[FOM] 508: Unique Undefinable Elements
Andrew Brooke-Taylor
andrewbt at gmail.com
Tue Dec 11 02:04:50 EST 2012
Dear FOMers,
I just wanted to mention to readers that similarly, Greg Hjorth
constructed a model with a unique element which is not definable in
L_{\omega_1, \omega} (JSL 75 no. 1 (2010), pp 269-274. Sorry, I'd
make it a link, but I only have hard copy access here).
Best wishes,
Andrew Brooke-Taylor
On 28 November 2012 05:29, Harvey Friedman <hmflogic at gmail.com> wrote:
> THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION
>
> *****************************************
>
> THIS POSTING IS ENTIRELY SELF CONTAINED
>
> *****************************************
>
> UNIQUE UNDEFINABLE ELEMENTS
>
> We present a structure (in a finite relational type) with a unique
> undefinable element. Furthermore, the unique undefinable element is
> undefinable even if we use the quantifier "there exist infinitely
> many" and quantitative variants. We show that the structures can be
> taken to be a graph. We also prove 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).
>
> 1. Introduction.
> 2. Cohen generic sets and forcing.
> 3. Cohen generic constructions.
> 4. Unique undefinable elements.
> 5. Interpretations into graphs.
> 6. Unique second order undefinable elements.
> 7. Dimension preservation.
>
> The manuscript is at
> http://www.math.osu.edu/~friedman.8/manuscripts.html #74, and has been
> submitted for publication.
>
> The unique undefinable element is called the God element. The work was
> motivated by theology, as discussed in the Introduction.
>
>
> *****************************************
>
> I use http://www.math.ohio-state.edu/~friedman/ for downloadable
> manuscripts. This is the 508th 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
>
> Harvey Friedman
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--
Dr Andrew Brooke-Taylor
JSPS Postdoctoral Research Fellow
Kobe University
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