# a < b, b < c implies a < c

A whole day and a night of sleep, and I thought of a strategy: show that A is similar to B; since b < c, if A is such, then the implication follows.

b < c means

1. B is equivalent to a proper subset of C, and
2. C is not equivalent to any subset of B

Since A is equivalent to the proper subset of B, is a subset of B, is essentially B – and C is not equivalent to such subsets of B – then a < c.

# That number called symbol

Algebra, not like in school, starts with sets. These elements associate to elements, set to set, a creative correspondence: chairs to students, objects to positive integers. Counting them, the highest ordinal number becomes the cardinal number, and this is what we see, naked, in every arithmetic operation thence.

The abstraction elevates to the sky: each number represents an abstraction of its class of sets; that is, 3 represents three of a set of anything; we free ourselves from leaks. We define equivalence and relations, addition, subtraction, multiplication and division.

Even the early problems ask for proofs. I’m tempted to jump back into discrete to plow through the proof tips section. But that would start a prerequisite cascade: I would begin with the first sentence of the first chapter, starting over, and exhaust months.