[FOM] 705: Large Cardinals and Continuations/18

[Harvey Friedman]

In http://www.cs.nyu.edu/pipermail/fom/2016-August/020048.html we left
out some very satisfying material concerning subsets of N =
nonnegative integers. Thus we have the new outline:

CONTINUATION THEORY
OUTLINE

1. INTRODUCTION.
2. INFINITE CONTINUATION OF FINITE SETS.
   2.1.  MAXIMAL CONTINUATION IN Q[-1,1]^k.
      2.1.1. TRANSLATION.
      2.1.2. EMBEDDING.
   2.2. STEP MAXIMAL CONTINUATION IN Q^k.
      2.2.1. TRANSLATION.
      2.2.2. EMBEDDING.
3. FINITE CONTINUATION OF FINITE SETS.
   3.1.  FINITE HEIGHT MAXIMAL CONTINUATION IN Q[-1,1]^k.
      3.1.1. TRANSLATION.
      3.1.2. EMBEDDING.
   3.2. FINITE HEIGHT STEP MAXIMAL CONTINUATION IN Q^k.
      3.2.1. TRANSLATION.
      3.2.2. EMBEDDING
4. CONTINUATION OF SETS OF NATURAL NUMBERS
   4.1. INFINITE SETS.
   4.2. FINITE SETS.
5. GREEDY k-CONTINUATION OF LINEARLY ORDERED DIRECTED GRAPHS.
6. FULL k-CONTINUATION  OF LINEARLY ORDERED DIRECTED GRAPHS.
   6.1. OMEGA-GRAPHS.
   6.2. FLODIGS - COUNTS.
   6.3. EMBEDDING.

where in its present form, 6.3 will probably be eliminated, not
meeting current standards. Concerning section 4: We find it convenient
to adhere to the single parameter k.

DEFINITION 4.1. A,B containedin N are k-isomorphic if and only if for
the unique increasing bijection h:A into B, for all
n_1,...,n_i,m_1,...,m_j in A, n_1 + ... + n_i < m_1 + ... + m_j iff
h(n_1) + ... + h(n_i) < h(m_1) = ... + h(m_j). Here 0 <= i,j <= n. The
<=k sums from A are the sums of lengths 0,...,k from A, repetitions
allowed. The powers of2n are the numbers 1,2,4,... .

DEFINITION 4.2. Let A containedin N. B is a full k-continuation of A
if and only if A containedin B containedin N, and <=k sums n from A
lie in B if and only if every <=k element subset of A U (B|<=n) is
k-isomorphic to an element of A.

PROPOSITION 4.1.1. Every set of the <=k sums from the powers of 2
starts k successive full k-continuations without 2^(8k)! - 1.

The above is equivalent to Con(SMAH) over ACA.

PROPOSITION 4.2.1. Every finite set of the <=k sums from the powers of
(8k)! starts k successive full k-continuations without 2^(8k)! - 1.

The above is equivalent to Con(SMAH) over ACA.

The base theory for these,ACA, can be considerably reduced but I need
to work with this some more (e.g., ACA' = for all n, n-th Turing jumps
of reals exist).

The next step is to move from these FOM postings on Continuation
Theory to the paper discussed at the beginning of
http://www.cs.nyu.edu/pipermail/fom/2016-August/020031.html. When it
is put on my website I will post the link.

***********************************************
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 705th 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

Harvey Friedman