[FOM] Inconsistency of Inaccessibility

[MartDowd at aol.com]

The paper _http://www.ijpam.eu/contents/2011-66-2/index.html_ 
(http://www.ijpam.eu/contents/2011-66-2/index.html)  (referred  to in my Sep. 23 posting) 
gives what seem to be strong arguments that the  existence of inaccessible 
cardinals should be added as an axiom to ZFC.  It  is very unlikely that 
this is inconsistent.
 
Martin Dowd
 
 
In a message dated 10/22/2011 9:52:47 A.M. Pacific Daylight Time,  
aakiselev at yahoo.com writes:

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/P
S_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/P
S_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/P
S_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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