An Introduction to Ordinals D. Salgado, N. Singer #blairlogicmath November 2017 D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 1 / 39 Introduction D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 2 / 39 History Bolzano defined a set as \an aggregate so conceived that it is indifferent to the arrangement of its members" (1883) Cantor defined set membership, subsets, powersets, unions, intersections, complements, etc. Given two sets A and B, Cantor says they have the same cardinality iff there exists a bijection between them Denoted jAj = jBj More generally, jAj ≤ jBj iff there is an injection from A to B Every set has a cardinal number, which represents its cardinality @0 Infinite sets have cardinalities @0; 2 ;::: Arithmetic can be defined, e.g. jAj + jBj = j(A × f0g) [ (B × f1g)j D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 3 / 39 That structure is derived from the order of the set The notions of cardinality only apply to unstructured sets and are determined by bijections Every set also has an order type determined by order-preserving bijections: f : A ! B is a bijection and 8x; y 2 A [(x < y) ! (f (x) < f (y))] Intuition Cantor viewed sets as fundamentally structured When we view the natural numbers, why does it make sense to think of them as just an “infinite bag of numbers"? The natural numbers is constructed in an inherently structured way: through successors D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 4 / 39 The notions of cardinality only apply to unstructured sets and are determined by bijections Every set also has an order type determined by order-preserving bijections: f : A ! B is a bijection and 8x; y 2 A [(x < y) ! (f (x) < f (y))] Intuition Cantor viewed sets as fundamentally structured When we view the natural numbers, why does it make sense to think of them as just an “infinite bag of numbers"? The natural numbers is constructed in an inherently structured way: through successors That structure is derived from the order of the set D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 4 / 39 Intuition Cantor viewed sets as fundamentally structured When we view the natural numbers, why does it make sense to think of them as just an “infinite bag of numbers"? The natural numbers is constructed in an inherently structured way: through successors That structure is derived from the order of the set The notions of cardinality only apply to unstructured sets and are determined by bijections Every set also has an order type determined by order-preserving bijections: f : A ! B is a bijection and 8x; y 2 A [(x < y) ! (f (x) < f (y))] D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 4 / 39 Cardinals and ordinals The cardinality of a set corresponds to its cardinal number; the order type of a set corresponds to its ordinal number Two sets can have the same cardinality but different order types The cardinality of N, jNj, is @0 The order type of N, Ord N, is ! Under order-preserving bijections, ordered sets are equivalent up to the labeling of the elements D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 5 / 39 Can they be placed in an order-preserving bijection? Naturals 0 1 2 3 ::: Primes 2 3 5 7 ::: Naturals and primes Consider the naturals 0; 1; 2; 3;::: and the primes 2; 3; 5; 7; 11;:::. Can they be placed in a bijection? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 6 / 39 Naturals 0 1 2 3 ::: Primes 2 3 5 7 ::: Naturals and primes Consider the naturals 0; 1; 2; 3;::: and the primes 2; 3; 5; 7; 11;:::. Can they be placed in a bijection? Can they be placed in an order-preserving bijection? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 6 / 39 Naturals and primes Consider the naturals 0; 1; 2; 3;::: and the primes 2; 3; 5; 7; 11;:::. Can they be placed in a bijection? Can they be placed in an order-preserving bijection? Naturals 0 1 2 3 ::: Primes 2 3 5 7 ::: D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 6 / 39 Naturals 0 1 2 3 4 ::: Integers 0 1 -1 2 -2 ::: Can they be placed in an order-preserving bijection? Why not? Naturals and integers What about the naturals 0; 1; 2; 3;::: and the integers :::; −2; −1; 0; 1; 2;:::? Can they be placed in a bijection? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 7 / 39 Can they be placed in an order-preserving bijection? Why not? Naturals and integers What about the naturals 0; 1; 2; 3;::: and the integers :::; −2; −1; 0; 1; 2;:::? Can they be placed in a bijection? Naturals 0 1 2 3 4 ::: Integers 0 1 -1 2 -2 ::: D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 7 / 39 Why not? Naturals and integers What about the naturals 0; 1; 2; 3;::: and the integers :::; −2; −1; 0; 1; 2;:::? Can they be placed in a bijection? Naturals 0 1 2 3 4 ::: Integers 0 1 -1 2 -2 ::: Can they be placed in an order-preserving bijection? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 7 / 39 Naturals and integers What about the naturals 0; 1; 2; 3;::: and the integers :::; −2; −1; 0; 1; 2;:::? Can they be placed in a bijection? Naturals 0 1 2 3 4 ::: Integers 0 1 -1 2 -2 ::: Can they be placed in an order-preserving bijection? Why not? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 7 / 39 What is !? ! is a mathematical object that we can explicitly define (we'll do it later) It represents the structure of any set when it's ordered like 0 < 1 < 2 < 3 < ··· An ordered set has order type ! when the set has a first element, a second element, and so on forever x N has order type !, but so does fx : x is primeg, f2 : x 2 Ng, and {−4; −3; −2; −1; 0; 1; 2; 3; :::g D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 8 / 39 Do they have the same order type? Define the order type of the new set to be ! + 1 What about f0; 1; 2; 3;:::; a; b; cg? Order types Let's look at some structured sets and figure out their order types! Consider the sets N = f0; 1; 2; 3;:::g and f0; 1; 2; 3;:::; ag. Do they have the same cardinality? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 9 / 39 Define the order type of the new set to be ! + 1 What about f0; 1; 2; 3;:::; a; b; cg? Order types Let's look at some structured sets and figure out their order types! Consider the sets N = f0; 1; 2; 3;:::g and f0; 1; 2; 3;:::; ag. Do they have the same cardinality? Do they have the same order type? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 9 / 39 Order types Let's look at some structured sets and figure out their order types! Consider the sets N = f0; 1; 2; 3;:::g and f0; 1; 2; 3;:::; ag. Do they have the same cardinality? Do they have the same order type? Define the order type of the new set to be ! + 1 What about f0; 1; 2; 3;:::; a; b; cg? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 9 / 39 Ordinals and hyperjumps There are only two kinds of ordinals: successor ordinals and limit ordinals Successor ordinals come \after" any given ordinal Limit ordinals come after you take a jump into \hyperspace" Also the supremum of an infinitely increasing set of ordinals Figure: The original USS Enterprise (NCC-1701) making the jump into \warp". D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 10 / 39 What do you think its order type is? The answer is ! · 2 = ! + ! Two sequential copies of ! What's the order type of f0; 00; 1; 10; 2; 20; 3; 30;:::g? What's the order type of f0; 1; 2; 3;:::; 00; 10; 20; 30;:::; 000; 100; 200; 300; 400g? Order types Consider the set f0; 1; 2; 3;:::; 00; 10; 20; 30;:::g. Does this have the same cardinality as N? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 11 / 39 The answer is ! · 2 = ! + ! Two sequential copies of ! What's the order type of f0; 00; 1; 10; 2; 20; 3; 30;:::g? What's the order type of f0; 1; 2; 3;:::; 00; 10; 20; 30;:::; 000; 100; 200; 300; 400g? Order types Consider the set f0; 1; 2; 3;:::; 00; 10; 20; 30;:::g. Does this have the same cardinality as N? What do you think its order type is? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 11 / 39 What's the order type of f0; 1; 2; 3;:::; 00; 10; 20; 30;:::; 000; 100; 200; 300; 400g? Order types Consider the set f0; 1; 2; 3;:::; 00; 10; 20; 30;:::g. Does this have the same cardinality as N? What do you think its order type is? The answer is ! · 2 = ! + ! Two sequential copies of ! What's the order type of f0; 00; 1; 10; 2; 20; 3; 30;:::g? D. Salgado, N. Singer (#blairlogicmath) An Introduction to Ordinals November 2017 11 / 39 Order types Consider the set f0; 1; 2; 3;:::; 00; 10; 20; 30;:::g.