# [FOM] 510: Unique Undefinable Elements Again

Harvey Friedman hmflogic at gmail.com
Wed Jan 9 17:06:37 EST 2013

```THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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THIS POSTING IS ENTIRELY SELF CONTAINED

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I left off the 'l' at the end of the link. The reference should read

"Below find the abstract and table of contents for paper #75 at
http://www.math.osu.edu/~friedman.8/manuscripts.html 70 pages.
Will submit for publication after people have a chance to comment."

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Well, a lot has happened with Unique Undefinable Elements, which
appears on the FOM at

http://www.cs.nyu.edu/pipermail/fom/2012-November/016806.html
http://www.cs.nyu.edu/pipermail/fom/2012-November/016809.html
http://www.cs.nyu.edu/pipermail/fom/2012-November/016821.html
http://www.cs.nyu.edu/pipermail/fom/2012-December/016830.html
http://www.cs.nyu.edu/pipermail/fom/2012-December/016833.html
http://www.cs.nyu.edu/pipermail/fom/2012-December/016840.html
http://www.cs.nyu.edu/pipermail/fom/2012-December/016853.html

As was revealed above, Hjorth published a solution to a grammatical
similar problem, but quite different in character: for the infinitary
language L_omega1,omega.

Hjorth cited Arnie Miller's problem list.

I looked up Arnie Miller's problem list, and found the following:

G.Fuhrken, A model with exactly one undefinable element, Colloquium
mathematicum,
19(1968), 183-185.

This was somewhat difficult to get a hold of, as it is not
electronically available, at least in Ohio.

Yes, this does prove the result I was claiming - by quantifier
elimination, and no recursion theoretic forcing. It is far more
elementary, and gives an example with a recursive theory.

I later found somewhat simpler examples using quantifier elimination
with somewhat stronger properties. Also the recursion theoretic
forcing method gives a model in which the unique undefinable element
is weak second order undefinable. And there are other things not in
Fuhrken.

So I substantially revised the paper and resubmitted. See Unique
Undefinable Elements,
http://www.math.osu.edu/~friedman.8/manuscripts.html #74.

Below find the abstract and table of contents for papers #74 and #75 at
http://www.math.osu.edu/~friedman.8/manuscripts.html.

UNIQUE UNDEFINABLE ELEMENTS
by
Harvey M. Friedman*
Distinguished University Professor of Mathematics,
Philosophy, Computer Science, Emeritus
Ohio State University
Columbus, Ohio 43210
January 9, 2013

* This research was partially supported by the John Templeton
Foundation grant ID #36297. The opinions expressed here are those of
the author and do not necessarily reflect the views of the John
Templeton Foundation.

Abstract. [Fu68] presents structures, in a finite relational type,
with unique undefinable elements and recursive theories. We present
somewhat simpler examples, with somewhat stronger properties. We also
present structures, in a finite relational type, whose unique
undefinable element is weak second order undefinable. We convert all
examples of a similar nature to corresponding examples in the form of
bipartite graphs and atomic inclusions. We show that "there is a
structure, in a finite relational type, with a unique second order
undefinable element" is not provable in ZFC (assuming ZFC is
consistent). We explore some properties of structures with unique
undefinable elements.

1. Introduction.
2. Unique undefinable elements.
3. Cohen generic sets and forcing.
4. Cohen generic constructions.
5. Unique weak second order undefinable elements.
6. Bipartite graphs and atomic inclusions.
7. Unique second order undefinable elements.
8. Dimension preservation.

A DIVINE CONSISTENCY PROOF FOR MATHEMATICS
by
Harvey M. Friedman*
Distinguished University Professor Emeritus
Mathematics, Philosophy, and Computer Science
The Ohio State University
Columbus, Ohio 43210
December 25, 2012

* This research was partially supported by the John Templeton
Foundation grant ID #36297. The opinions expressed here are those of
the author and do not necessarily reflect the views of the John
Templeton Foundation.

Abstract. We present familiar principles involving objects and classes
(of objects), pairing (on objects), choice (selecting elements from
classes), positive classes (elements of an ultrafilter), and definable
classes (definable using the preceding notions). We also postulate the
existence of a divine object in the formalized sense that it lies in
every definable positive class. ZFC (even extended with certain
hypotheses just shy of the existence of a measurable cardinal) is
interpretable in the resulting system. This establishes the
consistency of mathematics relative to the consistency of these
systems. Measurable cardinals are used to interpret and prove the
consistency of the system. Positive classes and various kinds of
divine objects have played significant roles in theology.

1. T1: Objects, classes, pairing.
2. T2: Extensionality, choice operator.
3. T3: Positive classes.
4. T4: Definable classes.
5. T5: Divine objects.
6. Interpreting ZFC in T5.
7. Interpreting a strong extension of ZFC in T5.
8. Without Extensionality.

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manuscripts. This is the 509th 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
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

Harvey Friedman
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