## A non fixed point by any other name…

June 15, 2011 at 22:00 | Posted in Uncategorized | Leave a commentThis 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 that are not fixed points of the 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 1Let be a singular cardinal. Then the following are equivalent:

- is not a fixed point, i.e., .
- where is a limit ordinal .
- There is a such that is progressive.

*Proof:* Suppose , and fix such that . Since is a limit cardinal, we know , so choose such that

Now fix such that

It is clear that must be a limit (as is), and the fact that

is completely trivial. For the last bit, we know

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

Now assume (2) holds, set , and define

Now note

and we have (3).

Moving on to the last implication, let us suppose and is progressive, i.e., . But then as well, and hence

Thus, we have

as required.

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