Infinity and Cardinality
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Infinity and Cardinality

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A test presentation, lacking rigor

A test presentation, lacking rigor

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Infinity and Cardinality Presentation Transcript

  • 1. Infinity and its Cardinality Robin K. Hill 6 February, 2012 University of Wyoming
  • 2. The Usual View of Infinity
    • Infinity has no bound; goes on forever: 1, 2, 3, 4, ..., 731, ..., 9394872398, ...
    •  
    • We have a symbol that we use informally:
    • We have a name for the set of natural numbers:  N
    • Is there a quantity that we can associate with it?
    •   (This is math---we can name and define anything we want, but we want some solid theoretical grounding.)
    •   We NEED named quantities as soon as we realize:
    • There are infinities bigger than the one we know!
    •  
    • So we call the quantity for the one we know, above,
    •  
  • 3. Relative Cardinalities
    • How do we know one set is bigger than another?
    • -- if we can take away as many things as there are in the second set and there's still something left in the first
    •  
    • Consider other "simple" infinite sets:
    •  
    • The even numbers E: 0, 2, 4, 6, 8, 10, 12, ...
    •  
    • The rational numbers Q:
    •  
    We can take E or Q out of N, and still have N just as big.
  • 4. Denumerable Sets
    • Anything we can line up with N is denumerable . And a set is infinite if it has a proper subset of same cardinality.
    •  
    • A musical reference: ``Amazing Grace''
    •  
    • D= {days we have to sing}
    • D - 10,000  365 = D!
    •  
    • Therefore D is infinite.
  • 5. The Real Numbers R
    • Can't be "lined up" with N.
    •  
    • 1) We try and fail, and
    • 2) We can prove it's impossible.
    • Cardinality of the Reals
    • In conclusion, let's ponder the Continuum Hypothesis:
    •  
    • I.e., there is no other infinite value between.
    What do you think? Is this hypothesis true? Let’s discuss.