Level 8: At the edge of mathematics

Infinity and strange math · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

This last level takes infinity to the places where modern mathematics was built and where it still meets its limits. You will tame "as close as you like" with the exact \(\varepsilon\)-\(\delta\) definition of a limit, test infinite sums, meet a horn you can fill but never paint, and learn why one simple question about infinity can never be answered from the usual axioms. Then you will count past infinity with ordinals, see infinitely small numbers made rigorous, and finish with infinity as a point you can draw.

Lesson	You will be able to
The formal definition of a limit	Use \(\varepsilon\) and \(N\) for sequences and \(\varepsilon\) and \(\delta\) for functions, and write proofs such as \(\lim_{x \to 3}(2x + 1) = 7\)
Convergence tests	Decide whether a series converges with the divergence, comparison, ratio and alternating series tests, and prove that \(\sum 1/n^2\) converges
Gabriel's horn	Show that the horn has finite volume \(\pi\) but infinite surface area, and resolve the painter's paradox
The continuum hypothesis	State the continuum hypothesis, explain what independence from the axioms means, and describe what Gödel and Cohen proved
Ordinals and cardinals	Count past infinity with \(\omega\), \(\omega + 1\) and \(\omega \cdot 2\), tell ordinals from cardinals, and run a Goodstein sequence
Infinitely small	Explain Leibniz's infinitesimals, Berkeley's objection, how limits replaced them and how Robinson's hyperreals made them rigorous
The point at infinity	Explain vanishing points, use homogeneous coordinates, and find where parallel lines and conics meet the line at infinity

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