Class 18: Spooky Infinities cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans University of Virginia

Plan Review: Countably Infinite Sets Any Bigger Sets? Preparation for Monday Uncountable Sets Problem Set 7 is due Friday Exam 2 is in two weeks (Nov 9)

Countably Infinite 3 A set is countable if and only if there exists a surjective function from ℕ to . A set is infinite, if there is no bijection between and any ℕ$ . A set is countably infinite if it is countable and it is infinite. A set is countably infinite if there exists a bijection between and ℕ.

Example: ℤ is countably infinite 4

Example: (ℕ×ℕ) is countably infinite 5

Example: ℚ is countably infinite 6

Tsil Definition (PS7) Base: the empty tsil () Constructor: for any tsil, , and object, postpend(, ) is a tsil 7 How many tsils?

Tsil-∅ Definition Base: the empty tsil () Constructor: for any tsil, postpend(, ∅) is a tsil 8

Power Sets 9 The power set of a set is the set of all subsets of . ∈ ⟺ ⊆ What is the cardinality of ()?

10 For all finite sets , > ?

11 For all finite sets , = 2|D|. Proven in Class 11

12 For all finite sets , > ||.

13 For all sets , is > ?

14 For all sets , > | |. Bogus non-proof: Proof by induction: ∷= ∀ sets where = . > . Base case: 0 : = ∅. = {∅}. = 1 > = 0. Inductive case: ∀ ∈ ℕ. ⇒ + 1 . for all sets where = + 1, ∃ where = , ∉ . = ∪ ⇒ > ⇒ + 1 > + 1 Since () includes all elements of () and includes , this means > ⇒ ( + 1). Therefore, always holds, so we can conclude for all sets , > | |.

15 For all sets , > | |. Bogus non-proof: Proof by induction: ∷= ∀ sets where = . > . Base case: 0 : = ∅. = {∅}. = 1 > = 0. Inductive case: ∀ ∈ ℕ. ⇒ + 1 . for all sets where = + 1, ∃ where = , ∉ . = ∪ ⇒ > ⇒ + 1 > + 1 Since () includes all elements of () and includes , this means > ⇒ ( + 1). Therefore, always holds, so we can conclude for all sets , > | |. 1 2 3 4 5 6 7 8 9 10 11 Which step is wrong?

Georg Cantor’s Shocking Proof (~1874) 16 Historical note: this isn’t actually Cantor’s proof, although it is commonly called Cantor’s Theorem. Cantor came up with the diagonalization argument we will see next; this proof is believed to have been first done by Hessenberg (1906).

17 For all sets , > | |.

18 For all sets , > | |.

Is ℕ countable? 19

Trick-or-Treat Protocols 20

“Trick or Treat” 21 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked

Illogical Threat 22 ∨ ∧ ⟹

“Trick or Treat” 23 Tricker initiates the protocol by making a threat and demanding tribute Victim either pays tribute (usually in the form of sugary snack) or risks being tricked Tricker must convince Victim that she poses a credible threat: prove she is a qualified tricker

Trick-or-Treat Trickers? Victim 24 Any problems with this?

Proof without Disclosure How can the tricker prove their trickability, without allowing the victim to now impersonate a tricker? 25

Challenge-Response Protocol 26 Prover: proves knowledge of by revealing (, ) . Verifier: convinces prover knows , but learns nothing useful about . Verifier: picks random . Need a one-way function: hard to invert, but easy to compute.

Example: RSA 27 Ee (M ) = Me mod n Dd (C ) = Cd mod n Correctness property: Ee (Dd ()) =

Trick-or-Treat Trickers? Victim 28

Trick-or-Treat Trickers? Victim 29 How does victim know e and n? Verify: j = jmod =

30 “Elsa #253224”, = 3482..., = 1234... signed by Tricker’s Buroo Verify: j = jmod = Verify Tricker’s Buroo signature on certificate

31 “virginia.edu”, = … = ... signed by Certificate Authority Verify and Decrypt: l j () = Verify signature on certificate Server

32 “virginia.edu”, = … = ... signed by Certificate Authority Verify and Decrypt: l j () = Verify signature on certificate Server

33

Countable Strings? 34 BitStrings = ∀ ∈ ℕ . {0, 1}m

Countable Infinite Strings? 35 InfiniteBitStrings = {0, 1}n

36 Class 9

Smallest Infinite Ordinal 37 = smallest infinite ordinal number ∞ + 1 = + 1 >

Smallest Infinite Ordinal 38 = smallest infinite ordinal number ∞ + 1 = ∞ = ∞ - 1 + 1 > ∀ ∈ ℕ . 2 > +

Countable Infinite Strings? 39 InfiniteBitStrings = {0, 1}n

Cantor’s Diagonalization Argument 40

41

Charge Enjoy Halloween But don’t be victimized by any unsubstantiated threats or spooky infinities! 42 Problem Set 7 due Friday Really spooky question: is there any set bigger than (ℕ) ?