# [FOM] 795: Beyond Perfectly Natural/1

Harvey Friedman hmflogic at gmail.com
Mon Mar 19 00:19:11 EDT 2018

```Emulation Theory is now Perfectly Natural, and we are moving to the
next higher standard. The next higher standard is: THEMATIC.

Emulation Theory is presently reasonably Thematic. However, the aim of
this series of postings is to fine tune, polish, and refine the THEMES
in Emulation Theory. The anticipated result is to make them RESONATE
more fully with various kinds of Mathematical Thinking.

In this quest, we have been led to a focus on two different kinds of
Mathematical Thinking.

COMBINATORIAL thinking
GEOMETRIC thinking

There are many other kinds of Mathematical Thinking, and also
important subdivisions of the above two kinds of Mathematical
Thinking, which we expect to EXPLOIT in our development of Emulation
Theory. E.g., Algebraic Thinking, Analytic Thinking, Topological
Thinking, and also Graph Thinking, Linear Thinking, etcetera.

So far, we have two up to date references on Emulation Theory. [1] is
a paper, with lots of proofs, and [2] is an abstract.

92. Concrete Mathematical Incompleteness: Basic Emulation Theory, July
1, 2017, 78 pages. To appear in the Putnam Volume, ed. Cook, Hellman.
Revised October 10, 2017.

99. Concrete Mathematical incompleteness Status 3/6/18 14 pages

In [1], we start with section 1.1, where we lay out a fairly general
form of Emulation Theory, and in section 1.2 and further, we focus
only on the (Q[0,1],<) environment, which is merely a countable dense
linear ordering with both endpoints. Thus we proceed purely order
theoretically in [1]. In [2], we only operate order theoretically,
using the (Q[0,n],<) environments. The difference here is only
notational - these environments are isomorphic, but with Q[0,n], we
can conveniently refer to distinguished elements 0,1,...,n.

In [1], we adhere to Implicitly Pi01 statements only that correspond
to SRP (a level of large cardinals). In [2], we discuss implicitly and
explicitly Pi01 statements, and also Inductive Equation Theory, and
correspondences with HUGE. But [2] is of course far less detailed than
[1] with regard to implicitly Pi01/SRP/Emulation Theory.

NEWS FLASH!! I MAY NOW SEE HOW INDUCTIVE EQUATION THEORY CAN BE OR
SHOULD BE CONSTRUED AS A PART OF GENERAL EMULATION THEORY. THIS WILL
BE TAKEN UP IN LATER POSTINGS, DEFINITELY NOT NOW!

##############

THEMATIC EMULATION THEORY/1

1. General Emulation Theory.
here
2. Order Emulation Theory.
here, and to be continued
3. Combinatorial Order Emulation Theory.
next posting
4. Geometric Order Emulation Theory.
next posting

1. GENERAL EMULATION THEORY

The most general conception that we have of Emulation Theory at
present is as follows.

Let W be a class of mathematical structures. Let E be a particular
mathematical structure from W. An Emulation S of E in W is a
mathematical structure S in W such that any pattern (of prescribed
kind) that arises in S already arises in E. A Maximal Emulation S of E
in W is an Emulation S of E in W where no proper extension (of
prescribed kind) is an Emulation of E in W.

We refer to the above as a General Emulation Setting. - more
specifically, the class W, the patterns considered, and the notion of
extension used.

We first observe that under the usual kinds of precise realizations of
the above, we have, using Zorn's Lemma, that

1) Every structure in W has a maximal emulation

In the kind of concrete realizations that we focus on here, there will
be no need for Zorn's Lemma. In fact there will be an effective
construction of a maximal emulation of every structure from W.

Let P be a (significant or interesting or desirable) property of
structures from W. I.e., P may or may not hold of a given structure
from W. We seek to determine whether the following holds.

2) For structures from W, some maximal emulation has property P

This depends very much on P. and the General Emulation Setting.

EXPOSITIONAL POINT: We avoid writing the seemingly more attractive

3) Every structure from W has a maximal emulation with property P

because of a subtle ambiguity. This might mean that for every
structure from W, there is an emulation with property P that is
maximal among emulations with property P. That is of course NOT what
we intend by 2).

Now what kind of properties P do we have in mind? Since we have only
focused on a very specific concrete General Emulation Setting, our
ideas of the kinds of P are rather limited at the moment.

We have been considering a certain kind of invariance under priorly
given binary relations. Generally speaking, these amount to what we
view as Symmetry Conditions.

We anticipate a much further development of Order Emulation Theory in
the sense of section 2 below - before we begin to investigate General
Emulation Theory.

NOTE: A formulation considerably less general than the above General
Emulation Theory, but far more general than our Order Emulation
Theory, was briefly considered in section 1.1 of [1]. We haven't even
started to seriously investigate that.

GENERAL CONJECTURE. Just as in Order Emulation Theory, we rather
naturally and quickly run out of axioms in ZFC in order to prove
perfectly natural theorems of kind 2), as long as the General
Emulation Setting is a little bit rich.

2. ORDER EMULATION THEORY

We now present the Order Emulation Theory that we have been
developing. We use arbitrary rational intervals.

DEFINITION 2.1. A rational interval is an interval of rationals in the
usual sense with endpoints from Q U {-infinity,infinity}. A bounded
rational interval is a rational interval whose endpoints are in Q. The
empty set and singleton rationals are rational intervals. We use I,J,K
for intervals in Q.

Since EmulationTheory uses only the usual linear ordering on the
rationals, there are really only three relevant choices of intervals:
Q[0,1], Q(0,1], Q(0,1). The degenerate intervals are of no
significance whatsoever, and Q[0,1) simply reverses Q(0,1]. Other
intervals in Q are order isomorphic to one of these three.

In [1] we exclusively used Q[0,1] and in [2] we used Q[0,n]. We now
prefer to use whatever is most convenient and leads to the most
transparent statements. Thus if we are in a context where there is
nothing to be gained by singling out distinguished intermediate
points, we would normally use Q[0,1], Q(0,1], or Q(0,1), avoiding the
unnecessary use of a letter. If on the other hand we find it
convenient to single out specific intermediate points, then we might
use, e.g., Q[0,n], with distinguished points 0,1,...,n at the ready.

DEFINITION 2.2.. S is an emulator of E containedin I^k if and only if
S containedin I^k and every element of SxS is order equivalent to an
element of ExE (as 2k-tuples). S is a maximal emulator of E
containedin I^k if and only if S is an emulator of E containedin I^k,
where no proper superset of S is an emulator of E containedin I^k.

THEOREM 2.1. (RCA_0) Every E containedin I^k has the same emulators as
does some finite subset of E.

Theorem 2.1 tells us that we can formulate our statements using only
finite E containedin I^k, and have the same effect as stating them for
all E containedin I^k. We would rather state then using finite E
containedin I^k, as this is more concrete.

ORDER EMULATION THEORY. What properties can we require of a maximal
emulator? I.e., for which properties P is it the case that for finite
subsets of I^k, some maximal emulator satisfies P?

The KIND of Order Emulation Theory being pursued is determined by the
KIND of properties of subsets of I^k that one is considering. Thus in
Combinatorial Order Emulation Theory, combinatorial properties are
used. In Geometric Order Emulation Theory, geometric properties are
used.

Order Emulation Theory starts with the following crucial triviality.

THEOREM 2.2. Every finite subset of I^k has a maximal emulator. This
is provable in RCA_0.

DEFINITION 2.3. S contianedin I^k is equivalent at x,y if and only if
x in S iff y in S.

So now we ask the obvious question which is in both Combinatorial and
Geometric Order Emulation theory: For finite subsets of I^k, is there
a maximal emulator which is equivalent at x,y?

THEOREM 2.3. Let x,y in I^k, where I is not a bounded closed interval.
The following are equivalent.
i. x,y are order equivalent.
ii. For finite subsets of I^k, some maximal emulator is equivalent at x,y.

The above can be gleaned from [1]. But for I = Q[0,1], we left the
determination open, although for I = Q[0,1] and dimension k = 2, we
have a complete determination by Maximal Emulation Singleton Use/2,
page 37, in [1].

In both Combinatorial Order EmulationTheory and Geometric Order
Emulation Theory, we consider relations R containedin I^k x I^k.

DEFINITION 2.4. R containedin I^k x I^k is ME usable if and only if
for finite subsets of I^k, some maximal emulation has x R y implies
(S(x) implies S(y)). R containedin I^k x I^k is invariantly ME usable
if and only if for finite subsets of I^k, some maximal emulation has x
R y implies (S(x) iff S(y)).

THEOREM 2.4. If R containedin I^k x I^k is ME usable then R is order preserving.

To Be Continued...

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 795th 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

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
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
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/23/16  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM
745: Philosophy of Incompleteness/7  1/11/16  12:40AM
746: Philosophy of Incompleteness/8  1/12/17  3:54PM
747: PA Incompleteness/2  2/3/17 12:07PM
748: Large Cardinals and Emulations/30  2/15/17  2:19AM
749: Large Cardinals and Emulations/31  2/15/17  2:19AM
750: Large Cardinals and Emulations/32  2/15/17  2:20AM
751: Large Cardinals and Emulations/33  2/17/17 12:52AM
752: Emulation Theory for Pure Math/1  3/14/17  12:57AM
753: Emulation Theory for Math Logic  3/10/17  2:17AM
754: Large Cardinals and Emulations/34  3/12/17  12:34AM
755: Large Cardinals and Emulations/35  3/12/17  12:33AM
756: Large Cardinals and Emulations/36  3/24/17  8:03AM
757: Large Cardinals and Emulations/37  3/27/17  2:39AM
758: Large Cardinals and Emulations/38  4/10/17  1:11AM
759: Large Cardinals and Emulations/39  4/10/17  1:11AM
760: Large Cardinals and Emulations/40  4/13/17  11:53PM
761: Large Cardinals and Emulations/41  4/15/17  4:54PM
762: Baby Emulation Theory/Expositional  4/17/17  1:23AM
763: Large Cardinals and Emulations/42  5/817  2:18AM
764: Large Cardinals and Emulations/43  5/11/17  12:26AM
765: Large Cardinals and Emulations/44  5/14/17  6:03PM
766: Large Cardinals and Emulations/45  7/2/17  1:22PM
767: Impossible Counting 1  9/2/17  8:28AM
768: Theory Completions  9/4/17  9:13PM
769: Complexity of Integers 1  9/7/17  12:30AM
770: Algorithmic Unsolvability 1  10/13/17  1:55PM
771: Algorithmic Unsolvability 2  10/18/17  10/15/17  10:14PM
772: Algorithmic Unsolvability 3  Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1  Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2  Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3  Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1  12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4  12/28/17  8:02PM
778: Goedel's Second: Proofs/5  12/30/17  2:40AM
779: End of Year Claims  12/31/17  8:03PM
780: One Dimensional Incompleteness/1  1/4/18  1:14AM
781: One Dimensional Incompleteness/2  1/6/18  11:25PM
782: Revolutionary Possibilities/1  1/12/18  11:26AM
783: Revolutionary Possibilities/2  1/20/18  9:43PM
784: Revolutionary Possibilities/3  1/21/18  2:59PM
785: Revolutionary Possibilities/4  1/22/18  12:38AM
786: Revolutionary Possibilities/5  1/24/18  12:15AM
787: Revolutionary Possibilities/6  1/25/18  4:09AM
788: Revolutionary Possibilities/7  2/1/18  2:18AM
789: Revolutionary Possibilities/8  2/1/18  9:02AM
790: Revolutionary Possibilities/9  2/2/18  3:07AM
791: Emulation Theory/Inductive Equations/1  2/8/18  11:52PM
792: Emulation Theory/Inductive Equations/2  2/25/18  12:58PM
793: Emulation Theory/Inductive Equations/3  2/27/18  12:11AM
794: Emulation Theory/Inductive Equations/4  3/6/18  9:22PM

Harvey Friedman
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