[FOM] Inconsistency of Inaccessibility
Alexander Kiselev
aakiselev at yahoo.com
Sat Oct 22 12:40:55 EDT 2011
Dear FOMers!
I should like to present the brief exposition of my previous works about
nonexistence (in ZF) of inaccessible cardinals.
This exposition consists of two papers, first of
them "Inconsistency of Inaccessibility" occupies
only few pages and can be seen in arXiv at site:
<http://arxiv.org/PS_cache/arxiv/pdf/1110/1110.3461v1.pdf>http://arxiv.org/PS_cache/arxiv/pdf/1110/1110.3461v1.pdf
The second paper "Appendix" supplements the
first one and it can be should laid aside for
some first times; it can be seen at arXiv site:
<http://xxx.lanl.gov/PS_cache/arxiv/pdf/1110/1110.4584v1.pdf>http://xxx.lanl.gov/PS_cache/arxiv/pdf/1110/1110.4584v1.pdf
The optimal and the most complete and detailed
form the proof of inaccessible cardinals
nonexistence have received in works
``Inaccessibility and Subinaccessibility", Part I
and Part II in 2008, 2010; these two works one can see at arXiv sites:
<http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.1956v4.pdf>http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.1956v4.pdf
http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.1447v2.pdf
and in Russian also at arXiv sites:
<http://arxiv.org/PS_cache/arxiv/pdf/1110/1110.0642v1.pdf>http://arxiv.org/PS_cache/arxiv/pdf/1110/1110.0642v1.pdf
http://arxiv.org/PS_cache/arxiv/pdf/1110/1110.0643v1.pdf
Here the following citation from the
beginning of first paper should be mentioned:
However, some criticism has been expressed that
these works [Part I and Part II] expose the
material which is too complicated and too
extensive and overloaded by the technical side of
the matter, that should be avoided even when it
uses in essence some new inevitable complicated
apparatus. According to these views every
result, even extremely strong, should be exposed
on few pages, otherwise it causes doubts in its
validity. So, the present work constitutes the
brief exposition of the whole investigation,
called to overcome such criticism.
Now some comments on the situation
should be stated. The common opinion is that
inaccessibles do exist in the Set Theory which is
sufficiently adequate. And it is really have to
be so, because the faith in inaccessibles
existence is the most ingenious attainment of
the mankind and it contains the greatest moments
of truth ( God himself is really the best inaccessible cardinal).
So, the principle of inaccessible cardinals
existence must not be destroyed by no means.
Therefore the nonexistence of inaccessible
cardinals within ZF and other affined theories
(and, more widely, within contemporary Set
Theory) confirms: not the inaccessible
cardinals nonexistence is fallacious, but the
theory ZF itself is nonadequate. And the
nonexistence of inaccessibles should be treated
as the "external inconsistence" of this theory itself.
- Therefore this theory should be confined
in its applications, and it should be corrected.
This correction should lie in the implementation
in this theory the notion of inaccessible
existence. It seems natural, that it should be
done by means of the following: the Time
phenomenon – that very notion, of which Set
Theory was deprived many centuries,
that already became absolute in all mathematical
world – should be redeemed bbackward in
mathematics. The way out of this crisis should
lie in the backward implementation the time
phenomenon in the body of the Set Theory, and
the more valuable it will be done the better.
Maybe, it should be done in fields
of ultraintuitionism of
Yessenin-Volpin, or of Vopenka (these theories
are the most appropriate for this purpose, as it
seems), maybe in the way of Nonstandard Mathematics, and so on.
Anyway, this state of matters must be
discussed. But the very first starting point of
the whole deal is the inaccessible cardinals
nonexistence in the contemporary Set Theory and this point cannot be avoided.
Sincerely yours,
Alexander Kiselev
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