FOM: Re: CH and RVM
Matt Insall
montez at rollanet.org
Mon Feb 21 14:07:55 EST 2000
Professor Shipman:
>It is just
> as interesting to put it the other way, that RVM settles the question of
> CH (very negatively -- the continuum must be at least as large as the
> first weak inaccessible).
*I* agree that it is interesting when put your way as well. However, there
are those who claim that Mathematicians are uninterested in CH for various
reasons. My point was only to indicate a reason that analysts and applied
matheaticians, who may have no inherent interest in set theoretic questions,
but only see them as tools for answering their own questions in their own
domain, CH is of interest *because* of its relationship to RVM, which
appears to me to be inherently an analysis question.
> This is not quite "as large as logically possible", because the
> continuum need not itself be weakly inaccessible, so there could be
> fewer than c cardinals below c.
Agreed. RVM does not imply that there are c cardinals between \aleph_0 and
c. In my post, I did not give my reasons for my conviction, and I did not
claim that RVM implies my conviction that there are c cardinals between
\aleph_0 and c.
>But I am not sure it is a valid
> application of Maddy's 'MAXIMIZE' principle to say there are a lot of
> cardinals, because a lot of cardinals means a dearth of 1-1
> correspondences between sets.
Yes. One can question my ``application'' of Professor Maddy's MAXIMIZE
principle. Yet, it seems to me that one must choose *what* to ``MAXIMIZE''
in applying this principle. I am willing to have some non-well-orderable
sets around [not(AC), of course], which is a form of Professor Maddy's
MAXIMIZE (since it is relatively consistent to have such non-well-orderable
sets), but also, for those cardinals I can identify easily, such as the
initial ordinals (i.e., the cardinals of the well-orderable sets), I want to
have as many as possible. Perhaps you can tell me why a large number of 1-1
correspondences would be preferable.
>This is why GCH implies the "maximizing"
> principle AC -- in the presence of so many 1-1 maps, we can construct
> choice functions.
Apparently, you already have chosen to prefer AC, so that you interpret the
idea of ``maximization'' in terms of ``more (constructible) choice
functions''. But since AC does not imply even CH, it is possible to have AC
*without* so many 1-1 correspondnces. Thus, although I mentioned the RVM
problem as a motivation (for analysts) to study CH, I do not advocate it. I
would like to have, if possible, instead, the axiom
c = \aleph_c,
meaning (a) The continuum is well-orderable, and (b) There are c initial
ordinals below c. I have not yet seen any reason to deny this possibility,
although I have admittedly not read all that has been done on ``large
cardinals''. If this axiom is not realizable, I will gladly abandon it.
However, it is my understanding that it is consistent to assume that c =
\aleph_{\alpha}, for any cardinal \aleph_{\alpha} not explicitly ruled out
by Königs Theorem, which states that ``c cannot be the sum of countably many
smaller cardinals''.
Matt Insall
Associate Professor
Mathematics and Statistics Department
University of Missouri - Rolla
insall at umr.edu
montez at rollanet.org
http:/www.umr.edu/~insall
http:/www.rollanet.org/~montez
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