[FOM] 510: More Undefinable Elements 2

T.Forster at dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Sun Dec 2 06:39:03 EST 2012

```Harvey, Do you know Wilfrid Hodges' model-theoretic analysis of Aquinas'
theory of the Trinity? All three members of the Trinity realise the same
1-type, but they realise distinct 2-types. There is a binary relation of
`procedes from'. I don't think he ever published it, but it might be worth
chasing up in the present context. It might further soften the hearts of
the Templeton people towards logic!

On Dec 1 2012, Harvey Friedman wrote:

>THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION
>
>*****************************************
>
>THIS POSTING IS A CONTINUATION OF
>http://www.cs.nyu.edu/pipermail/fom/2012-November/016806.html
>http://www.cs.nyu.edu/pipermail/fom/2012-November/016809.html
>
>*****************************************
>
>The Trinity - see
>http://www.cs.nyu.edu/pipermail/fom/2012-November/016809.html -
>suggests the following.
>
>A STRUCTURE WITH EXACTLY THREE UNDEFINABLE ELEMENTS x,y,z, EACH
>DEFINABLE FROM EACH OTHER, WHERE NO NONEMPTY SUBSET OF {x,y,z} IS
>DEFINABLE.
>
>More generally,
>
>A STRUCTURE WITH EXACTLY n UNDEFINABLE ELEMENTS x_1,...,x_n, EACH
>DEFINABLE FROM EACH OTHER, WHERE NO NONEMPTY SUBSET OF {x_1,...,x_n}
>IS  DEFINABLE.
>
> Here definability of elements and finite sets always means without
> parameters.
>
>This can be done by adaptation of the arguments in
>http://www.math.osu.edu/~friedman.8/manuscripts.html #74. As in there,
>we can get the structure to be a graph, and handle "there exists
>infinitely" and quantitative quantifiers.
>
>*****************************************
>
>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  11/27/12  3:30PM
>509: More Undefinable Elements   11/27/12  10:03PM
>
>Harvey Friedman
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```