1. No vector calculus / trig! No equations!
Problem: Can 5 test tubes be spun simultaneously in a 12-hole centrifuge?
No vector calculus / trig!
No equations!
Fundamental principles exposed!
Truth is guaranteed!
Easy to generalize!
High elegance / beauty!
What does “balanced” means?
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…

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

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

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

6. Historical Perspectives
Georg Cantor ( )
Created modern set theory
Invented trans-finite arithmetic
(highly controvertial at the time)
Invented diagolanization argument
First to use 1-to-1 correpondences 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. one-to-one correspondence
Problem: How can an infinity of infinities of new
guests be accommodated in a full infinite hotel? … 15
one-to-one correspondence 14 10 13 9 6 12 8 5 3 11 7 4 2 1

14. Problem: Are there more integers than natural #’s?
ℕ Ì ℤ
ℕ ¹ ℤ
So |ℕ|<|ℤ| ?
Rearrangement:
Establishes 1-1
correspondence
ƒ: ℕ « ℤ
Þ|ℕ|=|ℤ| ℤ -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 ℤ ℕ 1 2 3 4 5 6 7 8 9

15. Problem: Are there more rationals than natural #’s?1 2 3 5 4 6 7 2 1 3 5 4 6 7 3 2 1 5 4 6 7 4 2 1 3 5 6 7 5 2 1 3 4 6 7 6 2 1 3 5 4 7 7 2 1 3 5 4 6 8 2 1 3 5 4 6 7 …
ℕ Ì ℚ
ℕ ¹ ℚ
So |ℕ|<|ℚ| ?
Dovetailing:
Establishes 1-1
correspondence
ƒ: ℕ « ℚ
Þ|ℕ|=|ℚ| 1 2 3 4 6 5 7 55 17 18 19 20 21 16 15 14 13 22 5 6 7 12 23 28 4 3 8 11 24 27 1 2 9 10 25 26 1 2 3 4 5 6 7 8 …

16. Problem: Are there more rationals than natural #’s?24 25 26 27 28 29 39 1 7 2 7 3 7 4 7 5 7 6 7 7 8 7 …
ℕ Ì ℚ
ℕ ¹ ℚ
So |ℕ|<|ℚ| ?
Dovetailing:
Establishes 1-1
correspondence
ƒ: ℕ « ℚ
Þ|ℕ|=|ℚ| 7 23 22 30 1 6 2 6 3 6 4 6 5 6 6 7 6 8 6 … 6 12 13 14 15 31 38 1 5 2 5 3 5 4 5 5 21 6 5 7 5 8 5 … 5
Avoiding duplicates! 11 10 16 32 1 4 2 4 3 4 4 5 4 6 4 7 4 8 4 … 4 4 5 9 17 33 37 1 3 2 3 3 4 3 5 3 6 3 7 3 8 3 … 3 3 6 18 34 1 2 2 3 2 4 2 5 2 6 2 7 2 8 2 … 2 1 1 2 2 1 7 3 1 8 4 1 19 20 35 36 5 1 6 1 7 1 8 1 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 …

17. Problem: Are there more rationals than natural #’s?21 26 1 7 2 7 3 7 4 7 5 7 6 7 7 8 7 …
ℕ Ì ℚ
ℕ ¹ ℚ
So |ℕ|<|ℚ| ?
Dovetailing:
Establishes 1-1
correspondence
ƒ: ℕ « ℚ
Þ|ℕ|=|ℚ| 7 17 1 6 2 6 3 6 4 6 5 6 6 7 6 8 6 … 6 11 16 20 25 1 5 2 5 3 5 4 5 5 6 5 7 5 8 5 … 5 9 15 24 1 4 2 4 3 4 4 5 4 6 4 7 4 8 4 … 4 5 8 14 19 1 3 2 3 3 4 3 5 3 6 3 7 3 8 3 … 3 3 7 13 23 1 2 2 3 2 4 2 5 2 6 2 7 2 8 2 … 2 1 2 4 12 18 22 1 2 1 3 1 6 4 1 10 5 1 6 1 7 1 8 1 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 …

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

19. Dovetailing! Dovetailing Reloaded Dovetailing: ƒ:ℕ «ℤ -4 -4 -3 -3 -2-1 -1 1 1 2 2 3 3 4 4 1 2 3 4 5 6 7 8 … -1 -2 -3 -4 -5 -6 -7 -8 -9 ℤ ℕ 1 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!
Dovetailing! 1 2 3 4 5 6 7 8

20. Diagonalization Non-existence proof!
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: ℕ ℝ
ƒ(1) = 3 . 1 4 5 9 2 6 …
ƒ(2) =
ƒ(3) = 7 8
ƒ(4) =
ƒ(5) =
. . .
Diagonalization
Non-existence proof!
. . . Î ℝ
X = 0 . 2 1 9 3 4
But X is missing from our table! X ¹ ƒ(k) " kÎℕ
Þ ƒ not a 1-1 correspondence
Þ contradiction
ℝ is not countable!
There are more reals than rationals / integers!

21. Diagonalization Non-existence proof! ℕ ℝ ƒ(1) = 3 . 1 4 5 9 2 6 …
Problem 1: Why not just insert X into the table?
Problem 2: What if X=0.999… but 1.000… is already in table? ℕ ℝ
ƒ(1) = 3 . 1 4 5 9 2 6 …
ƒ(2) =
ƒ(3) = 7 8
ƒ(4) =
ƒ(5) =
. . .
Diagonalization
Non-existence proof!
. . . Î ℝ
X = 0 . 2 1 9 3 4
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.

23. Existence proofs can be easy! Non-existence proofs are often hard!
Must cover all possible (usually infinite) scenarios!
Examples / counter-examples are not convincing!
Not “symmetric” to existence proofs!
Ex: proof that you
are a millionare:
“Proof” that you
are not a millionare ?
Existence proofs
can be easy!
Non-existence proofs
are often hard!
P¹NP

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 log32=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