infinity: discovering it, taming it and... cantor!
DESCRIPTION
A mathematical grappling with infinity from Greek discovery to German insanity. Infinity in mathematics can be both a serious problem and a powerful tool - it can lead to mind blowing conclusions and threaten to undermine logic itslef. Where mathematical infinity began, how it was used, right up to Cantor's unusual ideas about infinity and God... and beyond.TRANSCRIPT
1. Title, introduction.
2. Maths. Other types of infinity. Truth.
3. Proof. The backbone of maths. Axioms.
4. Algebra. Language. Letters for unknowns.
5. Associative law. Start with numbers. Distinct amounts. Snakes, bees, and amazonian tribes. 1,2 many.
6. Roman numerals. M from 'myriad'.
7. Egyptian numerals. 1,000,000 is everything.
8. A googol. Million, billion, atoms in the universe, googolplex, many.
9. Apeiron. Anaximander. Ancient Greeks, sacred numbers and the allowed infinity.
10. Fractions.
11. Relations. Rational numbers and their density.
12. Pythagoras' theorem.
13. The problem of the missing diagonal.
14. Hippasus.
15. Sqrt2 is rational.
16. Algebraic manipulation. A is even!
17. More manipulation. B is even!
18. Contradiction. Sqrt2 is not a rational. A new type of number.
19. Decimal expansions. Algebraic irrationals. Infinity coiled up in unwritable numbers.
20. Transcendental irrationals.
21. Types of number.
22. Newton, Leibniz, Aristotle and potential infinities. Parmenides & Zeno.
23. The infinity field. Limits.
24. Infinite number of things making a finite amount. Where else? Harmonic series....
25. ...does not converge. Divergence and convergence. A sequence to try. 2.
26. A nice pictorial proof. 1,2, many.
27. Infinity tamed.
28. Infinity bites back.
29. The ill behaved series explained.
30. Wallis and his symbol.
31. Will it play nicely?
32. Hilbert's hotel infinity. Infinity mocks the operations.
33. Cantor. Set theory.
34. One to one correspondence means same size. No mention of number.
35. Cardinality of 4.
36. Infinite cardinal. Aleph null.
37. Galileo's paradox. This is now OK. Squares, odds, evens, multiples, primes all Aleph Null. Listing. What about integers?
38. Integers are Aleph Null.
39. Rationals are Aleph Null. What about the reals? What about between 0 & 1?
40. A list of real numbers. Cantor's diagonal.
41. The number that isn't on the list.
42. Cardinality of the reals is larger than Aleph Null. Are there larger infinities?
43. Power Sets. Power set of an infinite set is always larger.
44. Power set of the naturals is the same size as the continuum. What about a power set of the reals? And a power set of that?
45. The ascending Alephs! Transfinites to absolute infinity. Needs aleph one to be power set of the naturals.
46. The continuum hypothesis. Where Cantor goes mad.
47. Kronecker, poincare, inconsistent sets, madness, theology, god and the absolute infinite.
48. Godel & Cantor.
49. Infinity: one, two, many.