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