[FOM] Transfinite Recursion in Functional Analysis and Measure Theory

[Stephen G. Simpson]

Elaborating my remark about measure-theoretic regularity, I should probably
explain "LR-reducibility."  A Turing oracle Y is said to be LR-reducible to
a Turing oracle X if every real that is Martin-Löf random relative to X is
Martin-Löf random relative to Y.  Intuitively this means that X has at
least as much "de-randomization power" as Y.  This reducibility notion, due
to André Nies, has been very useful in various metamathematical analyses of
measure-theoretic concepts and theorems.

Now let me turn to transfinite recursion in functional analysis.  It is
well known that a mainstay of Banach space theory is the weak-star topology
on the dual of a Banach space.  Not so well known is the following:

   Let Z be a subspace of the dual of a separable Banach space.  Banach and
Mazurkiewicz
   observed that, although the weak- star closure of Z is the same as the
weak-star sequential closure
   of Z, it is not necessarily the case that every point of the weak- star
closure of Z is the weak- star
   limit of a sequence of points of Z. Indeed, the process of taking
weak-star limits of sequences
   may need to be iterated trans finitely many times in order to obtain the
weak-star closure.

And in fact, O. Carruth McGehee has constructed for each countable ordinal
alpha a weak-star dense subspace of the sequence space l_1 (= the dual of
the sequence space c_0^*) whose weak-star closure ordinal is exactly
alpha+1.

The above quotation is from a paper by Jim Humphreys and me:

    A. James Humphreys and Stephen G. Simpson, Separable Banach space
theory needs strong
    set existence axioms, Transactions of the American Mathematical
Society, 348, 1996, pp. 4231-4255

which is available at http://www.personal.psu.edu/t20/papers/convex.pdf .
In that paper we give a construction which is more elementary than
McGehee's, and we perform a reverse-mathematical analysis of the situation.

-- Steve Simpson


On Thu, Jan 31, 2019 at 9:47 AM Stephen G. Simpson <sgslogic at gmail.com>
wrote:

> Yes, Chapter V of my book covers mathematics and reverse mathematics in
> ATR0.  But there is little or no measure theory in that chapter.
>
> My paper "Mass problems and measure-theoretic regularity" (Bulletin of
> Symbolic Logic, 15, 2009, 385-409) includes degree-theoretic and
> reverse-mathematical analyses of regularity at transfinite levels of the
> Borel hierarchy.  (A set in Euclidean space is said to be regular if it
> includes an F_sigma set of the same measure.)  One of my results is that
> for all recursive ordinals alpha, the alpha-th Turing jump of 0 is
> LR-reducible to a Turing oracle X if and only if X is regularizing at level
> alpha+2 of the lightface Borel hierarchy.
>
> Stephen G. Simpson
> Research Professor
>
> Department of Mathematics
> 1326 Stevenson Center
> Vanderbilt University
> Nashville, TN 37240, USA
> web: www.math.psu.edu/simpson
> email: sgslogic at gmail.com
>
>
> On Wed, Jan 30, 2019 at 4:52 PM Jeffry Hirst <hirstjl at appstate.edu> wrote:
>
>> As pointed out by Kenny Easwaran, there is a connection between derived
>> sequences
>> and transfinite recursion.  In the reverse mathematics setting, the
>> arithmetical transfinite
>> recursion scheme (ATR0) has been shown to be equivalent to the existence
>> of derived
>> sequences.  The proof appears in
>>
>> Harvey Friedman and Jeff Hirst, Reverse mathematics of homeomorphic
>> embeddings,
>> The Annals of Pure and Applied Logic, 54, (1991) 229-253.
>>
>> and the reversal appears in
>>
>> Jeff Hirst, Derived sequences and reverse mathematics, Mathematical Logic
>> Quarterly,
>> 39 (1993), 443-449.
>>
>> Equivalences with ATR0 have been found for many theorems.  A good place
>> to start
>> is Chapter V of Steve Simpson’s book, Subsystems of Second Order
>> Arithmetic.
>>
>> -Jeff Hirst
>>
>> > On Jan 29, 2019, at 10:49 AM, Kenny Easwaran <easwaran at gmail.com>
>> wrote:
>> >
>> > I'm not sure that this is quite measure theory, but I believe that
>> > Cantor invented his concepts of the transfinite in order to prove the
>> > Cantor-Bendixson theorem. If you iterate the process of removing
>> > isolated points from a set, you must terminate in some countable
>> > number of steps, and each step involves removing at most countably
>> > many points. The remaining set is either perfect or empty. He noticed
>> > that in general, you need to iterate more than just finitely many
>> > times or omega many times, but can only need to iterate a countable
>> > number of times, and developed the theory of ordinals to do this.
>> >
>> > https://en.wikipedia.org/wiki/Derived_set_(mathematics)
>> >
>> > Kenny Easwaran
>> >
>> > On Tue, Jan 29, 2019 at 12:07 AM Adam Kolany <dr.a.kolany at wp.pl> wrote:
>> >>
>> >> Hi,
>> >>
>> >> I would appreciate examples of proofs in FA and MT where Transfinite
>> >> Recursion was used.
>> >>
>> >> Also "sensible" formulations of TR  in ZF set theory  would be welcome.
>> >>
>> >>
>> >> Can you help ?
>> >>
>> >>
>> >> regards,
>> >>
>> >> Adam Kolany
>> >>
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