[FOM] 741: Philosophy of Incompleteness/3
John Baldwin
jbaldwin at uic.edu
Sat Jan 7 13:18:48 EST 2017
In re Harvey's problem to find a philosophically interesting formulation of
V=L.
Many year ago I had a public debate with Saunders Maclane about the role of
modern logic
in mathematics.
Kaplansky was very pleased when I pointed out that various arguments
required diamond rather
than full V=L. He said, roughly. Great, that is a mathematical axiom.
Q1. What is a statement of V=L interesting to traditional mathematicians
Q2. How does Harvey's question change if V = L is replaced by diamond?
John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607
On Sat, Jan 7, 2017 at 7:33 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
> BULLETIN. Scott Aaronson agreed to all my suggestions. See page 26 of
> http://www.scottaaronson.com/papers/pnp.pdf
>
> We continue from 740: Philosophy of Incompleteness/2.
>
> There we discussed
>
> A. An existing mathematical question that is widely known and of wide
> interest, is shown to be neither provable nor refutable in ZFC.
>
> and noted that CH = continuum hypothesis is the closest approximation
> to A, but falls short since CH is widely known but not of wide
> interest. Also, we discussed the wide interest of A, that it would be
> deeply disturbing to many, given their attraction to mathematics as
> the "precious objectively ironclad" subject.
>
> As somewhat of a digression, we began to discuss a method for trying
> to entirely obliterate independence from ZFC from the face of
> mathematics.
>
> The hypothesis V = L is commonly referred to as the "axiom of
> constructibility".
>
> L CONJECTURE/1. Every mathematical statement presented before 2017 by
> a normal mathematician, that involves only sets in the first
> omega+omega levels of the cumulative hierarchy, is provable or
> refutable in ZFC + V = L.
>
> Here are two alternative formulations that don't require considering
> levels of the objects involved.
>
> L CONJECTURE/2. Every mathematical statement presented before 2017 by
> a normal mathematician is provable or refutable in ZFC + V = L +
> "there is no strongly inaccessible cardinal".
>
> L CONJECTURE/3. Every mathematical statement presented before 2017 by
> a normal mathematician is provable or refutable in ZFC + V = L +
> "there are infinitely many inaccessible cardinals".
>
> Of course, the only way we are going to prove such conjectures is to
> actually resolve all open questions presented by normal mathematicians
> before 2017 in the cited systems. For the vast bulk of open questions
> this simply boils down to resolving them in ZFC itself (or even in
> ZF). This is because, of the following well known fact going back to
> Shoenfeld and even back to Goedel for arithmetical sentences.
>
> THEOREM. If a Sigma-1-3 sentence is provable in ZFC + V = L then it is
> provable in ZF. If a Pi-1-3 sentence is refutable in ZFC + V = L then
> it is refutable in ZF. We can replace ZFC + V = L by ZFC + V = L +
> "there is no inaccessible cardinal".
>
> But the main point is that these L Conjectures have not been refuted
> and are plausible.
>
> Note that I said "normal mathematician". This emphatically excludes
> me, who has been in the business for about 50 years of "composing"
> mathematical statements all the way down from Pi-1-3 to, most
> recently, Pi-0-1, which are neither provable nor refutable in ZFC, ZFC
> + V = L, ZFC + V = L + "there are infinitely many inaccessible
> cardinals", ZFC + V = L + "there are no inaccessible cardinals", and
> many other systems.
>
> We will be discussing "mathematical composition" and relationships to
> musical composition later.
>
> In particular, what appears to be the case is that using V = L
> completely obliterates all of the independence results from ZFC that
> rely on the inner model method of Kurt Goedel and the forcing method
> of Paul Cohen. It also settles the status of the Axiom of Choice, as
> ZF + V = L proves the Axiom of Choice.
>
> Now before you begin to think that I have turned senile, relegated to
> covering very well known ground, let me raise this issue about V = L.
>
> V = L does not look like any kind of axiom, at least by the standards
> of the ZFC axioms. A full statement of it involves defining L = class
> of constructible sets, and that is a rather mathematically unfriendly
> thing to have to do. The general idea of L is very basic but the
> execution is unpleasantly awkward. It involves using transfinite
> recursion on ordinals and formalized first order definability. The ZFC
> axioms do not mention ordinals, or transfinite recursion. They do
> involve the formation of first order definitions, but not intellectual
> reflection on them, with any kind of formalized first order
> definability. To what extent can all of this be removed, to make V = L
> look more mathematical, more philosophical, or both? The mathematical
> and philosophical ideas are quite simple, but the execution is not, at
> least by any kind of serious standard.
>
> CHALLENGE. Find mathematically/philosophically attractive formulations of
> V = L.
>
> The best way I know at the moment how to go about this is kind of open
> ended and could use some serious research effort.
>
> PROTOTYPE/1. Every definable transitive class that is not a set, and
> forms a model of ZF under epsilon, is all of V.
>
> PROTOTYPE/2. Every definable class that is not a set, and forms a
> model of ZF under epsilon, satisfies the same sentences as V under
> epsilon.
>
> The problem with prototype/1 is not the "definable class", as that is
> simply axiomatized as a scheme like separation and replacement. It is
> with "forms a model of ZF". The good news of course is that there is
> no trace of ordinals and transfinite recursion here, and it is in
> absolute tune with the underlying philosophy of V = L. That the
> universe of sets is minimal. The "forms a model of ZF" requires some
> sort of formalized definability, which is something we wish to avoid -
> but is even worse here in that the relevant formalized definability
> does not even exist!
>
> Similarly with prototype/2, and there is not problem with the
> elementary equivalence at the end, also nicely handled as a scheme.
>
> PROBLEM. Find a sensationally simple finite fragment of ZF for which
> Prototype/1 and/or Prototype/2 is equivalent to V = L over ZF.
>
> In the next posting in this series, I will take up merits and demerits of
> V = L.
>
> ************************************************************************
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 741st 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 #1-699 can be found at
> http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
>
> 700: Large Cardinals and Continuations/14 8/1/16 11:01AM
> 701: Extending Functions/1 8/10/16 10:02AM
> 702: Large Cardinals and Continuations/15 8/22/16 9:22PM
> 703: Large Cardinals and Continuations/16 8/26/16 12:03AM
> 704: Large Cardinals and Continuations/17 8/31/16 12:55AM
> 705: Large Cardinals and Continuations/18 8/31/16 11:47PM
> 706: Second Incompleteness/1 7/5/16 2:03AM
> 707: Second Incompleteness/2 9/8/16 3:37PM
> 708: Second Incompleteness/3 9/11/16 10:33PM
> 709: Large Cardinals and Continuations/19 9/13/16 4:17AM
> 710: Large Cardinals and Continuations/20 9/14/16 1:27AM
> 700: Large Cardinals and Continuations/14 8/1/16 11:01AM
> 701: Extending Functions/1 8/10/16 10:02AM
> 702: Large Cardinals and Continuations/15 8/22/16 9:22PM
> 703: Large Cardinals and Continuations/16 8/26/16 12:03AM
> 704: Large Cardinals and Continuations/17 8/31/16 12:55AM
> 705: Large Cardinals and Continuations/18 8/31/16 11:47PM
> 706: Second Incompleteness/1 7/5/16 2:03AM
> 707: Second Incompleteness/2 9/8/16 3:37PM
> 708: Second Incompleteness/3 9/11/16 10:33PM
> 709: Large Cardinals and Continuations/19 9/13/16 4:17AM
> 710: Large Cardinals and Continuations/20 9/14/16 1:27AM
> 711: Large Cardinals and Continuations/21 9/18/16 10:42AM
> 712: PA Incompleteness/1 9/2316 1:20AM
> 713: Foundations of Geometry/1 9/24/16 2:09PM
> 714: Foundations of Geometry/2 9/25/16 10:26PM
> 715: Foundations of Geometry/3 9/27/16 1:08AM
> 716: Foundations of Geometry/4 9/27/16 10:25PM
> 717: Foundations of Geometry/5 9/30/16 12:16AM
> 718: Foundations of Geometry/6 101/16 12:19PM
> 719: Large Cardinals and Emulations/22
> 720: Foundations of Geometry/7 10/2/16 1:59PM
> 721: Large Cardinals and Emulations//23 10/4/16 2:35AM
> 722: Large Cardinals and Emulations/24 10/616 1:59AM
> 723: Philosophical Geometry/8 10/816 1:47AM
> 724: Philosophical Geometry/9 10/10/16 9:36AM
> 725: Philosophical Geometry/10 10/14/16 10:16PM
> 726: Philosophical Geometry/11 Oct 17 16:04:26 EDT 2016
> 727: Large Cardinals and Emulations/25 10/20/16 1:37PM
> 728: Philosophical Geometry/12 10/24/16 3:35PM
> 729: Consistency of Mathematics/1 10/25/16 1:25PM
> 730: Consistency of Mathematics/2 11/17/16 9:50PM
> 731: Large Cardinals and Emulations/26 11/21/16 5:40PM
> 732: Large Cardinals and Emulations/27 11/28/16 1:31AM
> 733: Large Cardinals and Emulations/28 12/6/16 1AM
> 734: Large Cardinals and Emulations/29 12/8/16 2:53PM
> 735: Philosophical Geometry/13 12/19/16 4:24PM
> 736: Philosophical Geometry/14 12/20/16 12:43PM
> 737: Philosophical Geometry/15 12/22/16 3:24PM
> 738: Philosophical Geometry/16 12/27/16 6:54PM
> 739: Philosophical Geometry/17 1/2/17 11:50PM
> 740: Philosophy of Incompleteness/2
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20170107/1d5b5f02/attachment-0001.html>
More information about the FOM
mailing list