[FOM] 761: Large Cardinals and Emulations/41

Sat Apr 15 16:54:23 EDT 2017

NOTE: THE NEW FINITE EMULATION IDEA DOES MAKE GOOD SENSE FOR EXPLICIT
Pi01 CORRESPONDING TO HUGE. THIS IS (AGAIN) EXPECTED TO CONCLUDE THE
PRESENTATION OF THE SCOPE OF EMULATION THEORY IN THIS PHASE. WE NOW
PLAN TO MOVE INTO WRITING MODE FOR THE REST OF 2017.

For the convenience of the reader, we will give the complete section
6, repeating the section 6 from Large Cardinals and Emulations/40.

EMULATION THEORY
Below will organize, as FOM postings, the contents of a ms. to be
placed on my website, and to appear in some form in a planned volume
in honor of Putnam. This ms. will contain proofs other than reversals.
Reversals will appear in a much expanded ms. later as a high priority,
and will form the book CONCRETE MATHEMATICAL INCOMPLETENESS, when
combined with the BOOLEAN RELATION THEORY ms. currently on my website.

1. INTRODUCTION.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020299.html
2. DROP EQUIVALENCE IN LINEAR ORDERINGS.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.html
3. MAXIMAL EMULATION IN Q[0,1]^k
http://www.cs.nyu.edu/pipermail/fom/2017-February/020300.html
      3,1, DROP EQUiVALENCE.
http://www.cs.nyu.edu/pipermail/fom/2017-February/020301.html
      3.2. USABILITY.
http://www.cs.nyu.edu/pipermail/fom/2017-March/020393.html
      3.3. EMBED USABILITY.
http://www.cs.nyu.edu/pipermail/fom/2017-March/020394.html
http://www.cs.nyu.edu/pipermail/fom/2017-March/020411.html
http://www.cs.nyu.edu/pipermail/fom/2017-March/020414.html
http://www.cs.nyu.edu/pipermail/fom/2017-April/020434.html
      3.4. TRANSLATION USABILITY.
http://www.cs.nyu.edu/pipermail/fom/2017-April/020435.html
4. STEP AND # MAXIMAL EMULATION IN Q^k.
Large Cardinals and Emulations/40
5. GREEDY AND UP GREEDY EMULATION IN Q^k.
Large Cardinals and Emulations/40
6. FINITE EMULATION.
Large Cardinals and Emulations/40
here

###########

6. FINITE EMULATION

Normally, every maximal r-emulation of E containedin Q[0,1]^k is
infinite. This tells us that. e.g.,

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

is implicitly Pi01 rather than explicitly Pi01. For explicit finite,
we need a statement of the form

FINITE MAXIMAL EMULATION. FME. For finite subsets of Q[0,1]^k, some
"weakly maximal r-emulation" is drop equivalent at (1,1/2,...,1/k),
(1/2,...,1/k,1/k).

We now define weak maximality. For comparison, we give the following
form of maximality:

S is a maximal r-emulation of E containedin Q[0,1]^k if and only if S
is an r-emulation of E containedin Q[0,1]^k, where every r-emulation S
U {x} of E containedin Q[0,1]^k is S.

DEFINITION 6.1. S is a finitely maximal r-emulation of E containedin
Q[0,1]^k if and only if S is a finite r-emulation of E containedin Q[0,1]^k,
where every r-emuation S U {x} x E of E^2 contianeidn Q[0,1]^2k is an
r-emulation of S x E containedin Q[0,1]^2k. We will also use this
definition with Q[0,1] replaced throughout by Q.

FINITE MAXIMAL EMULATION/1. FME/1. For finite subsets of Q[0,1]^k,
some finitely maximal r-emulation is drop equivalent at
(1,1/2,...,1/k), (1/2,...,1/k,1/k).

FINITE MAXIMAL EMULATION/2. FME/2. For finite subsets of Q^k,
some finitely maximal r-emulation contains its upper shift below t.

DEFINITION 6.2. S is a finitely greedy r-emulation of E containedin
Q^k if and only if S is a finite r-emulation of E containedin Q^k,
where every r-emuation S|<=p U {x} x E of E^2 contianeidn Q^2k is an
r-emulation of S|<=p x E containedin Q^2k.

FINITE GREEDY EMULATION/1. FGE/1. For finite subsets of Q^k,
some finitely greedy r-emulation contains its upper shift below k.

DEFINITION 6.3. S is a finitely up greedy r-emulation of E containedin
Q^k if and only if S is an r-emulation of E containedin Q^k,
where every r-emuation S|<=p U {x} x E of E^2 contianeidn Q^2k is an
r-emulation of S|<=p x E containedin Q^2k, x in Q^k<=.

DEFINITION 6.4. ush(S) is the upper shift of S. S[p] = {x: (p,x) in S}.

FINITE UP GREEDY EMULATION/1. FUGE/1. Every finite E containedin Q^k
has a finitely up greedy r-emulation S containing ush(S) below t.

FINITE UP GREEDY EMULATION/2. FUGE/2. Every finite E containedin Q^k
has a finitely up greedy k-emulation S containing ush(S) below 2k,
where S[k+(1/2] = ush(S)|[0] below k.

An a priori estimate can be placed on the size of the finitely maximal
r-emulation in terms of k and the size of the given finite subsets of
Q[0,1]^k and Q^k so that FME/1,2 and FUGE/1,2 become explicitly Pi01.

THEOREM 6.1. FME/1,2 and FUGE/1 are provably equivalent to Con(SRP) over EFA
(exponential function arithmetic). FUGE/2 is provably equivalent to
Con(HUGE) over EFA.

************************************************************************
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 761st 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  12 00:34:34 EST 2017
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

Harvey Friedman