[FOM] Insall on GCH

[E. Todd Eisworth]

[Insall]

By the way, I understand that my strong negation of (G)CH is not the one
supported by current arguments from analysts and topologists.  I still need
to learn more about why these analysts and toppologists prefer statements
like

2^{\aleph_0} = \aleph_2,


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I don't think analysts or topologists have any preference for the value of
the Continuum --- it's more a case of axioms.  For many people in topology
and analysis, set theory is used as a "black box" --- consistent axioms such
as diamond, CH, Martin's Axiom, and the Proper Forcing Axiom are all fairly
easy to use by those not intimately familiar with iterated forcing. In most
cases, the actual value of the continuum is irrelevant.

To say it another way, we can prove theorems from axioms that happen to
imply the Continuum is $\aleph_1$ or $\aleph_2$, but strong negations of GCH
don't (yet?) give us strong "proving power".

Best,

Todd