On the beginnings and motivations of certain branches of set theory

Question

I am planning to give a talk in my university on descriptive set theory, large cardinals and inner model theory. And the target audience are undergraduate students. I am trying to roughly explain what people are doing in these branches of set theory, as best as I can or know.

I have already decided on what theorems and definitions I want to mention, but as the audience hasn't likely seen much modern set theory, I have also decided to include a fair bit of history and motivations as well.

I know that there are some good materials in Kanamori's book, but I am in need of some article type thing to make it more interesting and also do it faster because of time restraints.

Just to be precise, these are some of the things I wish to mention:

• Some elementary large cardinals: weakly and strongly inaccessible/Mahlo/measurable
• Some stronger large cardinals: supercompact/strong/Woodin
• $V=L$, and maybe $V = L[U]$ and the role of mice [Maybe I'll do this just to mention the word "premouse". :)]
• Just mention the possiblity of Ultimate-$L$ [From some of Woodin's talks on youtube.]
• General descriptive set theory: introducing projective hierarchy, etc
• Infinite games and AD
• Some equiconsistency results

So in this light I wish to know if there are some articles or references where it is pointed out how inner model theory, large cardinals and descriptive set theory came about and what were their motivations.

Also I would really appreciate references which explain some of the materials that I have put in the above list.

Answer 1

There is a lot of good historical footnotes at the end of each chapter in Jech's book(s). If you're generally looking for timeline and references, that can be enough.

For the large cardinal historical reference you can check out the precursor of Kanamori's book:

Kanamori, A.; Magidor, M., The evolution of large cardinal axioms in set theory, Higher Set Theory, Proc. Oberwolfach 1977, Lect. Notes Math. 669, 99-275 (1978). ZBL0381.03038.

And generally, you can also take a look at Kanamori's introduction to the Handbook of Set Theory which is a treasure trove of history, as well as Kanamori's general bibliography which contains papers such as

Kanamori, Akihiro, Laver and set theory, ZBL06548031.

Kanamori, Akihiro, Cohen and set theory, Bull. Symb. Log. 14, No. 3, 351-378 (2008). ZBL1174.03001.

(His personal website contains a lot.)