[FOM] 514: Countable Elementary Extensions/again

Mon Jan 14 02:18:36 EST 2013

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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THIS POSTING IS ENTIRELY SELF CONTAINED
but follows up on http://www.cs.nyu.edu/pipermail/fom/2013-January/016884.html

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Here we restate
http://www.cs.nyu.edu/pipermail/fom/2013-January/016884.html with four
notions of enumeration rather than one. In the process, we correct a
glitch there.

Let K be a set of subsets of D^n, and m > n. We make four definitions.

i. An m dimensional exact enumeration of K is an S contained in D^m
such that K = {S_x: x in D^m-n}.

ii. An m dimensional inclusive enumeration of K is an S contained in
D^m such that K is contained in {S_x: x in D^m-n}.

iii. An m dimensional exact (inclusive) enumeration without repetition
is an exact (inclusive) enumeration S, where for all x,y in D^m-n, if
S_x = S_y is nonempty, then x = y.

Let M' be a countable elementary extension of M, in a finite
relational type. Consider the following three conditions on M,M':

1. THERE IS AN M DEFINABLE 2 DIMENSIONAL INCLUSIVE ENUMERATION OF THE M
DEFINABLE SUBSETS OF DOM(M).

2. THERE IS AN M' DEFINABLE 2 DIMENSIONAL INCLUSIVE ENUMERATION
(WITHOUT REPETITION) OF THE M
DEFINABLE SUBSETS OF DOM(M).

3. THERE IS AN M' DEFINABLE 2 DIMENSIONAL EXACT ENUMERATION (WITHOUT
REPETITION) OF THE M
DEFINABLE SUBSETS OF DOM(M).

4. THERE IS AN M' DEFINABLE 2 DIMENSIONAL INCLUSIVE (EXACT)
ENUMERATION (WITHOUT REPETITION) OF THE M'
DEFINABLE SUBSETS OF DOM(M).

Obviously 1 is impossible by Russell's Paradox.

THEOREM 1. Both forms of "2 is possible" are provably equivalent to
the consistency
of second order arithmetic, over WKL_0.

THEOREM 2. Both forms of "3 is possible" are provably equivalent to
"second-order arithmetic has an omega model" over ACA.

THEOREM 3. All four forms of "4 is possible" are provably equivalent
to the consistency
of ZFC, over WKL_0. Thus all four forms of "4 is possible" is independent of ZFC
(assuming ZFC does not prove its own inconsistency).

THEOREM 4. If M,M' satisfy either form of 2, then M,M' each interpret a model of
second order arithmetic. Any model of second order arithmetic
interprets some M,M' satisfying both forms of 2.

THEOREM 5. If M,M' satisfy either form of 3, then M,M' each interpret
an omega model of
second order arithmetic. Any omega model of second order arithmetic
interprets some M,M' satisfying both forms of 3.

THEOREM 6. If M,M' satisfy any of the four forms of 3, then M,M' each
interpret a model of ZFC.
Any model of ZFC interprets some M,M' satisfying all forms of 3.

We use countable structures here to emphasize concreteness. However,
Theorems 3,4 still hold without countability. Theorem 2 has the
following version without countability:

THEOREM 3'. Any of the four forms of "4 is possible" are provably
equivalent to the consistency
of ZFC, over ZF\P.

Also, if we

i. Allow the enumerations of the one dimensional sets to be of any dimension; or
ii. Require that for all n, the n dimensional sets have an n+1 enumeration.

then we get weaker and stronger versions. In either case, the same
results hold.

The statements "2 is possible" and "4 is possible", in all four forms,
are provably
equivalent, over WKL_0, to Pi01 sentences, via Goedel's completeness
theorem.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 514th 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 #1-449 can be found
in the FOM archives at
http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html

450: Maximal Sets and Large Cardinals II  12/6/10  12:48PM
451: Rational Graphs and Large Cardinals I  12/18/10  10:56PM
452: Rational Graphs and Large Cardinals II  1/9/11  1:36AM
453: Rational Graphs and Large Cardinals III  1/20/11  2:33AM
454: Three Milestones in Incompleteness  2/7/11  12:05AM
455: The Quantifier "most"  2/22/11  4:47PM
456: The Quantifiers "majority/minority"  2/23/11  9:51AM
457: Maximal Cliques and Large Cardinals  5/3/11  3:40AM
458: Sequential Constructions for Large Cardinals  5/5/11  10:37AM
459: Greedy CLique Constructions in the Integers  5/8/11  1:18PM
460: Greedy Clique Constructions Simplified  5/8/11  7:39PM
461: Reflections on Vienna Meeting  5/12/11  10:41AM
462: Improvements/Pi01 Independence  5/14/11  11:53AM
463: Pi01 independence/comprehensive  5/21/11  11:31PM
464: Order Invariant Split Theorem  5/30/11  11:43AM
465: Patterns in Order Invariant Graphs  6/4/11  5:51PM
466: RETURN TO 463/Dominators  6/13/11  12:15AM
467: Comment on Minimal Dominators  6/14/11  11:58AM
468: Maximal Cliques/Incompleteness  7/26/11  4:11PM
469: Invariant Maximality/Incompleteness  11/13/11  11:47AM
470: Invariant Maximal Square Theorem  11/17/11  6:58PM
471: Shift Invariant Maximal Squares/Incompleteness  11/23/11  11:37PM
472. Shift Invariant Maximal Squares/Incompleteness  11/29/11  9:15PM
473: Invariant Maximal Powers/Incompleteness 1  12/7/11  5:13AMs
474: Invariant Maximal Squares  01/12/12  9:46AM
475: Invariant Functions and Incompleteness  1/16/12  5:57PM
476: Maximality, CHoice, and Incompleteness  1/23/12  11:52AM
477: TYPO  1/23/12  4:36PM
478: Maximality, Choice, and Incompleteness  2/2/12  5:45AM
479: Explicitly Pi01 Incompleteness  2/12/12  9:16AM
480: Order Equivalence and Incompleteness
481: Complementation and Incompleteness  2/15/12  8:40AM
482: Maximality, Choice, and Incompleteness 2  2/19/12 7:43AM
483: Invariance in Q[0,n]^k  2/19/12  7:34AM
484: Finite Choice and Incompleteness  2/20/12  6:37AM__
485: Large Large Cardinals  2/26/12  5:55AM
486: Naturalness Issues  3/14/12  2:07PM
487: Invariant Maximality/Naturalness  3/21/12  1:43AM
488: Invariant Maximality Program  3/24/12  12:28AM
489: Invariant Maximality Programs  3/24/12  2:31PM
490: Invariant Maximality Program 2  3/24/12  3:19PM
491: Formal Simplicity  3/25/12  11:50PM
492: Invariant Maximality/conjectures  3/31/12  7:31PM
493: Invariant Maximality/conjectures 2  3/31/12  7:32PM
494: Inv Max Templates/Z+up, upper Z+ equiv  4/5/12  4:17PM
495: Invariant Finite Choice  4/5/12  4:18PM
496: Invariant Finite Choice/restatement  4/8/12  2:18AM
497: Invariant Maximality Restated  5/2/12 2:49AM
498: Embedded Maximal Cliques 1  9/18/12  12:43AM
499. Embedded Maximal Cliques 2  9/19/12  2:50AM
500: Embedded Maximal Cliques 3  9/20/12  10:15PM
501: Embedded Maximal Cliques 4  9/23/12  2:16AM
502: Embedded Maximal Cliques 5  9/26/12  1:21AM
503: Proper Classes of Graphs  10/13/12  12:17PM
504. Embedded Maximal Cliques 6  10/14/12  12:49PM
505: Function Transfer Theory 10/21/12  2:15AM
506: Finite Embedded Weakly Maximal Cliques  10/23/12  12:53AM
507: Finite Embedded Dominators  11/6/12  6:40AM
508: Unique Undefinable Elements  12/22/12  8:08PM
509: A Divine Consistency Proof for Mathematics  12/26/12  2:15AM
510: Unique Undefinable Elements Again  1/9/13  5:07PM
511: A Supernatural Consistency Proof for Mathematics   1/10/13  9:19PM
512: Countable Elementary Extensions  1/11/13  7:31PM
513: Five Supernatural Consistency Proofs for Mathematics

Harvey Friedman