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Re: Few questions on forcing, large cardinals
Posted:
Mar 19, 2013 3:18 AM
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On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> So isn't forcing simply Dirichlet box / pigeonhole principle? Without
> appeal to forcing, simply define arguments in symmetry then as to
> transfinite pigeonhole, what's the difference?
>
> How can large cardinals be defined in terms of V (the Universe) when V
> isn't a set? How are they cardinals if they aren't of sets?
>
> How can forcing's model be at once model and embedded in the model?
> Wouldn't it then be irregular?
>
> Are there any results in recursion/computation that can't simply (or
> not so simply) be framed in asymptotics without appeal to transfinite
> cardinals?
>
> Are there any results not of transfinite cardinals, solely due
> transfinite cardinals?
>
http://math.bu.edu/people/aki/14.pdf
"Rather, the focus is on the connections between the combinatorial
properties of the partial order of conditions and structural
properties of the extension."
"With forcing so expanded into the interstices of set theory and the
method so extensively amended from the beginning, what is the "it" of
Cohen's forcing and his individual achievement? Cohen discovered a
concrete and widely applicable means of operationally extending a
standard model of set theory to another without altering the ordinals.
The central technical innovation was the definable forcing relation,
through which satisfaction for the extension could be approached in
the ground model. Cohen's achievement was thus to be able to secure
properties of new sets without having all of their members in hand and
more broadly, to separate and then interweave truth and existence."
Quoting Cohen:
"One might say in a humorous way that the attitude toward my proof was
as follows. When it was first presented, some people thought it was
wrong. Then it was thought
to be extremely complicated. Then it was thought to be easy. But of
course it is easy in the sense that there is a clear philosophical
idea. There were technical points, you know, which bothered me, but
basically it was not really an enormously involved combinatorial
problem; it was a philosophical idea."
-- A. Kanimori, "Cohen and Set Theory", pp. 24-25
Symmetry in principle: all and nothing.
Regards,
Ross Finlayson
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