Ordinals

Ordinals: definition and properties and counting and non-commutativity. What are they used for? (Transfinite induction?). Need Chen for this. $\omega$ or \(\omega \) and some \[\int_0^1x^2=\frac{1}{3}.\]

Pathalogia

The origins of analysis: pathologies in context. Riemann integrable <=> discontinuous on a set of measure zero as a side effect. Lebesgue creates a measure beyond Borel's more natural definition, and B is contained in M. (Cantor set => ternary function => B subset of M) Dirichlet function as limit of limit of cosines. Dense G-delta and the like.

The Weierstrass Function

What is it useful for: are these things just pathologies? (Density of such objects: continuous, but not diff anywhere, in the set \(C[0,1]\) with Weiner measure. Work out what this is, and why it's a natural fit for such functions. How many such functions? Defined by values on a countable set: given \(f:Q\rightarrow Q\) continuous it extends. That is, if you have a function in \(C[0,1]\) it is in 1-1 correspondence with a function on a countable subset of its domain. So... no more than the number of sequences of real numbers, which is c.

Sets

ZFC and the infinite. Countability as the tipping point. A set containing every successor. Halmos on picking out the natural numbers from this requirement. Subsets of naturals giving the reals in a holy way. (Holy subsets...) Re-examine the notion that Feferman mentions: we don't have a grasp of "All functions from \(N\) to \(\{0,1\}\)". So we don't appreciate the depth of such a reservoir. Binary trees (infinite: countable nodes, uncountable paths). Common errors in this sort of reasoning: Davies re monkeys. 2-dimensional binary tree as showing \([0,1]\times[0,1]\) similar to \([0,1]\).

Astronomy

Astronomy: prosthapharaesis, logs, Kepler, etc. Eccentricities (minor -- translate these into kilometres or percentages). What sort of calculational load did Kepler bear? What tools did he have?