1. Problem: Can 5 test tubes be spun simultaneously in a 12-hole centrifuge? What does “balanced” mean? Why are 3 test tubes balanced? Symmetry! Can you merge solutions? Superposition! Linearity! ƒ(x + y) = ƒ(x) + ƒ(y) Can you spin 7 test tubes? Complementarity! Empirical testing… No vector calculus / trig! No equations! Truth is guaranteed! Fundamental principles exposed! Easy to generalize! High elegance / beauty!

2. Problem: Given any five points in/on the unit square, is there always a pair with distance ≤ ? 1 1 What approaches fail? What techniques work and why? Lessons and generalizations

3. Problem: Given any five points in/on the unit equilateral triangle, is there always a pair with distance ≤ ½ ? 1 1 1 What approaches fail? What techniques work and why? Lessons and generalizations

4. X = 2 X X X X … Problem: Solve the following equation for X: where the stack of exponentiated x’s extends forever. What approaches fail? What techniques work and why? Lessons and generalizations

5. What approaches fail? What techniques work and why? Lessons and generalizations x y Problem: For the given infinite ladder of resistors of resistance R each, what is the resistance measured between points x and y?

6. Historical Perspectives Georg Cantor (1845-1918) Created modern set theory Invented trans-finite arithmetic (highly controvertial at the time) Invented diagonalization argument First to use 1-to-1 correspondences with sets Proved some infinities “bigger” than others Showed an infinite hierarchy of infinities Formulated continuum hypothesis Cantor’s theorem, “Cantor set”, Cantor dust, Cantor cube, Cantor space, Cantor’s paradox Laid foundation for computer science theory Influenced Hilbert, Godel, Church, Turing

9. Problem: How can a new guest be accommodated in a full infinite hotel? ƒ(n) = n+1

10. Problem: How can an infinity of new guests be accommodated in a full infinite hotel? … ƒ(n) = 2n

11. 1 2 3 4 5 6 7 8 9 10 … 11 12 13 14 15 one-to-one correspondence Problem: How can an infinity of infinities of new guests be accommodated in a full infinite hotel?

14. Problem: Are there more integers than natural #’s? ℕ ℤℕ ℤ ℕ  ℤℕ  ℤ So | ℕ |<| ℤ | ? Rearrangement: Establishes 1-1 correspondence ƒ: ℕ  ℤ |ℕ|=|ℤ||ℕ|=|ℤ| -4-3-212340-4-3-212340 123467895 ℤ ℕ ℤ

15. Problem: Are there more rationals than natural #’s?1 2 3 4 6 57 1 1 2 1 1 1 3 1 5 1 4 1 6 1 72 2 2 2 1 2 3 2 5 2 4 2 6 2 73 3 2 3 1 3 3 3 5 3 4 3 6 3 74 4 2 4 1 4 3 4 5 4 4 4 6 4 75 5 2 5 1 5 3 5 5 5 4 5 6 5 76 6 2 6 1 6 3 6 5 6 4 6 6 6 77 7 2 7 1 7 3 7 5 7 4 7 6 7 78 8 2 8 1 8 3 8 5 8 4 8 6 8 7… … … … … … … … 12 4 567 8 9 10 11 12 131415 16 17181920 21 22 23 24 25 26 27 28 55 312345678 ℕ ℚℕ ℚ ℕ  ℚℕ  ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ  ℚ |ℕ|=|ℚ||ℕ|=|ℚ|

16. Problem: Are there more rationals than natural #’s?1 2 3 4 6 5 7 1 1 2 1 1 1 3 1 5 1 4 1 6 1 72 2 2 2 1 2 3 2 5 2 4 2 6 2 73 3 2 3 1 3 3 3 5 3 4 3 6 3 74 4 2 4 1 4 3 4 5 4 4 4 6 4 75 5 2 5 1 5 3 5 5 5 4 5 6 5 76 6 2 6 1 6 3 6 5 6 4 6 6 6 77 7 2 7 1 7 3 7 5 7 4 7 6 7 78 8 2 8 1 8 3 8 5 8 4 8 6 8 7… … … … … … … … 12 3 45 6 7 8 9 10 11 12131415 16 17 18 19 20 12345678 ℕ  ℚ ℕ  ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ  ℚ  | ℕ |=| ℚ | 21 2223 242526272829 30 31 32 33 34 35 36 37 38 39 Avoiding duplicates!

17. Problem: Are there more rationals than natural #’s?1 2 3 4 6 5 7 1 1 2 1 1 1 3 1 5 1 4 1 6 1 72 2 2 2 1 2 3 2 5 2 4 2 6 2 73 3 2 3 1 3 3 3 5 3 4 3 6 3 74 4 2 4 1 4 3 4 5 4 4 4 6 4 75 5 2 5 1 5 3 5 5 5 4 5 6 5 76 6 2 6 1 6 3 6 5 6 4 6 6 6 77 7 2 7 1 7 3 7 5 7 4 7 6 7 78 8 2 8 1 8 3 8 5 8 4 8 6 8 7… … … … … … … … 12 3 58 7 4 6 14 15 9 11162025 24 19 13 10 12 12345678 ℕ  ℚ ℕ  ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ  ℚ  | ℕ |=| ℚ | 17 2126 23 18 22

18. Problem: Why doesn’t this “dovetailing” work?1 2 3 4 6 5 7 1 1 2 1 1 1 3 1 5 1 4 1 6 1 72 2 2 2 1 2 3 2 5 2 4 2 6 2 73 3 2 3 1 3 3 3 5 3 4 3 6 3 74 4 2 4 1 4 3 4 5 4 4 4 6 4 75 5 2 5 1 5 3 5 5 5 4 5 6 5 76 6 2 6 1 6 3 6 5 6 4 6 6 6 77 7 2 7 1 7 3 7 5 7 4 7 6 7 78 8 2 8 1 8 3 8 5 8 4 8 6 8 7… … … … … … … … 12345678 There’s no “last” element on the first line! So the 2 nd line is never reached!  1-1 function is not defined!

19. Dovetailing Reloaded Dovetailing: ƒ: ℕ  ℤ 012345678… -2-3-4-5-6-7-8-9… To show | ℕ |=| ℚ | we can construct ƒ: ℕ  ℚ by sorting x/y by increasing key max(|x|,|y|), while avoiding duplicates: max(|x|,|y|) = 0 : {} max(|x|,|y|) = 1 : 0/1, 1/1 max(|x|,|y|) = 2 : 1/2, 2/1 max(|x|,|y|) = 3 : 1/3, 2/3, 3/1, 3/2...{finite new set at each step} Dovetailing can have many disguises! So can diagonalization! ℕ ℤ -4-3-212340-4-3-212340 123467895 12 34 6785 Dovetailing!

20. Theorem: There are more reals than rationals / integers. Proof [Cantor]: Assume a 1-1 correspondence ƒ: ℕ  ℝ i.e., there exists a table containing all of ℕ and all of ℝ : ƒ(1) =3.141592653… ƒ(2) =1.000000000… ƒ(3) =2.718281828… ƒ(4) =1.414213562… ƒ(5) =0.333333333…... 21934 X = 0.  ℝ But X is missing from our table! X  ƒ(k)  k  ℕ  ƒ not a 1-1 correspondence  contradiction  ℝ is not countable! There are more reals than rationals / integers! Diagonalization ℕℝ Non-existence proof!

21. Problem 1: Why not just insert X into the table? Problem 2: What if X=0.999… but 1.000… is already in table? ƒ(1) =3.141592653… ƒ(2) =1.000000000… ƒ(3) =2.718281828… ƒ(4) =1.414213562… ƒ(5) =0.333333333…... 21934 X = 0.  ℝ Table with X inserted will have X’ still missing! Inserting X (or any number of X’s) will not help! To enforce unique table values, we can avoid using 9’s and 0’s in X. ℕℝ Non-existence proof! Diagonalization

23. Non-Existence Proofs Must cover all possible (usually infinite) scenarios! Examples / counter-examples are not convincing! Not “symmetric” to existence proofs! Ex: proof that you are a millionaire: “Proof” that you are not a millionaire ? Existence proofs can be easy! Non-existence proofs are often hard! P  NP

24. Cantor set: Start with unit segment Remove (open) middle third Repeat recursively on all remaining segments Cantor set is all the remaining points Total length removed: 1/3 + 2/9 + 4/27 + 8/81 + … = 1 Cantor set does not contain any intervals Cantor set is not empty (since, e.g. interval endpoints remain) An uncountable number of non-endpoints remain as well (e.g., 1/4) Cantor set is totally disconnected (no nontrivial connected subsets) Cantor set is self-similar with Hausdorff dimension of log 3 2=1.585 Cantor set is a closed, totally bounded, compact, complete metric space, with uncountable cardinality and lebesque measure zero

25. Cantor dust ( 2D generalization): Cantor set crossed with itself

26. Cantor cube (3D): Cantor set crossed with itself three times