Exercises in Set Theory: Applications of Forcing
New solutions to exercises from Thomas Jech’s book “Set Theory”:
The exercises from this chapter was a good opportunity to play a bit more with the forcing method. Exercise 15.15 seemed a straightforward generalization of Easton’s forcing but turned out to be a bit technical. I realized that the forcing notion used in that exercise provides a result in ZFC (a bit like Exercises 15.31 and 15.32 allow to prove some theorems on Boolean Algebras by Forcing).
Remember that is the normal sequence built by application of the continuum function at successor step. One may wonder: is regular?
First consider the case where is limit. The case is clear ( is regular) so assume . If is an inacessible cardinal, it is easy to prove by induction that for all we have : at step we use that is uncountable, at successor step that it is strong limit and at limit step that it is regular. Hence and so is regular. If is not a cardinal then so is singular. If is a cardinal but not strong limit then there is such that . Since there is such that . Then . So and is singular. Finally, if is a singular cardinal, then again and is singular.
What about the successor case i.e. ? By Corollary 5.3 from Thomas Jech’s book any , we can show that is a regular cardinal. The Generalized Continuum Hypothesis says that . Since it holds in we can not prove in ZFC that for some , is singular.
The generic extension constructed in exercise 15.15 satisfies GCH and so it’s another way to show that can not be proved to be singular for some . However, it provides a better result: by construction, and so . Since “regular cardinal” is a notion we deduce that is a regular cardinal in .
Now the question is: is there any “elementary” proof of the fact that is regular i.e. without using the forcing method?
–update: of course, I forgot to mention that by KÃ¶nig’s theorem, so the singularity of would imply the failure of the continuum hypothesis for the cardinal and this is not provable in ZFC.