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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. .141592653589793238462643383279502884197169399375 10... 2. .582097494459230781640628620899862803482534211706 79... 3. .333333333333333333333333333333333333333333333333 33... 4. .718281828459045235360287471352662497757247093699 95... 5. .414213562373095048801688724209698078569671875376 94... 6. .5000000000000000000000000000000000000000000000006 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... 3. .3333333333... 4. .7182818284... 5. .4142135623... 6. .5000000000... 7. .8214808651... 28 Nov. 2012 7 No “Complete List of Real Numbers”! “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. • 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/InfiniteBottlesOfBeer_FullVer.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. 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. 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. 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. 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