# [FOM] Transfinite Recursion in Functional Analysis and Measure Theory

Jeffry Hirst hirstjl at appstate.edu
Wed Jan 30 15:24:38 EST 2019

```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,
>>
>>
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```