[FOM] 703: Large Cardinals and Continuations/16

[Harvey Friedman]

The structure of Continuation Theory is starting to take hold. The
systematic development is already rather extensive, and needs some
powerful organizational principles in order to stay coherent.

In this posting, we present an outline of the present development
which we hope is clarifying.

The lead statement, which is implicitly Pi01, using translation
between those particular open line segments in Q[-1,1]^k+1. Obviously,
section 1.1.1 itself is enough for most casual readers.

We have decided to describe the line segments used for the Lead as
part of the Lead. This can be done reasonably conveniently. This is
immediately followed by the Lead Template. Thus we have

LEAD. For finite subsets of Q[-1,0)^k+1, some maximal continuation in
Q[-1,1]^k+1 translates between the open line segments obtained by
dropping 1 from the points (0,1,...,k-1)/k, (1,2,...,k)/k positioned
in the hyperplane x_k+1 = 0.

LEAD TEMPLATE. Let L and L' be rational line segments in Q[-1,1]^k.
For finite subsets of Q[-1,0)^k, some maximal continuation in
Q[-1,1]^k translates between L and L'.

We will prepare the reader with the very familiar construction in
Q[-1,1]^3 that takes any point in the xy plane z = 0 and drops one
unit down, excluding the endpoints.

But to set the stage properly for explicitly finite statements, and
also statements corresponding to the HUGE hierarchy (and now higher),
it is natural to present a much greater variety of notions of
Continuation.

The statements corresponding to HUGE most naturally sit in the
LINEARLY ORDERED DIGRAPH context. See section 5.2, and for the
explicitly Pi01 forms, see section 6.2. In fact, natural stronger
versions are given there that lie between large large cardinal axioms
I3 and I2. So bigger than j:V(kappa) into V(kappa).. An initial foray
into the graph context is found in
http://www.cs.nyu.edu/pipermail/fom/2016-August/020031.html sections
4.5, but that corresponds to MAHLO.

We will present an outline for Continuation Theory, in its present
form. Each section with a two digit name has a key definition(s) that
drives that section. We will present those key definitions in the next
posting.

The Lead Statement, found in 1.1.1, has clear metaphorical power as
described most vividly (to date) in
http://www.cs.nyu.edu/pipermail/fom/2016-July/019966.html section 1,
Informal Symmetric Maximal Continuation.

Now what about 3 dimensions?

The reversal situation is too delicate for me to make a definite claim
about three dimensions. I need to be immersed in writing reversals
before I can see three dimensions clearly reversed. But I anticipate
being able to do this (and struggle hard if needed) - at least if I am
allowed k-continuations and not just continuations = 2-continuations.
Maybe something like 3-continuations will be enough.

Three dimensions has some major advantages. The metaphors with
seeds-to-plants, human conception to human maturity, and the big bang, etc., are
more compelling since we live in 3 dimensional space. Furthermore, the
two open line segments in question are obtained by dropping one unit
from (0,1/2) and (1/2,1) in the xy plane - very vivid.

CONTINUATION THEORY
OUTLINE

1. INFINITE CONTINUATION OF FINITE SETS.
   1.1.  MAXIMAL CONTINUATION IN Q[-1,1]^k.
      1.1.1. TRANSLATION.
      1.1.2. EMBEDDING.
   1.2. STEP MAXIMAL CONTINUATION IN Q^k.
      1.2.1. TRANSLATION.
      1.2.2. EMBEDDING
   1.3. FULL CONTINUATION IN Q[-1,1]^k.
      1.3.1. TRANSLATION.
      1.3.2. EMBEDDING.
   1.4. STEP FULL CONTINUATION IN Q^k.
      1.4.1. TRANSLATION.
      1.4.2. EMBEDDING.
   1.5. GREEDY CONTINUATION IN Q^k.
      1.5.1. TRANSLATION.
      1.5.2. EMBEDDING.
2. FINITE CONTINUATION OF FINITE SETS.
   2.1.  FINITE HEIGHT (LENGTH) MAXIMAL CONTINUATION IN Q[-1,1]^k (DQ[-1,1]^k).
      2.1.1. TRANSLATION.
      2.1.2. EMBEDDING.
   2.2. FINITE HEIGHT (LENGTH) STEP MAXIMAL CONTINUATION IN Q^k (DQ^k).
      2.2.1. TRANSLATION.
      2.2.2. EMBEDDING
   2.3. FINITE FULL CONTINUATION IN Q[-1,1]^k.
      2.3.1. TRANSLATION.
      2.3.2. EMBEDDING.
   2.4. FINITE STEP FULL CONTINUATION IN Q^k.
      2.4.1. TRANSLATION.
      2.4.2. EMBEDDING.
   2.5. FINITE GREEDY CONTINUATION IN Q^k.
      2.5.1. TRANSLATION.
      2.5.2. EMBEDDING.
3. OMEGA-GRAPH CONTINUATION OF OMEGA-GRAPHS.
4. FLOG CONTINUATION OF FLOGS.
5. GREEDY CONTINUATION OF FLODIGS.
   5.1. EMBEDDING.
   5.2. NEIGHBORHOOD EMBEDDING.
6. GREEDY FLODIG CONTiNUATION OF FLODIGS.
   6.1. EMBEDDING.
   6.2. NEIGHBORHOOD EMBEDDING.

The Lead Statement is in section 1.1.1. All of section 1 continues
finite sets to infinite sets, and is implicitly Pi01, corresponding to
the SRP hierarchy. All of section 2 is explicitly Pi02 and explicitly
Pi01, also corresponding to the SRP hierarchy. DQ is the dyadic
rationals. Length in DQ is analogous to height in Q.

Omega-graphs are linearly ordered graphs whose order type is omega.
Flogs are finite linearly ordered graphs. Flodigs are finite linearly
ordered digraphs.

All infinite sets, graphs, and digraphs used are countable.

***********************************************
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 703rd 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

Harvey Friedman