[FOM] Transfinite Recursion in Functional Analysis and Measure Theory

Fri Feb 1 12:33:03 EST 2019

 Sorry, my message contained a bad typo.  What I meant to say was:

     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.

-- Steve Simpson

On Fri, Feb 1, 2019 at 11:01 AM Stephen G. Simpson <sgslogic at gmail.com>
wrote:

> 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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