CONTINUATION TRANSLATION (general). RNCTg. Every A containedin
{0,...,(6k)!,(7k)!,(8k)!...,(k^2)!} has k successive rich/(6k)! k+
continuations translating from each (ik)!+...+(jk)! to
((i+1)k)!+...+((j+1)k)! by (ik)!-1, 7 <= i <= j < k-1.

There are obvious exponential type upper bounds on the continuations
involved in RNCTs and RNCTg. This puts RNCTs and RNCTg in explicitly
Pi01 form.

THEOREM 6.1. RNCTs, RNCTg are provably equivalent to Con(SRP) over EFA

7. FUTURE RESEARCH

A major open issue is to consider symmetries in maximal continuations
of varied kinds of mathematical objects, with appropriate notions of
"maximal", "continuation", and "symmetry".

More specifically, in section 1, which finite sets of translated line
segments can be used? Which finite sets of translated boxes can be
used? Which finite sets of embeddings can be used? What about more
general sets than lines and boxes? What about more general symmetries
than translations? E.g., isometries?

In sections 3,4, we have only used a particular operator #, and we
have not even begun to go beyond just the boxes that are used there
with those particular translations.

Using dimension 3 for section 1 to at least get well beyond ZFC seems
realistic, provided we are using multiple symmetries. This would make
the rough motivation through seeds growing into plants under a space
constraint more vivid, as your garden is indeed 3 dimensional.

8. LOGICAL NOTIONS

EFA  Exponential Function Arithmetic.
RCA_0  Our base theory for Reverse Mathematics. Recursive
Comprehension Axiom Scheme naught.
WKL_0  Our second main theory for RM. Weak Konig's Lemma naught.
ACA  Arithmetic Comprehension Axiom Scheme. Somewhat stronger than our
main third theory for RM.
ZFC  Zermelo Frankel Set Theory with the Axiom of Choice.
SRP  Stationary Ramsey Property. ZFC + {there exists lambda with the k-SRP}_k
SRP+  ZFC + "for all k, there exists lambda with the k-SRP".
SMAH  Strongly Mahlo. ZFC + {there exists a strongly k-Mahlo cardinal"_k.
SMAH+  ZFC + "for all k there exists a strongly k-Mahlo cardinal".
HUGE  ZFC + {there exists an k-huge cardinal}_k.
HUGE+  ZFC + "for all k, there exists a k-huge cardinal".

DEFINITION 8.1. 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".

***********************************************
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 694th 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
689: Large Cardinals and Continuations/9  6/11/16  2:51PM
690: Two Brief Sketches  6/13/16  1:18AM
691: Large Cardinals and Continuations/10  6/13/16  9:09PM
692: Large Cardinals and Continuations/11  6/15/16  10:22PM
693: Refuting the Continuum Hypothesis?/5  6/21/16  10:44AM

Harvey Friedman