Cantors Diagonal Proof and Uncountable Numbers To Infinity
Cantor’s Diagonal Proof and Uncountable Numbers: To Infinity and Beyond! Donald Byrd rev. 28 November 2012
Hilbert’s Hotel and Infinite Sets • Explanation of infinite sets by David Hilbert • Hotel with finite rooms, all occupied – Can’t accommodate a new guest • Hotel with infinite rooms, all occupied – Can accommodate a new guest – Move each guest from room n to n+1 • Hotel with infinite rooms, all occupied – Can accommodate infinite no. of new guests! – How? 28 Nov. 2012 2
One-to-One Correspondence • How can you compare size of collections of things if there are too many to count? – …for example, infinite collections • Georg Cantor (1891): – Put items in each set (collection) in a list – Try one-to-one correspondence • There are infinitely many integers, but… • Cantor proved there are more real nos. ! 28 Nov. 2012 3
One-to-One Correspondence between Infinite Sets (1) N -- 0 1 2 3 4 5 6 8 9 etc. Even Numbers ------0 2 4 6 8 10 12 14 16 etc. 28 Nov. 2012 Integers ------ 0 -1 1 -2 2 -3 3 -4 4 etc. 4
One-to-One Correspondence between Infinite Sets (2) N -- 0 1 2 3 4 5 6 8 9 etc. Even Numbers ------0 2 4 6 8 10 12 14 16 etc. 28 Nov. 2012 Integers ----- 0 -1 1 -2 2 -3 3 -4 4 etc. Squares ----- 0 1 4 9 16 25 36 49 64 etc. Positive Rationals ------ 1 1/2 2/1 1/3 3/1 1/4 2/3 3/2 4/1 etc. 5
“Complete List of Real Numbers”? 1. 2. 3. 4. 5. . 141592653589793238462643383279502884197169399375 10. . 582097494459230781640628620899862803482534211706 79. . 333333333333333333333333 33. . 71828459045235360287471352662497757247093699 95. . 414213562373095048801688724209698078569671875376 94. . . 6. . 50000000000000000000000006 28 Nov. 2012 00. . .
“Complete List of Real Numbers”? • Just real numbers between 0 and 1 is enough. • List might start like this: 1. . 1415926535. . . 2. . 5820974944. . . 333333. . . 4. . 718284. . . 5. . 4142135623. . . 6. . 500000. . . 7. . 8214808651. . . 28 Nov. 2012 7
No “Complete List of Real Numbers”! “Complete List” 1. . 1415926535. . . 2. . 5820974944. . . 333333. . . 4. . 718284. . . 5. . 4142135623. . . 6. . 500000. . . • . 8214808651. . . etc. Make a new number: 1. . 0415926535. . . 2. . 5720974944. . . 3323333333. . . 4. . 7181818284. . . 5. . 4142035623. . . 6. . 5000090000. . . 7. . 8214807651. . . etc. • New Number = 0. 0721097… isn’t in the list! • How do we know? 28 Nov. 2012 8
Different Sizes of Infinity • Proof by contradiction • Cantor’s conclusion: there are more reals between 0 and 1 than there are integers! • Integers are countable – …also even nos. , rational nos. , etc. • Reals (and larger infinities) are uncountable – No. of integers = aleph-0; of reals, aleph-1 • Amazed mathematicians • Led to set theory, new branch of math 28 Nov. 2012 9
Conclusion: Let’s Sing! (1) • Some versions of A Hundred Bottles of Beer for really long car trips Cf. http: //www. informatics. indiana. edu/donbyrd/Teach/Math/Infinite. Bottles. Of. Beer_Full. Ver. pdf • Basic transfinite version 1 Infinite bottles of beer on the wall, infinite bottles of beer; If one of those bottles should happen to fall, infinite bottles of beer on the wall. (etc. ) 28 Nov. 2012 10
Conclusion: Let’s Sing! (2) • Basic transfinite version 2 (generalization of ver. 1) Infinite bottles of beer on the wall, infinite bottles of beer; If finite bottles should happen to fall, infinite bottles of beer on the wall. (etc. ) 28 Nov. 2012 11
Conclusion: Let’s Sing! (3) • Larger-infinity version Uncountable bottles of beer on the wall, uncountable bottles of beer; If countable bottles should happen to fall, uncountable bottles of beer on the wall. (etc. ) 28 Nov. 2012 12
Conclusion: Let’s Sing! (4) • General transfinite version Aleph-n bottles of beer on the wall, aleph-n bottles of beer; If, where m < n, aleph-m bottles should happen to fall, aleph-n bottles of beer on the wall. (etc. ) • Transfinite and indeterminate version (by Richard Byrd) Infinite bottles of beer on the wall, infinite bottles of beer; If infinite bottles should happen to fall, indeterminate bottles of beer on the wall. (The End) 28 Nov. 2012 13
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