[FOM] 549: Conservative Growth - beyond triples

Harvey Friedman hmflogic at gmail.com
Mon Oct 6 01:31:24 EDT 2014

This research was partially supported by the John Templeton
Foundation grant ID #36297. The opinions expressed here are those of
the author and do not necessarily reflect the views of the John
Templeton Foundation.

In the posting #547
http://www.cs.nyu.edu/pipermail/fom/2014-September/018194.html on
Conservative Growth, I discussed triples in the theory of Conservative

I first want to correct an error in #547. The section on graph theory
needs to be changed. I can't use graphs in the sense of (V,E), where E
is irreflexive and symmetric. Instead, I need to use digraphs in the
sense of (V,E), where E is a subset of V^2. Instead of using ADJ(x), I
use OUT(x,G) = {y: x E y}. This makes the digraph environment very
close to the set theory environment, but there are very natural
conditions to place on the structures in the set theory environment
that would not be well motivated in the digraph environment.

I am preparing a detailed extended abstract to be placed at
#86 shortly. It isn't there as of 10/5/14.

This takes the theory beyond triples. I presented the triples in #547
in the interest of minimalism. That posting is now well contained in
the new #86.

I still separate the four venues as in #547:

1. Digraph Theory. This is the most rudimentary setting that we consider.
A digraph is a G = (V,E), where E is a subset of V^2.
2. Model Theory. This is the most general setting that we consider.
The treatment is based on signatures SIG consisting of finitely many
constant and relation symbols, and equality. No function symbols.
3. Set Theory. Here the treatment takes into account very basic
elemental features of set theory based, as usual, on epsilon and
4. As Theories. Here we present section 3 in terms of a formal set theory.

Note that venues 1,2 have no set theoretic infrastructure, and venue
3, has extremely minimal set theoretic infrastructure.

For each of the four above, I present conservative growth in much
greater generality - along a linear ordering (time). If the linear
ordering is of cardinality 3 then we have the triples of #547.

We show that for finite linear orderings, we climb the SRP hierarchy.

There are a few ways of presenting conservative growth along omega. A
first way climbs the SRP hierarchy. A second way is above the SRP
hierarchy but below omega Erdos cardinals (kappa arrows omega). A
third way puts you past Ramsey cardinals but below one measurable

We then come to using omega + 1. Here we get above a measurable
cardinal, and below two measurable cardinals. The same with the omega
+ n.

With omega + omega, we again have a few formulations. These correspond
to roughly 1 measurable cardinal to a form of hyper measurability,
stronger than, e.g., a measurable cardinal kappa with kappa many
measurable cardinals below.

With omega + omega + 1, we get above a measurable cardinal with a
normal measure concentrating on measurable cardinals.

This process continues along the hierarchy of stronger kinds of measurability.

I am developing some new forms of conservative growth that are yet
stronger. We continue to use venues 1,2 which have no set theoretic
infrastructure, and venue 3, which has extremely minimal set theoretic


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 549th 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 #1-527 can be found at the FOM posting

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM

Harvey Friedman

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