Post on 28-Dec-2015
transcript
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
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No equations!
Truth is guaranteed!
Fundamental principles exposed!
Easy to generalize!
High elegance / beauty!
Problem: Given any five points in/on the unitsquare, is there always a pair with distance ≤ ?
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• What approaches fail?• What techniques work and why?• Lessons and generalizations
Problem: Given any five points in/on the unit equilateral triangle, is there always a pair with distance ≤ ½ ?
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• What approaches fail?• What techniques work and why?• Lessons and generalizations
X = 2X
XXX
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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
• 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 measuredbetween points x and y?
Historical PerspectivesGeorg 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
Problem: How can a new guest be accommodated in a full infinite hotel? ƒ(n) = n+1
Problem: How can an infinity of new guests be accommodated in a full infinite hotel?
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ƒ(n) = 2n
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one-to-one correspondence
Problem: How can an infinity of infinities of new guests be accommodated in a full infinite hotel?
Problem: Are there more integers than natural #’s?
ℕ ℤ ℕ ℤ
So | |<| |ℕ ℤ ?
Rearrangement:Establishes 1-1correspondence ƒ: ℕ ℤ
| |ℕ =| |ℤ
-4 -3 -2 -1 1 2 3 40-4 -3 -2 -1 1 2 3 40
1 2 3 4 6 7 8 95
ℤ
ℕℤ
Problem: Are there more rationals than natural #’s?
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ℕ ℚ ℕ ℚ
So | |<| |ℕ ℚ ?
Dovetailing:Establishes 1-1correspondence ƒ: ℕ ℚ| |ℕ =| |ℚ
Problem: Are there more rationals than natural #’s?
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ℕ ℚ ℕ ℚ
So | |<| |ℕ ℚ ?
Dovetailing:Establishes 1-1correspondence ƒ: ℕ ℚ| |ℕ =| |ℚ
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Avoiding duplicate
s!
Problem: Are there more rationals than natural #’s?
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ℕ ℚ ℕ ℚ
So | |<| |ℕ ℚ ?
Dovetailing:Establishes 1-1correspondence ƒ: ℕ ℚ| |ℕ =| |ℚ
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Problem: Why doesn’t this “dovetailing” work?
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There’s no “last” element on the first line!
So the 2nd line is never reached!
1-1 functionis not defined!
Dovetailing ReloadedDovetailing: ƒ: ℕ ℤ
0 1 2 3 4 5 6 7 8 …-1 -2 -3 -4 -5 -6 -7 -8 -9 …
To show |ℕ|=|ℚ| we can construct ƒ:ℕℚ by sorting x/y by increasing key max(|x|,|y|), while avoiding duplicates:
max(|x|,|y|) = 0 : {}max(|x|,|y|) = 1 : 0/1, 1/1max(|x|,|y|) = 2 : 1/2, 2/1max(|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!
ℕℤ
-4 -3 -2 -1 1 2 3 40-4 -3 -2 -1 1 2 3 40
1 2 3 4 6 7 8 95
1 23 4
6 7 85Dovetailin
g!
Theorem: There are more reals than rationals / integers.
Proof [Cantor]: Assume a 1-1 correspondence ƒ: ℕ ℝi.e., there exists a table containing all of andℕ all of :ℝ
ƒ(1) = 3 . 1 4 1 5 9 2 6 5 3 …ƒ(2) = 1 . 0 0 0 0 0 0 0 0 0 …ƒ(3) = 2 . 7 1 8 2 8 1 8 2 8 …ƒ(4) = 1 . 4 1 4 2 1 3 5 6 2 …ƒ(5) = 0 . 3 3 3 3 3 3 3 3 3 …. . . . . .
2 1 9 3 4X = 0 . ℝBut X is missing from our table! X ƒ(k)kℕ
ƒ not a 1-1 correspondence contradiction ℝ is not countable!There are more reals than rationals / integers!
Diagonalization
ℕ ℝ Non-existence proof!
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 . 1 4 1 5 9 2 6 5 3 …ƒ(2) = 1 . 0 0 0 0 0 0 0 0 0 …ƒ(3) = 2 . 7 1 8 2 8 1 8 2 8 …ƒ(4) = 1 . 4 1 4 2 1 3 5 6 2 …ƒ(5) = 0 . 3 3 3 3 3 3 3 3 3 …. . . . . .
2 1 9 3 4X = 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.
ℕ ℝ Non-existence proof!
Diagonalization
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!
PNP
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 + … = 1Cantor set does not contain any intervalsCantor 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=1.585Cantor set is a closed, totally bounded, compact, complete metric space, with uncountable cardinality and lebesque measure zero
Cantor dust (2D generalization): Cantor set crossed with itself
Cantor cube (3D):Cantor set crossed withitself three times