A non fixed point by any other name…

June 15, 2011 at 22:00 | Posted in Uncategorized | Leave a comment

This is a silly post, but looking ahead to Conclusion 3.2A reminded me of one of the frustrations I had when first trying to read The Book. One of the major themes of pcf theory is that it tells us quite a lot about cardinal arithmetic at singular cardinals {\mu} that are not fixed points of the {\aleph} function. The problem is that this is equivalent to lots of other conditions, and very often Shelah will use a different formulation of this property in the statement of a theorem, so it isn’t clear right away that what he’s really talking about is a cardinal that isn’t a fixed point. This little proposition is just an elementary exercise in set theory, but it’s useful to keep in mind when reading The Book. (And let’s hope I haven’t flubbed it since I’m typing from home while babysitting…)

Proposition 1 Let {\mu} be a singular cardinal. Then the following are equivalent:

  1. {\mu} is not a fixed point, i.e., {\mu<\aleph_\mu}.
  2. {\mu=\aleph_{\xi+\zeta}} where {\zeta} is a limit ordinal {<\aleph_\xi}.
  3. There is a {\tau<\mu} such that {(\tau,\mu)\cap{\sf Reg}} is progressive.

Proof: Suppose {\mu<\aleph_\mu}, and fix {\delta<\mu} such that {\mu=\aleph_\delta}. Since {\mu} is a limit cardinal, we know {|\delta|^+<\mu}, so choose {\xi<\delta} such that

\displaystyle  |\delta|^+=\aleph_\xi. \ \ \ \ \ (4)

Now fix {\zeta} such that

\displaystyle  \delta=\xi+\zeta. \ \ \ \ \ (5)

It is clear that {\zeta} must be a limit (as {\delta} is), and the fact that

\displaystyle  \mu=\aleph_\delta=\aleph_{\xi+\zeta} \ \ \ \ \ (6)

is completely trivial. For the last bit, we know

\displaystyle  |\zeta|\leq |\delta|<|\delta|^+=\aleph_\xi, \ \ \ \ \ (7)

and we are done with proving that (1) implies (2)

Now assume (2) holds, set {\tau=\aleph_\xi}, and define

\displaystyle  \mathfrak{a}=(\tau,\mu)\cap{\sf Reg}. \ \ \ \ \ (8)

Now note

\displaystyle  |\mathfrak{a}|\leq |(\tau,\mu)|=|\zeta|<\tau<\tau^+=\min(\mathfrak{a}), \ \ \ \ \ (9)

and we have (3).

Moving on to the last implication, let us suppose {\tau<\mu} and {\mathfrak{a}=(\tau,\mu)\cap{\sf Reg}} is progressive, i.e., {|\mathfrak{a}|\leq\tau}. But then {|(\tau,\mu)\cap{\sf Card}|\leq\tau} as well, and hence

\displaystyle  |\mu\cap{\sf Card}|\leq\tau. \ \ \ \ \ (10)

Thus, we have

\displaystyle  \mu\leq\aleph_{\tau^+}<\aleph_\mu, \ \ \ \ \ (11)

as required.


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