15-251 Great Theoretical Ideas in Computer Science Uncountability and Uncomputability January 29th, 2015
Our heros for this lecture father of set theory father of computer science 1845-1918 1912-1954 Uncountability Uncomputability
Our heros for this lecture father of set theory father of computer science (ZFC) 1912-1954 Uncountability Uncomputability
Infinity in mathematics Pre-Cantor: “Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics” - Carl Friedrich Gauss Post-Cantor: Infinite sets are mathematical objects just like finite sets.
Some of Cantor’s contributions > The study of infinite sets > Explicit definition and use of 1-to-1 correspondence - This is the right way to compare the cardinality of sets > There are different levels of infinity. - There are infinitely many infinities. > even though they are both infinite. | N | < | R | > even though . | N | = | Z | N ( Z > The diagonal argument.
Reaction to Cantor’s ideas Most of the ideas of Cantorian set theory should be banished from mathematics once and for all! - Henri Poincaré
Reaction to Cantor’s ideas I don’t know what predominates in Cantor’s theory - philosophy or theology. - Leopold Kronecker
Reaction to Cantor’s ideas Scientific charlatan. - Leopold Kronecker
Reaction to Cantor’s ideas Corrupter of youth. - Leopold Kronecker
Reaction to Cantor’s ideas Wrong. - Ludwig Wittgenstein
Reaction to Cantor’s ideas Utter non-sense. - Ludwig Wittgenstein
Reaction to Cantor’s ideas Laughable. - Ludwig Wittgenstein
Reaction to Cantor’s ideas No one should expel us from the Paradise that Cantor has created. - David Hilbert
Reaction to Cantor’s ideas If one person can see it as a paradise, why should not another see it as a joke? - Ludwig Wittgenstein
How do we count a finite set? A = { apple, orange, banana, melon } What does mean? | A | = 4 There is a 1-to-1 correspondence between and { 1 , 2 , 3 , 4 } A apple 1 orange 2 banana 3 melon 4
How do we count a finite set? A = { apple, orange, banana, melon } B = { 200 , 300 , 400 , 500 } What does mean? | A | = | B | apple 1 500 orange 2 200 banana 3 300 melon 4 400
How do we count a finite set? A = { apple, orange, banana, melon } B = { 200 , 300 , 400 , 500 } What does mean? | A | = | B | apple 500 orange 200 banana 300 melon 400 iff there is a 1-to-1 correspondence | A | = | B | between and . A B
3 important types of functions A B injective, 1-to-1 f : A → B is injective if A , → B a 6 = a 0 = ) f ( a ) 6 = f ( a 0 ) surjective, onto A B f : A → B is surjective if A ⇣ B ∀ b ∈ B, ∃ a ∈ A s.t. f ( a ) = b bijective, 1-to-1 correspondence A B f : A → B is bijective if A ↔ B f is injective and surjective
Comparing the cardinality of finite sets A B | A | ≤ | B | A , → B A B | A | ≥ | B | A ⇣ B A B | A | = | B | A ↔ B
Sanity checks | A | ≤ | B | i ff | B | ≥ | A | A , → B i ff B ⇣ A | A | = | B | i ff | A | ≤ | B | and | A | ≥ | B | A ↔ B i ff A , → B and A ⇣ B A ↔ B i ff A , → B and B , → A If | A | ≤ | B | and | B | ≤ | C | then | A | ≤ | C | If A , → B and B , → C then A , → C
One more definition | A | < | B | not | A | ≥ | B | There is no surjection from A to B. There is no injection from B to A. There is an injection from A to B, but there is no bijection between A and B.
These are the right definitions for infinite sets as well!
All is OK with infinite sets | A | ≤ | B | i ff | A | ≤ | B | A , → B i ff B ⇣ A | A | = | B | i ff | A | ≤ | B | and | B | ≤ | A | A ↔ B i ff A , → B and A ⇣ B Cantor Schröder A ↔ B i ff A , → B and B , → A Bernstein If | A | ≤ | B | and | B | ≤ | C | then | A | ≤ | C | If A , → B and B , → C then A , → C
Let me show you some interesting consequences.
Examples of equal size sets | N | = | Z | N = { 0 , 1 , 2 , 3 , 4 , . . . } Z = { . . . , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , . . . } 4 5 0 1 2 3 6 7 8 . . . f ( n ) = ( − 1) n +1 l n m 2 0 , 1 , − 1 , 2 , − 2 , 3 , − 3 , 4 , − 4 , . . . List the integers so that eventually every number is reached.
Examples of equal size sets | N | = | Z | Does this make any sense? N ( Z Surely . ⇒ | A | < | B | ? | N | < | Z | A ( B = Does renaming the elements of a set change its size? Let’s rename the elements of : Z { . . . , banana, apple, melon, orange, mango, . . . } Let’s call this set . How can you justify ? | N | < | F | F Bijection is nothing more than renaming.
Examples of equal size sets | N | = | S | N = { 0 , 1 , 2 , 3 , 4 , . . . } S = { 0 , 1 , 4 , 9 , 16 , . . . } f ( n ) = n 2
Examples of equal size sets | N | = | P | N = { 0 , 1 , 2 , 3 , 4 , . . . } P = { 2 , 3 , 5 , 7 , 11 , . . . } f ( n ) = n ’th prime number.
Countable sets | N | = | A | if: A is infinite, and you can list the elements as a 0 , a 1 , a 2 , . . . ( a i 6 = a j for i 6 = j ) in a well-defined way. Definition: A is countably infinite if | N | = | A | . A is countable if A is finite or | N | = | A | .
Countable sets Definition: A is countably infinite if | N | = | A | . A is countable if A is finite or | N | = | A | . What if A is infinite, but | A | < | N | ? No such set exists! So really A is countable if | A | ≤ | N | .
Countable? | N | = | Z × Z | ? (0 , 0) . . (1 , 0) . (1 , 1) (0 , 1) ( − 1 , 1) ( − 1 , 0) ( − 1 , − 1) (0 , − 1) … … (1 , − 1) (0 , 0) (2 , − 1) (2 , 0) (2 , 1) (2 , 2) (1 , 2) . . (0 , 2) .
Countable? | N | = | Q | ? -4 -3 -2 -1 0 1 2 3 4 Between any two rational numbers, there is another one. Can’t just list them in the order they appear on the line. a Any rational number can be written as a fraction . b a ( map to ) ( a, b ) Z × Z ⇣ Q b ⇒ | Q | ≤ | Z × Z | = | N | = Clearly . So . | N | ≤ | Q | | N | = | Q |
Countable? | N | = |{ 0 , 1 } ∗ | ? { 0 , 1 } ∗ = the set of finite length binary strings. ε 0 1 00 , 01 , 10 , 11 000 , 001 , 010 , 011 , 100 , 101 , 110 , 111 · · ·
Countable? | N | = | Σ ∗ | ? = the set of finite length words over . Σ ∗ Σ Same idea. CS method to show a set is countable : ( | A | ≤ | N | ) A Show | A | ≤ | Σ ∗ | i.e. Σ ∗ ⇣ A
CS method in action Is countable? Q [ x ] Q [ x ] = polynomials with rational coefficients. Take Σ = { 0 , 1 , . . . , 9 , x, + , − , ∗ , /, ˆ } Every polynomial can be described by a finite string over . Σ e.g. x ˆ3 − 1 / 4 x ˆ2 + 6 x − 22 / 7 So Σ ∗ ⇣ Q [ x ]
Seems like every set is countable… Nope! That would be boring!
Cantor’s Theorem Theorem: For any non-empty set A , | A | < |P ( A ) | . S = { 1 , 2 , 3 } P ( S ) = { ∅ , { 1 } , { 2 } , { 3 } , { 1 , 2 } , { 2 , 3 } , { 1 , 3 } , { 1 , 2 , 3 }} |P ( S ) | = 2 | S | P ( S ) ↔ { 0 , 1 } | S | S = { 1 , 2 , 3 } → { 1 , 3 } 1 0 1 ← 0 0 0 ← → ∅ binary strings of length | S |
Cantor’s Theorem Theorem: For any non-empty set A , | A | < |P ( A ) | . So: | N | < |P ( N ) | . | N | < |P ( N ) | < |P ( P ( N )) | < |P ( P ( P ( N ))) | < · · · (an infinity of infinities)
Proof by diagonalization Assume for some set . |P ( A ) | ≤ | A | A So . Let be such a surjection. A ⇣ P ( A ) f Example { 3 , 7 , 9 } 1 { 2 , 5 } 2 S = { 1 , 4 , . . . } { 1 , 2 , 3 } N ⇣ P ( N ) 3 . 4 . . S s Define S = { a ∈ A : a / ∈ f ( a ) } ∈ P ( A ) . Since is onto, s.t. . f ( s ) = S ∃ s ∈ A f But this leads to a contradiction: Why is this called a if then s ∈ S diagonalization argument? s / ∈ S if then s / ∈ S s ∈ S
Proof by diagonalization Example { 3 , 7 , 9 } 1 { 2 , 5 } 2 S = { 1 , 4 , . . . } { 1 , 2 , 3 } N ⇣ P ( N ) 3 . 4 . . … 1 2 3 4 5 f(1) 0 0 1 0 0 f(2) 0 1 0 0 1 … f(3) 1 1 1 0 0 f(4) 1 1 1 0 0 f(5) 0 0 0 1 1 . . S is defined so that . . . . S cannot equal any f(a) 1 0 0 1 0 … f(s)= S
Uncountable sets So |P ( N ) | > | N | . Definition: A set is uncountable if it is not countable, A i.e. . | A | > | N | Some examples: P ( N ) , P ( P ( N )) , . . .
Uncountable sets Let be the set of binary strings of infinite length. { 0 , 1 } ∞ {0,1,2,3,4,5,6,7,8,9, … } 0000000000 … ← → ∅ 1111111111 … → N ← 1010101010 … → { even natural numbers } ← . . . { 0 , 1 } ∞ is uncountable, i.e. |{ 0 , 1 } ∞ | > | N | { 0 , 1 } | S | ↔ P ( S ) because . { 0 , 1 } ∞ ↔ P ( N ) (just like ) (Recall is countable.) { 0 , 1 } ∗
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