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Set of real numbers

  1. Jul 22, 2011 #1
    If the set of natural numbers is [itex] \aleph [/itex]
    and when we write a real number we have 10 choices for each position 0-9
    so can we say that there are [itex] 10^{\aleph} [/itex] real numbers ?
     
  2. jcsd
  3. Jul 22, 2011 #2
    Hi cragar! :smile:

    You probably mean [itex]\aleph_0[/itex], right?

    Yes, that is correct, the real numbers have cardinality [itex]10^{\aleph_0}[/itex]. Also note that

    [tex]2^{\aleph_0}=10^{\aleph_0}=\aleph_0^{\aleph_0}[/tex]

    But, I also must give you a warning. Saying that you have "10 choices for each position 0-9" is not exactly true, there are technical details. For example 1.00000000... and 0.9999999... are the same numbers, so some choice yield the same number. Also, choice like ...9999999.999999.... are not allowed: we must only have a finite number of 1-9 in front of the dot.

    These technical matters can be fixed however.
     
  4. Jul 22, 2011 #3
    how is this true [tex]2^{\aleph_0}=10^{\aleph_0}=\aleph_0^{\aleph_0}[/tex]
     
  5. Jul 22, 2011 #4
    Well, to give an intuitive explanation. You showed that the real numbers have cardinality [itex]10^{\aleph_0}[/itex], but you used decimal representation here. We can also use binary representation. In that way, you have numbers of the form 111.0101101 for example. So you have to choose 0 or 1 a countable number of times. So by the same reasoning, the real numbers have cardinality [itex]2^{\aleph_0}[/itex].
    When using hexadecimal, you'll obtain [itex]16^{\aleph_0}[/itex] as cardinality of the reals. So

    [tex]2^{\aleph_0}=3^{\aleph_0}=...=10^{\aleph_0}=...[/tex]
     
  6. Jul 22, 2011 #5
    I seen the proof where the set has 2^n subsets . like for example if i have a sub set {3,2} this would mean I would put a one in the 3rd position and a 2 in the second position and zeros in the rest. but i thought this was a proof where we couldn't repeat numbers. We didn't start with a multiset. So are you saying the reals are all of the subsets of the naturals.
     
  7. Jul 22, 2011 #6
    Well, the reals aren't the set of all subsets of the naturals, but they certainly have as much elements!!
     
  8. Jul 22, 2011 #7
    when you use binary for your list count, what do you mean by your 0 or 1 .
     
  9. Jul 22, 2011 #8
    Just use the binary system: for example 10=2, 11=3, 100=4, 0.1=1/2, etc.
     
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