20 Mar 2019•0 j'aime•417 vues

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Overview of Cantor's infinity theorems, notion of cardinality, the continuum hypothesis and some basic proofs.

Oren Ish-AmSuivre

- 1. To Infinity and Beyond! Cantor’s Infinity theorems Oren Ish-Am
- 2. Georg Cantor (1845-1918) His theories were so counter-intuitive that met with much resistance from contemporary mathematicians (Poincaré, Kronecker) They were referred to as “utter nonsense” “laughable” and “a challenge to the uniqueness of God”. Invented Set Theory – fundamental in mathematics. Used in many areas of mathematics including topology, algebra and the foundations of mathematics. Before Cantor – only finite sets in math - infinity was a philosophical issue Cantor, a devout Lutheran, believed the theory had been communicated to him by God
- 3. Set Theory – basic terminology Set – collection of unique elements – order is not important A = 3, , B = 8, Member: 𝒂 ∈ 𝑨 means 𝑎 is a member of the set 𝐴 ∈ A 8 ∈ B Subset: 𝑺 ⊂ 𝑨 means 𝑆 contains some of the elements of the set 𝐴 S = 3, S ⊂ A Union: 𝑪 = 𝑨 ∪ 𝑩 means 𝐶 contains all the elements 𝐴 and all elements of 𝐵 C = 3, , 8,,
- 4. Set Theory – basic terminology Set Size – |𝑨| is the number of elements in 𝐴 Empty Set: {} or 𝝓 is the empty set – set with no items. 𝜙 = 0 |A| = 3 |B| = 2 Natural Numbers: ℕ = {1,2,3…} Rational Numbers: ℚ = 1 3 , 101 23 , 6 6 , 10 5 , … Real Numbers : ℝ = ℚ ∪ all numbers that are not rationals like π,e, 2 …
- 5. Functions – basic terminology על חח"ע חח"עועל
- 6. Counting (it’s as easy as 1,2,3) How can we tell the size of a set of items? Just count them: 21 3 4 A B For finite sets, if A is a proper subset of B (A ⊂ B) Then the size of A is smaller then the size of B ( A < B )
- 7. Counting (it’s not that easy…) Surely there are less squares than natural numbers (1,2,3,…) – right? What can be say about an infinite subset of an infinite set? Let’s try another counting method - comparison 1 2 3 4 5 6 7 8 9 … … Can we count infinite sets? For example - what is the “size” of the set of all squares?
- 8. Assume we have a classroom full of chairs and students We know all chairs are full and - there are students standing. Without having to count, we know there are more students then chairs! Counting (it’s not that easy…) Similarly, if there is exactly one student on each chair – the set of students is the size of the set of chairs. This type of mapping is called bijection ( חדחדועל ערכי )
- 9. Cardinality )עוצמה( Definition: Two sets have the same cardinality iff there exists a bijection between them We can now measure the size of infinite sets! Let’s try an example: The set of natural numbers and the set of even numbers 2 4 6 8 10 12 14 16 …f(x) 1 2 3 4 5 6 7 8 …x A simple function exists: 𝑓 𝑥 = 2𝑥, a bijection. Therefore the two sets have the same cardinality.
- 10. Cardinality )עוצמה( How about natural number and integers? 1 -1 2 -2 3 -3 4 …f(x) 0 1 2 3 4 5 6 7 …x 0 𝑓 𝑥 = 𝑥/2 𝑥 ∈ odd − 𝑥 2 𝑥 ∈ even Cantor was sick of Greek and Latin symbols Decided to note this cardinality as ℵ 𝟎 This is the cardinality of the natural numbers, called countably infinite
- 11. Cardinality of ℚ The rationales are dense; there is a fraction between every two fractions. There are infinite fractions between 0 and 1 alone! In fact, there are infinite fractions between any two fractions! Could we find a way to match a natural number to each fraction? Ok, we get it, so ℵ0 + 𝑐𝑜𝑛𝑠𝑡 = ℵ0 and ℵ0 ⋅ 𝑐𝑜𝑛𝑠𝑡 = ℵ0 What about the rationals - ℚ?
- 12. Cardinality of ℚ Naturals 1 2 3 4 5 6 7 8 9 10 numerator 1 1 2 3 2 1 1 2 3 4 denominator 1 2 1 1 2 3 4 3 2 1 We map a natural number to a numerator and denominator (fraction) Eventually, we will reach all possible fractions – Bijection! Hurray!
- 13. Countably Infinite - ℵ 𝟎 Challenge: find the bijection from natural numbers to fractions. This is the cardinality of All finite strings All equations All functions you can write down a description of All computer programs. Seems like one can find a clever bijection from the natural numbers to any infinite set….right? Are we done here? Well… not so fast…
- 14. Cardinality of Reals - ℝ List might start like this: 1. .1415926535... 2. .5820974944... 3. .3333333333... 4. .7182818284... 5. .4142135623... 6. .5000000000... 7. .8214808651... Cantor had a hunch that the Reals were not countable. Proof by contradiction – let’s assume they are and make a list: This method is called Proof By Diagonalization
- 15. Cardinality of Reals - ℝ New Number = 0.0721097… isn’t in the list! Contradiction! The reals are not countable – more than one type of infinity!! “Complete List” 1. .1415926535... 2. .5820974944... 3. .3333333333... 4. .7182818284... 5. .4142135623... 6. .5000000000... 7. .8214808651... etc. Make a new number: 1. .0415926535... 2. .5720974944... 3. .3323333333... 4. .7181818284... 5. .4142035623... 6. .5000090000... 7. .8214807651... etc.
- 16. Cardinality of Reals Easy! Just de-interleave the digits. For example 0.1809137360 … → 0.1017664 … , 0.893301 … Therefore - ℵ is the cardinality of any continuum in ℝ 𝑛 Cantor: “I see this but I do not believe!” 0 1 Cantor called this cardinality ℵ (or ℵ1 as we will soon see). ℵ is the cardinality of the continuum –all the points on the line: 0,1 , (−∞, ∞) both have cardinality ℵ. Can you find the bijection? What about the Unit Square? Can we map 0,1 → 0,1 × [0,1]?
- 17. Cardinality of Reals This leads to some interesting “paradoxes” like the famous Banach-Tarsky: Given a 3D solid ball, you can break it into a finite number of disjoint subsets, which can be put back together in a different way to yield two identical copies of the original ball. The reconstruction can work with as few as five “pieces”. Not pieces in the normal sense – more like sparse infinite sets of points
- 18. Cardinality of Reals ספור אין או סוף אין? Implications of ℵ0 < ℵ: Most real numbers do not have a name. They will not be the result of any formula or computer program. Can we say they actually “exist” if they can never be witnessed?? This has actual implications in mathematics! We need to add the Axiom of Choice
- 19. More Power! So are we done now? Countable infinity and continuum? Power Set: For a set 𝑆 the power set 𝑃(𝑆) is the set of all subsets of 𝑆 including the empty set (𝜙 or {}) and 𝑆 itself. Example: 𝑆 = 1,2,3 𝑃 𝑆 = 𝜙, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , 1,2,3 Challenge: prove that for any finite set 𝑆, 𝑃 𝑆 = 2 𝑆 (hint – think binary) What about infinite sets? Here Cantor came up with a simple and ingenious proof that shows for any infinite set 𝐴: 𝑃 𝐴 > |𝐴|
- 20. Cantor’s Theorem Theorem: Let 𝑓 be a map from set 𝐴 to 𝑃(𝐴) then 𝑓: 𝐴 → 𝑃(𝐴) is not surjective ()על על חח"ע חח"עועל
- 21. Cantor’s Theorem (proof from “the book”) Proof: a. Consider the set 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝑥 ∉ 𝑓 𝑥 }. Set of all items from 𝐴 which are not an element of the subset they are mapped to by 𝑓 a. Note that 𝐵 is a subset of 𝐴 the therefore B ∈ 𝑃(𝐴) b. Therefore there exists an element 𝑦 ∈ 𝐴 such that 𝑓 𝑦 = 𝐵 c. Two options: 𝑦 ∈ 𝐵 in this case by definition of 𝐵, 𝑦 ∉ 𝑓 𝑦 = 𝐵: contradiction! 𝑦 ∉ 𝐵 in this case 𝑦 ∉ 𝑓(𝑦) which means 𝑦 ∈ 𝐵: contradiction! So such a function 𝑓 cannot exist and there can not be a bijection: 𝑃 𝐴 > 𝐴 1 2 3 4 5 . . . {1,2} {3,17,5} {8} {17,4} {1,2,5} . . . y . B={2,3,…} . A P(A)
- 22. Cardinality of Reals – connection to subsets of ℕ Challenge: Prove that 2 ℕ = |ℝ| (hint – think binary. Again) Let’s look at [0,1]: The binary representation of all numbers in [0,1] are all the subsets of ℕ: 0.11000101… is the set {1,2,6,8,…}
- 23. Other cardinalities Why not go further? Define ℶ = 2ℵ the cardinality of the power set of ℝ. This is the cardinality of all functions 𝑓: ℝ → ℝ We can continue forever increasing the cardinality: ℵ0, ℵ1 = 2ℵ0, ℵ2 = 2ℵ1, … , ℵ 𝑛+1 = 2ℵ 𝑛, … There is no largest cardinality! Wait – what about the increments? Is there a cardinality between the integers and the continuum? This was Cantor’s famous Continuum Hypothesis
- 24. Continuum Hypothesis CH is independent of the axioms of set theory. That is, either CH or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent. Continuum Hypothesis (CH): There is no set whose cardinality is strictly between that of the integers and the real numbers. Stated by Cantor in 1878 is became one of the most famous unprovable hypotheses and drove Cantor crazy. Only in 1963 it was proved by Paul Cohen that….
- 25. Cardinality of Cardinalities Just one last question to ask – exactly how many different cardinalities are there? By iterating the power set we received the sets ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , … With the cardinalities ℵ0, ℵ1, ℵ2, ℵ3, … There are at least ℵ0 different cardinalities. Are we good Vincent?
- 26. 𝑩 = 1,17, 4,9 , 102,4,22 , 2 , 3,6 , 96,3,21 , … Cardinality of Cardinalities Ok, hold on to your hats… Let’s look at the set 𝐴 that contains all the power sets we just saw: 𝐴 = ℕ , 𝑃 ℕ , 𝑃 𝑃 ℕ , P P 𝑃 ℕ , … And now let’s look at the set 𝑩 = ⋃𝑨 which is the union of all the elements of all the sets in 𝐴. Here is what 𝐵 may look like Elements from ℕ Elements from 𝐏 ℕ Element from 𝐏 𝐏 ℕ 𝐵’s cardinality is as large as the largest set in 𝐴
- 27. Cardinality of Cardinalities The set 𝐵 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 𝐵 = 𝑃(⋃𝐴) - that cardinality is not in the ℵ 𝑛 list In fact – for any set 𝑆 of sets, the set 𝑆 𝑏𝑖𝑔𝑔𝑒𝑟 = 𝑃 ⋃𝑆 will have a cardinality not in 𝑆! No set, no matter how large cannot hold all cardinalities. So the collection of possible cardinalities is… is not even a set because it does not have a cardinality. Mathematicians call this a proper class, something like the set of all sets (which is not permitted in set theory to avoid things like Russel’s paradox).
- 28. Thank You!

- Give slide with basic definitions – Set, member, Union, Subset, Intersection. And other set notation
- Add animation to cross off the main diagonal for repetition