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Implications of Cantorian Transfinite Set Theory on Creation

Explore Georg Cantor's set theory for transfinite numbers and the concept of infinite infinities, including Hilbert's Hotel. Learn about rational and irrational numbers, infinite sets, and the impact on mathematical and philosophical thought.

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Implications of Cantorian Transfinite Set Theory on Creation

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  1. Implications of Cantorian Transfinite Set Theory on Creation

  2. A graduate student at Trinity Computed the  But the number of digits Gave him the figits, So he quit math and took up divinity.

  3. Georg Ferdinand Ludwig Philipp CantorBorn: 3 March 1845 in St Petersburg, RussiaDied: 6 Jan 1918 in Halle, Germany Georg Cantor developed the set theory for transfinite numbers.  1 0 2 3 o ? Georg a Protestant, the religion of his father. Georg's mother was a Roman Catholic

  4. Cantorian Infinities David Hilbert described Cantor's work as:“...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” "I see it but I don't believe it.” Georg Cantor on his own theory. “…the infinite is nowhere to be found in reality” David Hilbert. Hilbert http://www-gap.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Cantor.html

  5. Cantorian Infinities… • is not a number – it is a process. • is approached, never achieved. limn

  6. Cantorian Infinities… 0 The set of all counting numbers C={1, 2, 3, 4, …} has cardinality* 0 . Other sets have cardinality* 0 if each element in the set can be placed on a one-to-one correspondence to C. * The number of elements in a set.

  7. The Hottentots (Gamow) • One • Two • Three • Many

  8. Cantorian Infinities… 0 The set of even numbers has cardinality* 0 E={2, 4, 6, 8, …} Why? Because of the correspondence:

  9. Hilbert’s Hotel… 0 0 Rooms – all full. One more person comes. No Problem! Send guest in room 1 to room 2, guest 2 to room 3, etc. This frees room 1 for the new guest.

  10. Hilbert’s Hotel… 0 0 Rooms – all full. 0 more person comes. No Problem! Send guest in room 1 to room 2, guest 2 to room 4, 3 to 6, etc. This frees all the odd rooms for the new guests.

  11. Hilbert’s Hotel… 0 0 Rooms – all full. 0 guests leave. How many rooms are left occupied? • Guests from all rooms leave. 0 - 0 = 0 • Guests from rooms 4 and higher leave. 0 - 0 = 4 • Guests from all the even rooms leave, or, guest from every tenth room leaves. 0 - 0 = 0

  12. Hilbert’s Hotel… http://www.buzzle.com/editorials/9-9-2002-26002.asp

  13. Hilbert’s Hotel… http://www.buzzle.com/editorials/9-9-2002-26002.asp

  14. Cantorian Infinities… 0 Numerator A number is rational number if it can be expressed as the ratio of two integers. The set of rational numbers, R, has cardinality 0.     Denominator        

  15. Cantorian Infinities… 1 The set of all subsets of a set with 0 elements is of cardinality 1. This is a “bigger” infinity. Example: the set of irrational numbers between 0 and 1 (points on a line) is of cardinality 1.

  16. Cantorian Infinities… 1 Proof by counterexample: Suppose a mapping exists: Choose any other digit other than the one circled – say the number after. The number 0.86314… is not in the table. Contradiction!

  17. Cantorian Infinities… 1 The number of points on a line, 1, is the same as the number of points in a square - or in a cube. Consider a unit interval and a unit square. (1,1) For every point, P, in the square, there is a unique corresponding point on the line segment, and visa versa. P y P 0 z 1 (0,0) x The point on the line is z=0.132456754…. Taking every other digit, corresponds to x=0.12574… and y=0.346754…

  18. Cantorian Infinities… 2 n+1, is the set of all subsets of n . Q: What is an example of 2? A: All the squiggles that can be drawn on a plane.

  19. Cantorian Infinities… 2 n+1 is the set of all subsets of n. Q: What is an example of 3? A: Like a fifth spatial dimension, this is beyond comprehension.

  20. Georg Cantor and Pope Leo XIII • There exists no biggest transfinite number. • googol, • a = o , or a • The set of all transfinite numbers does not exist. • “From me, Christian Philosophy will be offered for the first time the true theory of the infinite.” Cantor Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite Joseph W. Dauben Journal of the History of Ideas, Vol. 38, No. 1 (Jan. - Mar., 1977), pp. 85-108

  21. William Lane Craig

  22. William Lane Craig “…since the actual infinite cannot exist and infinite temporal regress of events is an actual infinite, we can be sure that an infinite temporal regress of events cannot exist, that is to say, the temporal regress of events is finite. Therefore, since the temporal regress of events is finite, the universe began to exist.”

  23. William Lane Craig Some absurdities of an infinite past: • “To try to instantiate an actual infinite progressively in the real world would be hopeless, for one could always add one more element.” (0 versus .) • Tristram Shandy Paradox (Russell):If Tristram Shandry rote his autobiography 365 times as slow as he lived life, he would finish in an infinite past. (Careful here! Don’t confuse 0 with .) More…

  24. N years 0 t N days Bummer Tristram will fall further and further behind. Tristram Shandy Paradox • Writing from t = 0 onward:

  25. N years Cool t N days No matter how big N, there is always time for Tristram to finish: even if N = 0. Tristram Shandy Paradox • Writing from the past to end at t = 0: This is Absurd!!

  26. William Lane Craig “… an infinite temporal regress is absurd.” Thus: Time and the universe are finite. The universe must have been created ex nihilo. There are, of course, other scientific/mathematical arguments that reach the same conclusion: The Prime Mover (first cause) & Big Bang Cosmology.

  27. Finis

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