[FOM] 742: Philosophy of Incompleteness/4

[Harvey Friedman]

In the previous postings in this series, 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.

We don't quite have A, but CH seems to come closest, being still
widely known but at this time not of wide interest for a number of
reasons previously discussed. Con(ZFC) is another kind of maximal stab
at A, but it is more commonly viewed as a metamathematical or logical
question, is not really so widely known as a mathematical question,
and is not really of wide interest as a mathematical question,
although it is of universal interest among mathematicians and many
others who have an interest in foundations of mathematics. OBVIOUSLY,
something like the independence of RH or e+pi is irrational from ZFC
is what we are really seeking. More later about how we want to proceed
in light of the situation with A.

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.

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".

These seem to me to be rather plausible. Again, more later about how
we want to proceed in light of these. And also the merits and demerits
of adopting ZFC + V= L and such extensions.

CHALLENGE. Find mathematically/philosophically attractive formulations of V = L.

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.

PROBLEM. Find a sensationally simple finite fragment of ZF for which
Prototype/1 and/or Prototype/2 is equivalent to V = L over ZF.

The Prototypes don't address the Challenge as well as we would like,
as it intellectually reflects on language - definability.

However, the Prototypes are similar to the axioms of ZFC in that there
is no mention of ordinals.

Jensen's Diamond was formulated by Jensen as a consequence of ZFC + V
= L that gets to the combinatorial essence of al least why ZFC + V = L
refutes Souslin's Hypothesis. However, Diamond cannot quite replace V
= L for present purposes because it is known that ZFC + Diamond does
not prove the existence of a Kurepa tree. However, there is a sharper
version, Strong Diamond, that suffices.
https://en.wikipedia.org/wiki/Diamond_principle (John Baldwin
http://www.cs.nyu.edu/pipermail/fom/2017-January/020230.html
anticipated this planned discussion of diamond).

QUESTION. How confident are we that we don't have to strengthen
Diamond yet again in order to replace V = L in the L Conjectures
above?

Sure versions of Diamond are more mathematical than versions of the
Prototype. However, However, versions of V = L are mush more
foundational or philosophical. They communicate a general idea in the
direction of: given an axiom system, asserting that the universe is
generated by the axioms of that axiom system. Of course, when talking
in any generality about this kind of "minimality", we need to be
careful.

QUESTION. Is there an interesting general notion of the "minimality
axiom" for a wide variety of systems, where in the case of set theory
and ZF in particular, this is V = L?

The Philosophical Idea behind V = L is sufficiently clear that we
really need to have at least some working answer to the PROBLEM above
or a workaround.

V = L EQUIVALENT/1. V=L(1eq,n). Every definable transitive class under
epsilon, that is not a set, and forms a model of those axioms of ZF
with at most n occurrences of symbols, not counting parentheses,
includes all sets.

V = L EQUIVALENT/2. V=L(2eq,n). Every definable class under epsilon,
that is not a set, and forms a model of those axioms of ZF with at
most n occurrences of symbols, not counting parentheses, satisfies the
same sentences as the class of all sets under epsilon.

You can pick your own complexity measure here where there are only
finitely many formulas of a given complexity, up to an appropriate
equivalence, as I have done above. Or one where you know how to define
satisfaction for any given complexity class, although that moves
squarely into the territory of the logician and away from the common
thinker.

Obviously, from the logician's point of view, we normally would count
the number of parameters and the number of quantifiers or number of
alternations of quantifiers. But those considerations are for the
expert.

Before you repel in disgust at V=L(1eq,n) and V=L(2eq,n), here is the
narrative to consider. What we really want to say is the
philosophically coherent Prototype/1, Prototype/2, above, but this
cannot be stated as a first order scheme. So we replace this with
something we can state as a first order scheme and that is sharper.

The n needed in V= L(1eq,n), V=L(2eq,n) for it to be equivalent to V =
L over ZF is not very big, I think. But getting something optimal or
close to optimal is going to take some work.

In my PROBLEM above I didn't have in mind this kind of mindless count
n. Instead, something intelligible, and that is likely quite
preferable.

OK, now that we have discussed formulations of V = L, and, after all,
there is the usual formulation by transfinite recursion using FODO, we
take up merits and demerits.

MERITS. It obliterates all independence results from ZF in the sense
of the three L Conjectures. Mathematicians generally find independence
unwanted nuisances, even threatening to their personal relationship
with mathematics, and generally approve of the idea of getting rid of
them. That is, getting rid of them if it doesn't affect their work and
the mathematics that they know and care about. Furthermore, the
mathematicians have never encountered any mathematical object that
they had any reason to believe was outside L. In the case of
encountered countable objects, they know or can be made aware of that
these are demonstrably in L. They have heard from set theorists that V
= L precedes the existence of medium to large large cardinals, and
that is a big negative for them. For the mathematicians, that is a big
positive for them, as such enormous objects are very much out there,
and not having to worry about them is a really good thing. Not having
to think about independence and not having to think about medium/large
large cardinals is an attractive combination.

DEMERITS. Why would one believe that all sets are in fact
constructible, or even that all real numbers are constructible? By
comparison, the axioms of ZFC look obvious. And if we assume that even
all real numbers are constructible, then maybe this throws away real
numbers that I am talking about when I prove something about all real
numbers. Also, I hear that set theorists categorically reject V = L so
shouldn't we defer to them on this matter, or at least not go against
what they advise and think? OK, it is more convenient to have V = L so
as to get rid of weird stuff we don't care about or care much about
anyway, but there is TRUTH, and you can't disrespect TRUTH for
convenience.

************************************************************************
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 742nd 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  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM

Harvey Friedman