[FOM] 711: Large Cardinals and Continuations//21

Sun Sep 18 10:42:03 EDT 2016

THIS POSTING IS SELF CONTAINED

We continue to fine tune Continuation Theory. As we initiated in ?, we
view both continuations and emulations as important. We make the
following assessment.

1. Continuations have compelling metaphorical content. E.g., seeds
grow into plants. Emulations also have significant metaphorical
content, but not as compelling as continuations.
2. On the other hand, emulations are more fundamental mathematically.
In fact, for much but not all of the development, continuations are
just emulations that are supersets, and maximal continuations are just
maximal emulations that are supersets. In some contexts, emulations
are enough simpler than continuations that we use only emulations.
3. In particular, in any context where we are using the very strong
form of maximality called *greedy*, the emulation approach is used and
the continuation approach is not. And in series contexts (series of
sets), we use "emulation towers".
4. At present, our preferred way to get to HUGE and beyond is to use
greedy emulations in digraphs. This is implicitly Pi01. For explicitly
Pi01, more effort is needed to meet current standards.

We have changed the outline from
http://www.cs.nyu.edu/pipermail/fom/2016-September/020071.html some to
reflect 1-4 above.

CONTINUATION THEORY
OUTLINE

1. INTRODUCTION.
2. INFINITE CONTINUATION/EMULATION.
   2.1.  MAXIMAL CONTINUATION/EMULATION IN Q[-1,1]^k.
      2.1.1. TRANSLATION.
      2.1.2. EMBEDDING.
   2.2. STEP MAXIMAL CONTINUATION/EMULATION IN Q^k.
      2.2.1. TRANSLATION.
      2.2.2. EMBEDDING.
3. CONTINUATION/EMULATION TOWERS OF FINITE SETS.
   3.1.  HEIGHT MAXIMALITY IN Q[-1,1]^k.
      3.1.1. TRANSLATION.
      3.1.2. EMBEDDING.
   3.2. HEIGHT STEP MAXIMALITY IN Q^k.
      3.2.1. TRANSLATION.
      3.2.2. EMBEDDING
4. GREEDY EMULATION OF LINEARLY ORDERED DIGRAPHS.
   4.1. EMULATION.
   4.2. EMULATION/<=.
5. GREEDY EMULATION TOWERS OF SETS OF NONNEGATIVE INTEGERS.
   5.1. INFINITE SETS.
   5.2. FINITE SETS.
6. GREEDY EMULATION TOWERS OF LINEARLY ORDERED DIGRAPHS.
   6.1. OMEGA-GRAPHS.
   6.2. FLODIGS - COUNTS.
   6.3. EMBEDDING.

NOTE: HUGE and beyond is achieved in 4.2, where we give extremely strong
implicitly Pi01. HUGE and beyond is achieved in 6.3 with explicitly
Pi01, but the present form of  6.3 does not meet
current standards. There is hope for simplification.

In section 2, we straightforwardly give continuation and emulation
forms. The essential difference is that with continuations, we have to
treat negatives in Q[-1,1] differently than nonnegatives. With
emulations, there is no different treatment for the negatives and the
nonnegatives. Here, as always, the more general symmetry conclusions
are simpler to state for the emulation forms than the continuation
forms. However, in the Lead Statements, there is no difference in the
conclusion: the continuation or the emulation translates between the
drops of (k,...,0)/k and (k-1,...,0,0)/k. NOTE: In section 2.2 with
step maximality, Q replaces Q[-1,1] and Q|<0 replaces Q[-1,0).

In section 2, there is considerable material on translations between
line segments and boxes. The drop of a point is a special kind of line
segment and box. We know exactly what drop translations can be
imposed, but we don't know in full generality what line or box
translations can be imposed.

In section 3, with Continuation Towers, we start with E containedin
Q[-1,0)^k and consider finite sequences of successive height maximal
continuations starting with a height maximal continuation of E. These
start with a height maximal continuation of E. This sequence is
automatically a tower. With Emulation Towers, we start with E
containedin Q[-1,1]^k, and consider finite sequences of successive
height maximal emulations which are required to form a tower. These
start with a height maximal emulation of E. I.e., a maximal emulation
tower. In particular, this first term does not have to be a superset
of E.  Note that emulation towers are thus a little more awkward than
continuation towers, but are still worth using. NOTE: In section 3.2
with height step maximality, Q replaces Q[-1,1] and Q|<0 replaces
Q[-1,0).

In section 4, we are back to the situation in section 2 where there is
just one continuation or emulation. Here continuations are enough more
awkward that we only use emulations. We first define full emulations,
which are maximal in the sense that if you add a new edge and keep the
same vertices, you must no longer be an emulation. Greedy emulations
are where all initial segments are full emulations. In emulations/<=,
the constraints are on the up edges only, which are the edges (v,w), v
<= w. This extra freedom is what is needed to get to HUGE and beyond.

In section 5, we are back to Emulation Towers as in section 3,
However, this time we are in one dimension, and even the infinite case
needs to have Towers (as opposed to section 2). We can get away with
one dimension because we are using addition. In sections 2,3,4,6,
there is no addition - everything is purely order theoretic. We
structure the definitions just as we have for greedy emulation towers
in section 4. We get something quite tangible for using addition - we
can avoid using any symmetry and instead make purely order theoretic
conclusions. Translation Symmetries can also be introduced here
instead, and there is a point to this. The purely order theoretic
conclusions correspond to MAHLO, whereas  Translation Symmetries
correspond to SRP.

In section 6, we use towers of omega-graphs and towers of finite
graphs. The omega-graph formulations in section 6.1 are related to
section 5.1, corresponding also to MAHLO, but without using addition.
The finite graph formulations in section 6.2 are related to section
5.2, corresponding also to MAHLO, but without using addition. Section
6.3 uses embeddings. The more basic use corresponds to SRP. However,
there is a form like section 4.2 which uses omega-graphs and which
uses finite graphs, that corresponds to HUGE. However, we have not
tracked down various opportunities for simplification.

The next step is expected to be preparing the manuscript with all of
the formal details of the claims, and with many proofs of the forward
directions. I.e., derivations of many of the Propositions from the
(consistency of) certain large cardinal hypotheses. This needs to be
followed by a manuscript with key reversals, giving us proofs of
unprovability in ZFC. After this, a complete treatment in the form of
a book on Concrete Incompleteness, that also includes the present book
length manuscript on Boolean Relation Theory currently on my website.

***********************************************
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 711th 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM

Harvey Friedman