[FOM] 762: Baby Emulation Theory/Expositional
Harvey Friedman
hmflogic at gmail.com
Mon Apr 17 00:13:23 EDT 2017
We finished the presentation of the first phase of Emulation Theory
with http://www.cs.nyu.edu/pipermail/fom/2017-April/020444.html . Here
we elaborate on the EXPOSITIONAL points made at the beginning of
http://www.cs.nyu.edu/pipermail/fom/2017-April/020443.html .
We discuss some new BABY statements that I have found in Emulation
Theory that are provable well below ZFC, but have very straightforward
extensions to the Lead statements of Emulation theory that require
large cardinals to prove. As I said in
http://www.cs.nyu.edu/pipermail/fom/2017-April/020443.html this
appears to be a major EXPOSITIONAL advance. First you make the simple
jump from nothing to the new Baby Statements. Then you make the simple
jump from the new Baby Statements to the Lead Statements.
The idea is that each of these two jumps is irresistible, whereas
making the jump all the way at once might still be resistable by some.
Most readers are familiar with these very simple definitions already,
and they are perhaps .1% as involved as what you routinely find in
typical "easy to state" theorems of modern mathematics. They only
involve < on the rationals - and NO arithmetic operations.
Q[0,1] is Q intersect [0,1]. x,y in Q^k are order equivalent iff their
terms are in the same numerical order (e.g., (5,3,1,6) and (4,2.-3,8)
are order equivalent). S is an emulation of E containedin Q[0,1]^k if
and only if S containedin Q[0,1]^k and S^2, E^2 have the same elements
(2k-tuples) up to order equivalence. Maximal emulations are emulations
that are not a proper subset of emulations. Any set S is equivalent at
any points x,y if and only if (x in S iff y in S). These are ALL of
the definitions needed for the Baby Statements.
All of the preceding definitions are (MATHEMATICALLY) FREE (OF COST),
except possibly "emulation". However, a little reflection on
emulations reveals that it clearly represents the idea that emulations
have the same tiny patterns. There is the even more elemental notion
"S,E have the same elements up to order equivalence" where there are
no interactions between elements.
In the Putnam Volume paper I will call these statement below MAXIMAL
EMULATION POINT/1-6, or MEP/1-6. Here I call them BS, which is not
exactly such a good name for posterity.
BABY STATEMENT/1. BS/1. For finite subsets of Q[0,1]^2, some maximal
emulation is equivalent at (1/2,1/3), (1/3,1/4).
BABY STATEMENT/2. BS/2. For finite subsets of Q[0,1]^2, some maximal
emulation is equivalent at (1,1/2), (1/2,1/3).
BABY STATEMENT/3. BS/3. For finite subsets of Q[0,1]^k, some maximal
emulation is equivalent at (1/2,1/3,...,1/(k+1)),
(1/3,1/4,...,1/(k+2)).
BABY STATEMENT/4. BS/4. For finite subsets of Q[0,1]^k, some maximal
emulation is equivalent at (1,1/2,...,1/k), (1/2,,1/3,...,1/(k+1)).
For r-emulations, we use: S^r,E^r have the same elements (rk-tuples)
up to order equivalence. Here are the strongest of our Baby
Statements:
BABY STATEMENT/5 BS/5. For finite subsets of Q[0,1]^k, some maximal
r-emulation is equivalent at (1/2,1/3,...,1/(k+1)),
(1/3,,1/4,...,1/(k+2)).
BABY STATEMENT/6 BS/6. For finite subsets of Q[0,1]^k, some maximal
r-emulation is equivalent at (1,1/2,...,1/k), (1/2,,1/3,...,1/(k+1)).
It is immediate that these statements remain equivalent if we drop "finite".
BS/1,3,5 have straightforward proofs that combine the fact that we are
in a purely order theoretic context with the classical finite Ramsey
theorem of 1930. With BS/1, the finite Ramsey theorem involved can be
done by hand without using the 1930 work. For BS/2,4,6, some
additional ideas are needed. But we still need only ACA'.
We now make a second irresistible jump that takes us far beyond ZFC.
Any set S is drop equivalent at x,y in Q[0,1]^k if and only if
i. x_k = y_k
ii. For all 0 <= p < x_k, S is equivalent at (x_1,...,x_k-1,p),
(y_1,...,y_k-1,p).
We think of x,y as raindrops in Q[0,1]^k falling from the same height
to the ground, encountering the same membership in S.
MAXIMAL EMULATION DROP /1. MED/1. For finite subsets of Q[0,1]^2, some
maximal emulation is drop equivalent at (1/2,1/3), (1/3,1/3).
MAXIMAL EMULATION DROP/2. MED/2. For finite subsets of Q[0,1]^2, some
maximal emulation is drop equivalent at (1,1/2), (1/2,1/2).
MAXIMAL EMULATION DROP/3. MED/3. For finite subsets of Q[0,1]^k, some
maximal emulation is drop equivalent at (1/2,1/3,...,1/(k+1)),
(1/3,...,1/(k+1),1/(k+1)).
MAXIMAL EMULATION DROP/4. MED/4. 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)).
MAXIMAL EMULATION DROP/5. MED/5. For finite subsets of Q[0,1]^k, some
maximal r-emulation is drop equivalent at (1/2,1/3,...,1/(k+1)),
(1/3,...,1/(k+1),1/(k+1)).
MAXIMAL EMULATION DROP/6. MED/6. 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).
We know how to prove MED/1,3,5 in ACA', with more ideas than for
BS/1-6. However, we know that MED/4,6 are provably equivalent to
Con(SRP) over WKL_0. The only proof that we have of MED/2 involves
some significant dose of set theory, and we expect that in dimension k
= 3 with r-emulations, large cardinals are needed.
************************************************************************
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 762nd 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
Harvey Friedman
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