Mathematics and logic

retrospect and prospects

Edición de la obra Mathematics and logic

1. Infinity of primes 2. Arbitrarily long sequences of successive integers, all not primes 3. Number of primes between 1 and n 4. Euler’s formula yields primes for x=0,1,2,3,…39 5. Irrational numbers: Algebraic, Transcendental (transcends operations of ordinary arithmetic) 6. Irrationality of square root of 2 7. Covering intervals 8. Euler’s constant C: 9. Approximating irrationals by rational numbers 10. Cantor’s existence proof of transcendental numbers 11. Non-constructibility of cube root of 2 12. Impossibility of finding center of circle with straightedge alone 13. Impossibility of covering modified chessboard with dominoes 14. Impossibility of decomposing cube into smaller cubes all of different size 15. Sperner’s Lemma: enumeration of patterns, fixed-point theorem follows 16. 292 ways of changing a dollar 17. The number system 18. The number of ways of partitioning a number into sums 19. The number of ways of partitioning a number into squares 20. Coin tossing: probability of m heads in n tosses 21. DeMoivre - Laplace Theorem 22. Axioms of probability theory equivalent to axioms of measure theory 23. Independent events implies normal distribution 24. Permutation group and solution of algebraic equations 25. Group of residues modulo p, Wilson’s Theorem 26. Homology group (a factor group) 27. Vectors, matrices, and geometry 28. Special theory of relativity as an example of geometric view in physics 29. Transformations, flows, and ergodicity 30. Iteration and composition of transformations: Markov chains 31. Consider two real valued functions both defined and continuous on the surface of a sphere. There must exist at least one point such that at this point and its antipode, both functions assume the same value. 32. Continuous, nowhere differentiable function 33. Convolution integrals: Heaviside calculus 34. Groups: braids. Does an algorithm exist to decide if two braids are equivalent? Yes, but general word problem in group theory is unsolved. 35. Gödels’s Theorem, Gödel numbering 36. Turing machine 37. Proof of independence of 5th postulate in plane geometry 38. Existence of sets satisfying axioms of set theory (including axiom of choice) but in which the continuum is of a “very high” power. Then sets intermediate between aleph-null and power of the continuum exist. 39. Maxwell’s equations 40. Ehrenfest game 41. Queues 42. Game theory by von Neumann 43. Information theory