New paper on fundamental sequences

martdowd at aol.com martdowd at aol.com
Sat Jul 31 11:28:53 EDT 2021

```FOM.

I have a new manuscript,
P-schemes
https://www.researchgate.net/publication/353605998

Here's the abstract.

B-schemes were introduced by the author in a previous manuscript.  This
paper introduces a new class of schemes, called P-schemes.  In 1964 H.
Pfeiffer defined schemes Sigma-n-p for integers n,p.  It is a question
of interest whether these are P-schemes.  It is shown that Sigma-n-1 is
a P-scheme, whence Sigma-n-2 exists.  This is ongoing research, and
efforts will continue to show that Sigma-n-p is a P-scheme for all p.

I have posted the results I have obtained so far to raise interest in the
subject.  It is quite possible that answers to questions I have raised can
be found in Pfeiffer's thesis, and I will continue to translate it.  I
have some concerns about what I have translated so far.  Pfeiffer claims
that Sigma-n-p has what I call property ^P3; the proof is given in the
proofs of Hilfsatz 3_p and Hilfsatz 4_p.  I prove that the designators
are maximal, and Pfeiffer does not seem to do this, although this may be
clear from other facts.  My proof uses some facts which I don't yet know
how to generalize to larger p.

Thanks to an email from Marcus Schaefer, I found out that a print copy of
Pfeiffer's thesis was available through AbeBooks.  I ordered it, and
digitized it.  If anyone wants the print copy (particularly for a
library), send me an email.

In his "Introduction", Pfeiffer mentions the "Axiom der Hauptfolgen",
which is considered in H. Bachmann's 1955 book "Transfinite Zahlen".
Do any FOM members know of more recent references on this subject?
Jech's book on the axiom of choice doesn't seem to consider it.

In the process of digitizing Pfeiffer's thesis, I wrote some software for
cleaning scanned images in preparation for OCR.  This is available at
https://www.researchgate.net/publication/353003117
I also made search-able the German-English math dictionary available
at archive.org.  This is available at
https://www.researchgate.net/publication/352998687
Unfortunately, I can't put Pfeiffer's thesis on ResearchGate.

Martin Dowd

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