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Cantor ’ s Diagonal Proof and Uncountable Numbers: To Infinity and Beyond!

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

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Cantor ’ s Diagonal Proof and Uncountable Numbers: To Infinity and Beyond!

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  1. Cantor’sDiagonalProof and Uncountable Numbers:To Infinity and Beyond! Donald Byrd rev. 28 November 2012

  2. 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?

  3. 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.!

  4. One-to-One Correspondence between Infinite Sets (1) Even N Numbers Integers --- ------------ ----------- 0 0 0 1 2 -1 2 4 1 3 6 -2 4 8 2 5 10 -3 6 12 3 8 14 -4 9 16 4 etc. etc. etc.

  5. One-to-One Correspondence between Infinite Sets (2) Even Positive N Numbers Integers Squares Rationals --- ------------ ----------- ---------- ------------ 0 0 0 0 1 1 2 -1 1 1/2 2 4 1 4 2/1 3 6 -2 9 1/3 4 8 2 16 3/1 5 10 -3 25 1/4 6 12 3 36 2/3 8 14 -4 49 3/2 9 16 4 64 4/1 etc. etc. etc. etc. etc.

  6. “Complete List of Real Numbers”? 1. .14159265358979323846264338327950288419716939937510... 2. .58209749445923078164062862089986280348253421170679... 3. .33333333333333333333333333333333333333333333333333... 4. .71828182845904523536028747135266249775724709369995... 5. .41421356237309504880168872420969807856967187537694... 6. .50000000000000000000000000000000000000000000000000... 7. .82148086513282306647093844609550582231725359408128... 8. .48111745028410270193852110555964462294895493038196... 9. .44288109756659334461284756482337867831652712019091... 10. .45648566923460348610454326648213393607260249141273... 11. .72458700660631558817488152092096282925409171536436... • .78925903600113305305488204665213841469519415116094... etc., etc.

  7. “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...

  8. No “Complete List of Real Numbers”! • Make a new number: • 1. .0415926535... • 2. .5720974944... • 3. .3323333333... • 4. .7181818284... • 5. .4142035623... • 6. .5000090000... • .8214807651... • etc. “Complete List” 1. .1415926535... 2. .5820974944... 3. .3333333333... 4. .7182818284... 5. .4142135623... 6. .5000000000... • .8214808651... etc. • New Number = 0.0721097… isn’t in the list! • How do we know?

  9. 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

  10. 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.)

  11. 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.)

  12. 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.)

  13. 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)

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