1. Ch. 11: Cantor’s Infinity!

2. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rationals” R = {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size?

3. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rationals” R = {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size? OLD-FASHIONED DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if either (1) they are both finite and have the same number of members, or (2) they are both infinite. According to this definition, all of the sets above have the same size.

4. N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rationals” R = {all real numbers} “the real numbers” Which of these sets is the largest? Do they all have the same size? OLD-FASHIONED DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if either (1) they are both finite and have the same number of members, or (2) they are both infinite. According to this definition, all of the sets above have the same size.

5. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size…

6. How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers?

7. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING!

8. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING! When comparing two infinite sets, Your situation is similar to this child’s. You don’t know how to separately count each set … so you should try matching!

9. What else could “same size” possibly mean? Think about how you decide whether two sets have the same size… How can a child who can’t yet count to seven decide whether there are equal numbers of cars and drivers? BY MATCHING! When comparing two infinite sets, Your situation is similar to this child’s. You don’t know how to separately count each set … so you should try matching! We have the same number of fingers!

10. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. We have the same number of fingers.

11. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers.

12. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers”

13. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” WRONG ANSWERS: “Yes, because they are both infinite” “No, because N contains all of E’s members plus more.”

14. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” YES! n 2n Formula:

15. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! We have the same number of fingers. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” E = {2, 4, 6, 8, 10, 12, …} “the even natural numbers” YES! n 2n Formula: How strange that an infinite set can have the same size as a subset of itself!

16. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! N = {1, 2, 3, 4, 5, 6, …} “the natural numbers” Z = {…, –3, –2, –1, 0, 1, 2, 3, …} “the integers” Q = {all quotients “a/b” of integers with b≠0} “the rationals” R = {all real numbers} “the real numbers” Main Goal: to decide which pairs of these sets have the same size

17. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} “the integers”

18. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} “the integers” WRONG ANSWERS: “Yes, because they are both infinite” “No, because Z goes to infinity in two directions.”

19. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! WRONG ANSWERS: “Yes, because they are both infinite” “No, because Z goes to infinity in two directions.” “No, because my first attempt “n n” is not a valid one-to-one correspondence.” Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} “the integers”

20. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} “the natural numbers” Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} “the integers”

21. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… (even n) n/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

22. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! (even n) n/2 BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

23. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) n/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

24. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) n/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

25. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) n/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

26. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) n/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

27. Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) n/2

28. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. YES! (even n) (odd n) n/2 -(n-1)/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

29. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. The matching looks simpler if we re-order the members of Z… YES! (even n) (odd n) n/2 -(n-1)/2 Q Q: Do these sets have the same size: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z={…,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}

30. N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1, 2, -2, 3, -3, 4, -4, 5,…} MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. IMPORTANT: From now on, to decide whether two sets have the same size, your only job will be to determine whether their members can be matched with a one-to-one correspondence! BIG IDEA: Match the evens in N with the positives in Z… …which leaves the odds in N free to match with the negatives in Z. (even n) (odd n) n/2 -(n-1)/2 The matching looks simpler if we re-order the members of Z…

31. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1, 2, -2, 3, -3, 4, -4, 5,…} (We just proved that Z is countable)

32. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} Z = {0, 1, -1, 2, -2, 3, -3, 4, -4, 5,…} (We just proved that Z is countable) 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … To prove that an infinite set is countable, we must find an infinite listing of its members {1 st, 2 nd, 3 rd, …} which is organized so as to eventually include each member.

33. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} E = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18,…} (We previously proved that E is countable) 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … To prove that an infinite set is countable, we must find an infinite listing of its members {1 st, 2 nd, 3 rd, …} which is organized so as to eventually include each member.

34. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). To prove that an infinite set is countable, we must find an infinite listing of its members {1 st, 2 nd, 3 rd, …} which is organized so as to eventually include each member. Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size?

35. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). To prove that an infinite set is countable, we must find an infinite listing of its members {1 st, 2 nd, 3 rd, …} which is organized so as to eventually include each member. Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational?

36. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … Does this pattern eventually include every fraction?

37. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, … Does this pattern eventually include every fraction? NO, it only includes the positive fractions with numerator 1.

38. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? Idea: insert the other numerators 2/3,2/4, 3/4,2/5, 3/5, 4/5,2/6, 3/6, 4/6, 5/6, FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, …

39. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? Idea: insert the other numerators (but skip the ones that aren’t reduced). 2/3,2/4, 3/4,2/5, 3/5, 4/5,2/6, 3/6, 4/6, 5/6, FIRST ATTEMPT: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, …

40. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, … Does this eventually include all of the fractions?

41. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? Does this eventually include all of the fractions? No, it only includes positive fractions whose numerators are smaller than their denominators. SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, …

42. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? SECOND ATTEMPT: 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, … Idea: Insert the reciprocals. 2/1,3/1,3/2,4/1,4/3,5/1,5/2,5/3,5/4,6/1,6/5,

43. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? THIRD ATTEMPT: 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, 2/5, 5/2, 3/5, 5/3, … Idea: Insert the reciprocals. Does this eventually include all of the fractions?

44. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? THIRD ATTEMPT: 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, 2/5, 5/2, 3/5, 5/3, … Idea: Insert the reciprocals. Does this eventually include all of the fractions? No, but all that remains is to insert zero and intersperse the negatives! -1/1,-1/2,-2/1,-1/3,-3/1,0,and so on…

45. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? FOURTH ATTEMPT: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … Does this eventually include all of the fractions?

46. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is Q countable? (the set of rational numbers) In other words, do the sets Q and N have the same size? Can we list {1 st rational, 2 nd rational, 3 rd rational, …} is a manner that’s organized so as to eventually include each rational? FOURTH ATTEMPT: 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … Does this eventually include all of the fractions? YES! YES! THEOREM: The set of rational numbers, Q, is countable. 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th …

47. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF:

48. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Like a computer spreadsheet, this grid extends indefinitely right and down.

49. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Next, organize the cells of this grid into an infinite list by meandering through it: The list goes: 1/1, 2/1, 2/2, 1/2, 1/3, 2/3, 3/3, 3/2, 3/1, 4/1, 4/2, 4/3, 4/4, 3/4,…

50. THEOREM: The set of rational numbers, Q, is countable. ANOTHER PROOF: First arrange all of the positive fractions into an infinite grid: Next, organize the cells of this grid into an infinite list by meandering through it: The list goes: 1/1, 2/1, 2/2, 1/2, 1/3, 2/3, 3/3, 3/2, 3/1, 4/1, 4/2, 4/3, 4/4, 3/4,… Finally, insert zero and intersperse the negative, as before. That’s it!

51. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers

52. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is R countable? (the set of real numbers) In other words, do the sets R and N have the same size? Can we list {1 st real, 2 nd real, 3 rd real, …} is a manner that’s organized so as to eventually include each real? Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers

53. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is R countable? (the set of real numbers) In other words, do the sets R and N have the same size? Can we list {1 st real, 2 nd real, 3 rd real, …} is a manner that’s organized so as to eventually include each real? Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers FIRST ATTEMPT: Insert √2 and π at the front of my listing of the rational numbers: √2, π, 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … Does this pattern eventually include every real number?

54. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is R countable? (the set of real numbers) In other words, do the sets R and N have the same size? Can we list {1 st real, 2 nd real, 3 rd real, …} is a manner that’s organized so as to eventually include each real? Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers FIRST ATTEMPT: Insert √2 and π at the front of my listing of the rational numbers: √2, π, 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … Does this pattern eventually include every real number? NO!

55. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is R countable? (the set of real numbers) In other words, do the sets R and N have the same size? Can we list {1 st real, 2 nd real, 3 rd real, …} is a manner that’s organized so as to eventually include each real? Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers FIRST ATTEMPT: Insert √2 and π at the front of my listing of the rational numbers: √2, π, 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … Does this pattern eventually include every real number? NO! Since this attempt failed, does that mean that R is NOT countable?

56. MODERN DEFINITION OF “SAME SIZE”: A pair of sets is said to have the same size if their members can be matched with a one-to-one correspondence. DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). DEFINITION: An infinite set is called countable if it has the same size as N (the set of natural numbers). Question: Is R countable? (the set of real numbers) In other words, do the sets R and N have the same size? Can we list {1 st real, 2 nd real, 3 rd real, …} is a manner that’s organized so as to eventually include each real? Summary: The following sets are countable: E = the even natural numbers Z = the integers Q = the rational numbers FIRST ATTEMPT: Insert √2 and π at the front of my listing of the rational numbers: √2, π, 0, 1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 1/3, -1/3, 3/1, -3/1, 2/3, -2/3, 3/2, -3/2, … 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th … Does this pattern eventually include every real number? NO! Since this attempt failed, does that mean that R is NOT countable? NO, a more clever attempt might still succeed!

57. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. (so we’ll call it “uncountable”) Georg Cantor

58. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. (so we’ll call it “uncountable”) Georg Cantor Cantor proved that no listing {1 st real, 2 nd real, 3 rd real,…}, no matter how cleverly organized, could ever succeed in listing all of the real numbers. Every attempted listing is doomed in advance to be incomplete!

59. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor (kind of like how Euclid described a concrete procedure for identifying a prime number that is missing from any given finite list of prime numbers)

60. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: We seek a concrete procedure for looking at this (or any such) infinite listing of real numbers and identifying a real number that is missing from it! ………

61. Georg Cantor WARM UP WITH FINITE EXAMPLES Here is a list of exactly five 5-digit numbers: 18901 69072 12501 78543 30727 Can you find a 5-digit number that’s NOT on this list?

62. Georg Cantor WARM UP WITH FINITE EXAMPLES Here is a list of exactly five 5-digit numbers: 18901 69072 12501 78543 30727 Can you find a 5-digit number that’s NOT on this list? YES, EASILY

63. Georg Cantor WARM UP WITH FINITE EXAMPLES Here is a list of exactly five 5-digit numbers: 18901 69072 12501 78543 30727 Can you find a 5-digit number that’s NOT on this list? What if only the red “diagonal” digits are visible…

64. Georg Cantor WARM UP WITH FINITE EXAMPLES Here is a list of exactly five 5-digit numbers: 1**** *9*** **5** ***4* ****7 Can you find a 5-digit number that’s NOT on this list? Do you still have enough information to find a number that’s not on the list? What if only the red “diagonal” digits are visible… …and the rest are smudged out?

65. Georg Cantor WARM UP WITH FINITE EXAMPLES Here is a list of exactly five 5-digit numbers: 1**** *9*** **5** ***4* ****7 Can you find a 5-digit number that’s NOT on this list? Do you still have enough information to find a number that’s not on the list? What if only the red “diagonal” digits are visible… …and the rest are smudged out? YES! Just choose a number whose digits are: (not 1)(not 9)(not 5)(not 4)(not 7) Like: 27038 (there are lots of choices that work.)

66. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of exactly six 6-digit numbers, with only diagonal digits visible, like this: 4***** *2**** **0*** ***9** ****9* *****7 Can you always find a 6-digit number that’s NOT on this list?

67. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of exactly six 6-digit numbers, with only diagonal digits visible, like this: 4***** *2**** **0*** ***9** ****9* *****7 Can you always find a 6-digit number that’s NOT on this list? YES! Here you would choose a number whose digits are: (not 4)(not 2)(not 0)(not 9)(not 9)(not 7) Like: 276384 (there are lots of choices that work.)

68. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of exactly seven 7-digit numbers, with only diagonal digits visible, like this: 4****** *2***** **0**** ***9*** ****9** *****7* ******0 Can you always find a 7-digit number that’s NOT on this list?

69. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of exactly seven 7-digit numbers, with only diagonal digits visible, like this: 4****** *2***** **0**** ***9*** ****9** *****7* ******0 Can you always find a 7-digit number that’s NOT on this list? YES!

70. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******5*… Can you always find a decimal expression that’s NOT on this list? …

71. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******5*… Can you always find a decimal expression that’s NOT on this list? … YES! Here you would choose a decimal expression whose digits are: 0.(not 4)(not 2)(not 0)(not 9)(not 9)(not 7)(not 5)… Like: 0.2763846… (there are lots of choices that work.)

72. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******5*… Can you always find a decimal expression that’s NOT on this list? … YES! Here you would choose a decimal expression whose digits are: 0.(not 4)(not 2)(not 0)(not 9)(not 9)(not 7)(not 5)… Like: 0.2763846… (there are lots of choices that work.) Careful! these two numbers might be the same! Do you see how?

73. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******59999… Can you always find a decimal expression that’s NOT on this list? … YES! Here you would choose a decimal expression whose digits are: 0.(not 4)(not 2)(not 0)(not 9)(not 9)(not 7)(not 5)… Like: 0.276384600000… (there are lots of choices that work.) Careful! these two numbers might be the same! Do you see how?

74. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******59999… Can you always find a decimal expression that’s NOT on this list? … YES! Here you would choose a decimal expression whose digits are: 0.(not 4)(not 2)(not 0)(not 9)(not 9)(not 7)(not 5)… Like: 0.276384600000… Easy to fix: just make your number NOT end in 000… or 999… Careful! these two numbers might be the same! Do you see how?

75. Georg Cantor WARM UP WITH FINITE EXAMPLES Given any list of infinitely many decimal expressions (each which has infinitely many digits), with only diagonal digits visible, like this: 0.4*******… 0.*2******… 0.**0*****… 0.***9****… 0.****9***… 0.*****7**… 0.******5*… Can you always find a decimal expression that’s NOT on this list? … YES! Here you would choose a decimal expression whose digits are: 0.(not 4)(not 2)(not 0)(not 9)(not 9)(not 7)(not 5)… Like: 0.2763846… Let’s get back to Cantor’s proof…

76. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: ………

77. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: ………

78. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: A real number that’s missing from Aunt Clair’s list is built like this: M = 0.(not 1)(not 3)(not 4)(not 0)(not 7)(not 7) …. Like: M=0.258163… (there are lots of choices that work.) ………

79. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: A real number that’s missing from Aunt Clair’s list is built like this: M = 0.(not 1)(not 3)(not 4)(not 0)(not 7)(not 7) …. Like: M=0.258163… (there are lots of choices that work.) How do you know that M is different from Aunt Clair’s first number? Her second? Her third? ………

80. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. STRUCTURE OF PROOF: Cantor described a concrete procedure for identifying a real number that is missing from any given listing of real numbers. Georg Cantor Imagine Aunt Clair’s listing begins like this: A real number that’s missing from Aunt Clair’s list is built like this: M = 0.(not 1)(not 3)(not 4)(not 0)(not 7)(not 7) …. Like: M=0.258163… (there are lots of choices that work.) To be safe, always choose digits other than 0 and 9, or at least make sure M doesn’t end with 000… or 999… so that M is not the same as any other decimal expression. To be safe, always choose digits other than 0 and 9, or at least make sure M doesn’t end with 000… or 999… so that M is not the same as any other decimal expression. ………

81. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. PROOF: Given any infinite listing of real numbers: 1 st, 2 nd, 3 rd, 4 th, … We can construct a number M = 0.d 1 d 2 d 3 d 4 d 5 … that’s missing from this list. How? We simply choose the n th digit (d n ) of M to be different from the n th digit (after the decimal point) of the n th real number on the list (also choose d n ≠ 0,9). Thus, no listing of real numbers could every be complete! Georg Cantor

82. CANTOR’S THEOREM: The set of real numbers, R, is NOT countable. PROOF: Given any infinite listing of real numbers: 1 st, 2 nd, 3 rd, 4 th, … We can construct a number M = 0.d 1 d 2 d 3 d 4 d 5 … that’s missing from this list. How? We simply choose the n th digit (d n ) of M to be different from the n th digit (after the decimal point) of the n th real number on the list (also choose d n ≠ 0,9). Thus, no listing of real numbers could every be complete! Georg Cantor I hope you like my proof.