For some reason, this lemma remained elusive in my attempts to find it on the web. I couldn't find it in Hrbacek & Jech either.
I understand every part of the proof above except for the last sentence. How does beta = kappa follow? Many thanks.
soft question: reading this book is very hard for me :( the previous page took a solid 3 hours to fill in the blanks of the proofs, and the author sometimes assumes knowledge that was never stated. Any resources that are good supplements to Jech? very lost on how I should approach reading this book. any advice is helpful
The sentence before last shows there is a one-to-one mapping of $\kappa$ into $\lambda\times \beta,$ so $\kappa \le \lambda \cdot |\beta|.$ Since $\lambda < \kappa,$ it follows that $|\beta| \ge \kappa$ (otherwise we'd have $\lambda\cdot |\beta| <\kappa).$ Since $\beta_\xi < \kappa$ for all $\xi <\lambda,$ $\beta = \sup_{\xi < \lambda}\beta_\xi\le \kappa.$ We have $\beta \le \kappa$ and $|\beta|\ge \kappa,$ so it follows that $\beta =\kappa.$
As for books, the book by Enderton tends to get good reviews, and I've heard good things about Just and Weese as well (as well as Hrbacek/Jech that you mention). This is just what I've heard from others: I don't really know these first hand, other than Hrbacek and Jech which I've gone through bits and pieces of. The first book I learned from was Smullyan and Fitting, which I found quite pedagogical (although with a lot of annoying errors) but it does things a bit differently so might not serve the purpose of supplementing a more standard text. My second was Kunen (the older version), which was great, though the second chapter is rough going at first (it's probably actually best to skip it or read it selectively and then read it in full before the forcing chapters). Perhaps you can also browse the blurbs on the teach yourself logic guide (I think all the books I mentioned are written up there).