# [FOM] 752: Emulation Theory for Pure Math/1

Harvey Friedman hmflogic at gmail.com
Tue Mar 14 00:57:50 EDT 2017

```It is now clear to me that Emulation Theory is best presented in
somewhat different ways for different audiences.

The particular way I have chosen to present Emulation Theory to a
"general audience" has been

mathematical realization. I then move down slowly and deliberately to
fill in more mathematical details.
2. I use Emulations, which is purely combinatorial, based on order of
tuples in a fixed dimension. (S is an r-emulation of E containedin
Q[0,1]^k if and only if E^r,S^r have the same elements up to order
equivalence).
3. I use Drop Equivalence as a viivid form of Symmetry, which is cast
in simple geometric terms, with the metaphor of two falling raindrops
from the same height.

That's all there is to the basics of Emulation Theory. This reaches
independence from the usual ZFC axioms for mathematics. The Theory
features generalizations of 2 and 3 in ways that are entirely
standard, most straightforwardly with 3.

STATEMENT: For subsets of Q[0,1]^k, some maximal r-emulation is drop
equivalent at (1,1/2,...,1/n), (1/2,...,1/n,1/n).

Here raindrops are at (1,1/2,...,1/n), (1/2,...,1/n,1/n), falling down
by taking the last coordinate 1/n down to 0.

Emulation Theory is, in a certain interesting sense, more fundamental
than any other deep mathematical theory in that it only involves an
enumerated dense linear ordering with endpoints - and not even
addition, multiplication, division, or minus. We don't use the actual
enumeration, but just that there is one. For definiteness, including a
definite enumeration, we can realize this any starting with two points
0 < 1. We then put a point between any two adjacent points, resulting
in 0 < (0,1) < 1. We then put a point between any two adjacent points,
resulting in 0 < (0,(0,1)) < (0,1) < ((0,1),1) < 1. And so forth. This
is more elementary and fundamental than even addition, multiplication,
division, minus. Conceptually, it is not mandatory to even use this
pairing operation, as the idea of placing new points between adjacent
old points is very vivid.

HOWEVER, this is definitely NOT the best way to present Emulation
Theory to many communities. It doesn't really leverage off of the
notions that these communities use every second of every minute of
every day of every year. Such notions basic for one community may not
even be known to other even adjacent communities, Or even if they are
known or explainable, they might not seem even to be natural.

In this posting and in the next one, I will start a recasting of
Emulation Theory in terms that are more fundamental to these two
communities: Pure Math (here), and Math Logic (next posting).

Also, it is clear that communities vary concerning the Top/Down vs.
Bottom/Up dichotomy, and in many of these communities, individuals
vary widely concerning this dichotomy. So I must start giving both
Top/Down and Bottom/Up dichotomies.

Now for the Pure Math community. I don't see any improvement on
Maximal Emulations. There is definitely an improvement on Maximal
Emulations for Math Logic, using "universal properties" that I will
take up in the next posting.

But Drop Equivalence definitely has a preferred alternative for Pure
Math (and for Math Logic). For Pure Math it is preferable to leverage
off of the notion of Invariance. This is even better for Pure Math
than Falling Raindrops.

There are various related notions of Invariance and Symmetry floating
around in Pure Math, but there is the general idea that it is very
fundamental.

Invariance in a point set sitting in an ambient space is an
appropriate setting in Pure Math.

DEFINITION. Let S be a set and f be a function from a subset of S into
S. S is f-invariant if and only if for all x in the domain of f, x
lies in S if and only if f(x) lies in S.

There are some variants and differences in terminology.The if and only
if is more commonly used in pure math than the implies. And of course
a very commonly loved situation is where the domain of f is exactly S.
Of course, this is incorporated in the above Definition by merely
extending f to all of S by the identity. Thus S is f-invariant if and
only if S is f*-invariant where f* is this extension to all of S.

It may also be useful, in Emulation Theory later, and in pure math, to
generalize this Definition to general functions f whose domain may not
be in S. E.g.,

DEIFNITION'. Let S be a set and f be a function. S is f-invariant if
and only if for all x in the domain of f, x lies in S if and only if
f(x) lies in S.

After defining Emulations (above see 2) using order equivalence, and
order equivalence as an obvious equivalence relation with trivial
examples, we do a little bit of Top/Down, as Pure Math can tolerate
this level of top/down:

PROTOTYPE. For subsets of Q[0,1]^k, some maximal emulation is f-invariant.

And then we say that Emulation Theory, basic part, is geared to
determining the truth and falsity of the Prototype for various simple
basic functions f from subsets of Q[0,1]^k into Q[0,1]^k.

So then the task at hand is to organize the results in terms of basic
families of functions f to be used in the Prototype. I then make this
definition:

DEFINITION. f is ME usable if and only if f maps some subset of some
Q[0,1]^k into Q[0,1]^k, where the Prototype holds.

unexpectedly interesting, and not quite solved. However, not
challenging ZFC.

Then I give the f that is associated with Drop Equivalence, first in
two dimensions, then in 3 dimensions, then in k dimensions. Then
apologize for being so special, and start giving some general classes
of f that work. Then it starts to look exactly like a general theory
with deep open questions. I.e.,

In two dimensions: f(1,p) = f(1/2,p), 0 <= p < 1/2. Already some
difficulties in proving Prototype.
In three dimensions: f(1,1/2,p) = (1/2,1/3,p), 0 <= p < 1/3. Probably
requires large cardinals.
In k dimensions: f(1,1/2,...,1/(k-1),p) = (1/2,...,1/k,p), 0 <= p <
1/k. Definitely requires large cardinals.

THEN and only then, do I define Drop Equivalence. First I actually
present the f for f-invariance.

NOTE: This presentation of f-invariance followed by displaying the f
will NOT work well for non Pure Math.
NOTE: For Math Logic, both of these would work pretty well, but MUCH
BETTER would be the approach in the next posting through universal
properties and strong indiscernibles.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 752nd 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

Harvey Friedman
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