What is written below is a proof due to Nathan Bowler. The attached file titled "acyclic.pdf" is written by Nathan Bowler himself, and I've posted it as such, i.e. what is posted IS the original copy of it. However the review pdf document is my own review about that proof.
The following is a copy of the message of Bowler to which he attached the 'acyclic.pdf' file:
Nathan Bowler 04/07/14 at 4:36 PM
To Thomas Forster
CC Randall Holmes Zuhair al Johar
I think the attached document shows that any stratified formula can be rephrased as an acyclic formula (assuming we have enough comprehension to build finite sets {x_1 … x_k} given the elements x_1 … x_k).
Best wishes,
Nathan
Note: Permission was taken beforehand tonight from Nathan to publish this proof on this website: This is the written permission given here:
Nathan Bowler
To Zuhair Abdul Ghafoor Al-Johar Feb 3 at 11:18 PM
Dear Zuhair,
of course I don't mind if you put this result up on your website.
Best wishes,
Nathan
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On Sat, Feb 3, 2018 at 7:47 PM, Zuhair Abdul Ghafoor Al-Johar <zaljohar@yahoo.com> wrote:
Dear Nathan,
I don't know if you had published your proof of every stratified formula having an acyclic equivalent, [albeit using "impredicative formulation"].My guess is that you didn't.
I want to present this proof in a pdf file in my website and of course, I'll attribute it to YOU. Actually what I want to post is a copy and past of your e-mail response exactly as it is.
Do you permit that?
Zuhair