[FOM] 700: Large Cardinals and Continuations/14

[Harvey Friedman]

The state of the art in Continuation Theory is
http://www.cs.nyu.edu/pipermail/fom/2016-July/019966.html

However, we do have some improvements/amplifications for section 2.4 there.

We now prefer to state Definition 2.3 in a weaker form and also more
abstractly. Also we have firmed up results in section 2.4.

NEW 2.4. MAXIMAL CONTINUATION TRANSLATION

Recall Definition 2.3.

DEFINITION 2.3. Let S containedin Q[-1,1]^k. S translates from A
onto B if and only if there is a translation from S intersect A onto S
intersect B.

(Definition 2.3 is now changed, and is preceded by the definition of a
translation from A to B, which requires that A,B be subsets of some
Q[-1,1]^k. Since translations are unique (if they exist), we refer to
it as "their translation").

DEFINITION 2.4.1. A,B are translation usable if and only if the
following holds universally. For finite subsets of
Q[-1,0]^k, some maximal continuation in Q[-1,1]^k translates from A
onto B.

We can now restate MCTs, which is our lead statement at the beginning
of the Abstract, in the following form.

MAXIMAL CONTINUATION TRANSLATION (specific). MCTs. For all k, Q[-1,0)
x {(0/k,...,(k-1)/k)}, Q[-1,0) x {(1/k,...,k/k)} are translation
usable.

Here is a necessary condition for translation usability.

THEOREM 2.4.1. If A,B are translation usable their translation
preserves order equivalence over the negative rationals.

This necessary condition is, in many cases, sufficient.

Note that the two displayed sets in MCTs are both boxes and lines segments.
First we focus on boxes.

DEFINITION 2.4.2. A box is a product of at least 1 and finitely many
intervals in Q[-1,1]. The intervals of a box are these factors. The
sides of a box are its non degenerate intervals.

MAXIMAL CONTINUATION TRANSLATION (box). MCTb. Two boxes with the same
sides of length < 1 are translation usable if an only if their
translation preserves order equivalence over the negative rationals.

DEFINITION 2.4.3. A line segment is the segment between two points in
some Q[-1,1]^k, with or without the two points.

MAXIMAL CONTINUATION TRANSLATION (lines). MCTl. Two line segments, in
the direction of a vector of 0's and 1's, are translation usable if
and only if their translation preserves order equivalence over the
negative rationals.

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This is the 700th in a series of self contained numbered
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Harvey Friedman