[FOM] 764: Large Cardinals and Emulations/43

Thu May 11 00:26:33 EDT 2017

Building on the new definition of an emulation of E containedin M^k,
where M is any relational structure, presented in
http://www.cs.nyu.edu/pipermail/fom/2017-May/020481.html, we now have
a major breakthrough in the explicitly finite Emulation Theory.

For comparison, let us start with the original maximal emulation
definition, followed by a trivial restatement:

S is a maximal emulation of E containedin M^k if and only if S is an
emulation of E containedin M^k which is not a proper subset of an
emulation of E containedin M^k

S is a maximal emulation of E containedin M^k if and only if S is an
emulation of E containedin M^k, where every emulation S U {x} of E
containedin M^k is S.

NEW DEFINITION. S is a relatively maximal emulation of E containedin
M^k if and only if S is an emulation of E containedin M^k, where every
emulation S U {x} of E containedin M^k is an emulation of S
containedin (M,E)^k.

Note that the new definition is trivial if we replace (M,E) by M.
I.e., it just refuses to S being an emulation of E containedin M^k.

THEOREM. Let M be of finite type. Then every finite E has a finite
relatively maximal emulation.

Proof: Let M,E be as given. Define emulations S_0,S_1,... of E
containedin M^k as follows. S_0 = M. S_i+1 is any S_i U. {x} that is
an emulation of E containedin M^k, but not an emulation of S_i
containedin (M,E)^k if there is one; otherwise undefined. This clearly
must terminate, as emulation for any structure of finite type is an
equivalence relation with finitely many equivalence classes. QED

Now recall

MED/1. For finite subsets of Q[0,1]^k, some maximal emulation is drop
equivalent at (1,1/2,...,1/k),(1/2,...,1/k,1/k).

NEW MED/1. For finite subsets of Q[0,1]^k, some finite relatively
maximal emulation is drop equivalent at
(1,1/2,...,1/k),(1/2,...,1/k,1/k).

We can also use droppable, and also our two finiteness conditions
(using prunings and iterated prunings) - where the only difference
between the old form and the new form is the replacement of

maximal emulation

with

finite relatively maximal emulation

So this gives explicitly Pi02 Incompleteness at the level of SRP. Easy
a priori upper bounds give us explicitly Pi01 Incompleteness also at
the level of SRP.

With the finite forms, we can use [0,1] in the reals instead of Q[0,1]:

NEW MED/2. For finite subsets of [0,1]^k, some finite relatively
maximal emulation is drop equivalent at
(1,1/2,...,1/k),(1/2,...,1/k,1/k).

where we are of course using M = ([0,1],<) instead of (Q[0,1],<).

************************************************************************
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 764th 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
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

Harvey Friedman