[FOM] Possible nonexistence of repeat points

[martdowd at aol.com]

FOM:

I have a new paper on lower bounds on the smallest repeat point.
The paper is titled
  B-schemes
It is temporarily available at
  https://www.researchgate.net/publication/337001387_B-schemes

This gives a system of fundamental sequences of length
$\psi_\Omega^{\Omega^+}(0)$.  In the paper
 An Ordinal Larger Than the Bachmann-Howard Ordinal
which I posted previously the length was $\psi_2^\Omega(0)$.
The previous paper used the notion of ``built-up'' systems as defined by
Bachmann, and further developed by Schmidt.  The new paper uses a different
and seemingly more tractable notion, that of a "B-scheme".  Maximal
indices are shown to exist, and a notation system is given for the closure
ordinal.  The previous paper gave a number of further results for the
shorter scheme, but such are omitted in the new paper.

The system is a prefix of one defined in an earlier paper.
A system of the same length was defined in a 2016 paper of Buckholtz
(see the references in the new paper).

Using the the notation system the lower bound of the previous paper on the
smallest repeat point is improved.

Additional question which are left open include whether the method can be
pushed through for a scheme of length $\psi_{\epsilon_0^{\Omega^+}}^\Omega(0)$.

Arbitrarily long schemes were constructed in the previous paper (using a
variation of a method given by Schmidt).  These are shown to be B-schemes.
This construction might be of interest, regarding the following question.

It would be of interest to prove a version of theorem 46 of
 https://www.researchgate.net/publication/
  333058424_Schemes_Ordinal_Functions_and_Repeat_Points
to give lower bounds using families of schemes.  I have some ideas for this,
which I'm leaving to a future paper.

Martin Dowd

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