[FOM] Transfinite Recursion in Functional Analysis and Measure Theory

[Stephen G. Simpson]

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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