[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:
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:
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:
and in Russian also at arXiv sites:
      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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