[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.
>
>*****************************************
>
>I use http://www.math.ohio-state.edu/~friedman/ for downloadable
>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
>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 11/27/12 3:30PM
>509: More Undefinable Elements 11/27/12 10:03PM
>
>Harvey Friedman
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