June 26, 2025 Finished the chapter on Integers and Rationals. It's interesting how we defined both… basically as subsets of N×N, suitably modulo-ed out by equality conditions. For example (2,3) = (1,2) as integers and (1,2) = (2,4) as rationals! And then there's the slick notation of -n and p/q, immediately convenient after defining the subtraction and division operations respectively. The algebraic and order properties also become natural….

We saw with the last exercise of the chapter how the principle of infinite descent makes the complexity of Naturals, Integers, Rationals visible!

Next up: We are finally going to start constructing the reals by adding to rationals the the so called limit of sequences close to irrational numbers like √2. We already saw how it is indeed possible to get hold rations arbitrarily close to √2!

June 20, 2025 Solved the Cardinal Arithmetic problem! Induction and finite union (induction again) to the rescue. Although I haven’t noted much in green this week, I like how last couple of sections challenged my ability to prove abstractly…

Green Notes: These are special writings in the notebook where I am solving the exercises and basically a side companion for all the sketching. Basically green notes are just the insights I continue to gather along the way… end notes of some reflections…. some connections….

June 19, 2025 Finished Chapter 3 exercises of last section today. One important cardinal arithmetic exercise is pending. I thought about it a lot a couple of days back and gave it a pause. I probably have a new idea or at least some headspace to try it again tomorrow.

Next up: Integers and Rationals. Let’s go!

June 14, 2025 Finished reading first three chapters. Beautiful content and writing related to Peano axioms. Further content was also good. Very important exercises and food for thought in general and practice of basic theory of sets, functions e.t.c. There is a general resonance with the depth of thinking (ofcourse the book challenges me more than I challenge myself) but there are regular instances where my thinking depth is rewarded via exercises or otherwise, in the spirit of understanding the content deeply! Just finished reading last section of chapter on Sets (which also includes functions), the Cardinality of sets. Exercises of this section are pending. There was a challenging exercise in last section which I did manage to almost completely prove. I needed help with one part though. I will write the complete proof neatly sometime soon. I also skipped one exercise after that question which asserts that there is only one kind of natural number system (obeying Peano axioms) (up to bijections that is). I will work this at the end of exercises for last section may be.