[FOM] 741: Philosophy of Incompleteness/3

[Harvey Friedman]

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