[FOM] possible non-existence of repeat points
martdowd at aol.com
martdowd at aol.com
Tue Mar 12 23:48:49 EDT 2019
FOM:
In
https://cs.nyu.edu/pipermail/fom/2018-January/020758.html
I provided the link
www.hyperonsoft.com/scof.pdf
to
SCHEMES, ORDINAL FUNCTIONS, AND REPEAT POINTS
This has been substantially revised; the link above is to the new version.
It has been submitted. The earlier version will not appear, since apparently
Academic Publisher's IJPAM has stopped publishing.
Theorem 55 states that if $\Sigma$ is a "T-extending O-scheme with suitable
$x_0$", and satisfying some additional restrictions, then
$\theta_\Sigma\leq\theta_T$, where $\theta_\Sigma$ is the closure ordinal
of $\Sigma$ (starting at a modified fixed point enumerator) and $\theta_T$
is the least $\theta$ such that there is no "T-separating" set at $\theta$.
Aspects of the question of the existence in $L[A]$ of O-schemes
over $\Omega$ satisfying restrictions of interest, with closure ordinals
arbitrarily large below $\Omega$, are considered. The question is of
interest in $L$, since to obtain the non-existence of repeat points,
O-schemes in $L$ with arbitrarily large closure ordinals would be of interest,
and furthermore this is an important question in itself.
Also given are some results on some specific schemes. In section 8
a scheme $\Sigma_{CT}$ is defined, whose closure ordinal is the (analog of
the) Bachmann-Howard ordinal (over $\Omega=\mu^+$, where for applications to
repeat points $\mu$ is a measurable cardinal), which will be denoted
$\theta_{CT}$. A self-contained treatment of results of Gerber
https://doi.org/10.1007/BF01360719
is given. This results in a notation system for $\theta_{CT}$.
Using this notation system a scheme $\Sigma_{CTT}$ is defined. Let
$\theta_{CTT}$ denote its closure ordinal. It is shown that the
smallest repeat point is at least as large as an ordinal related to this.
It is of interest to show the existence of maximal elements for
$\Sigma_{CTT}$. Another question of interest is whether Gerber's
lemma 1.5 (lemma 20 of my paper) holds for $\Sigma_\CTT$; some partial
results are given.
Whether maximal elements exist, and whether Gerber's lemma can be avoided,
are both questions of interest. The methods of Bachmann's 1950 paper are of
interest (these were used by Isles in his 1971 paper). This is being
left to further research. The question arises how $\Sigma_\CTT$ compares
to the schemes (Bachmann collections) defined by Isles.
In connection with this latter research, I have translated Bachmann's paper
"Die Normalfunktionen und das Problem der
ausgezeichneten Folgen von Ordnungszahlen"
(Normal Functions and
the Problem of the Distinguished Sequences
of Ordinal Numbers).
I received permissiom from the editors of
"Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich"
to post my translation. It is currently available at
http://arxiv.org/abs/1903.04609
The translation could doubtless benefit from improvement; any suggestions
would be welcome. If anyone would like a copy of the latex source send me
an email.
Martin Dowd
Postscript: "Pidgeon TeX" can be viewed via MathJax. You can download
a posting such as this and convert it to an html file with a MathJax wrapper,
and open it in a web browser. If you want paragraphs, etc., some editing might be
required. See "www.hyperonsoft.com/fom2.html" for an example.
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