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I Information and Cardinality

  1. Jan 7, 2017 #1
    To select an element from countably infinite set (list set of integers) you need to provide finite amount of information. To specify an element in continuum in general case you have to provide infinite amount of information: any real number is specified as countable-infinite number of digits. So here is a pattern:

    [itex]\aleph_0[/itex] - finite information
    [itex]\aleph_{continuum}[/itex] - countably infinite number of digits,

    So the amount of information is 1 degree less than the cardinality of set.

    Now, lets deny continuum hypotesis and lets assume continuum = [itex]\aleph_2[/itex], so there is 1 cardinality between countable and continuum (I've heard that Goedel believed in it), and that cardinality is obviously [itex]\omega_1[/itex]

    What information is required to completely specify an element in [itex]\omega_1[/itex] ?
     
  2. jcsd
  3. Jan 7, 2017 #2

    fresh_42

    Staff: Mentor

    Shouldn't it be ##\aleph_1## instead of ##\omega_1## by its definition? But anyway. Whether you put it in a pseudo information theory language, or leave it where it belongs to, namely ZFC, makes no difference. Since it is undecidable in ZFC, it stays as such in all other languages based upon ZFC.
     
  4. Jan 7, 2017 #3

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Rational numbers are real. Not all real numbers require an infinite number of digits.
     
  5. Jan 8, 2017 #4
    yes, but in general case you have to provide an infinite number of digits
    most of reals are random numbers.
     
  6. Jan 8, 2017 #5
    Thank you for the correction.
    But then question "what amount of information is required to specify a countable ordinal?" (which is an element of ##\omega_1## ) must be also undecidable, because answering it you also provide one or another "answer" to CH?
     
  7. Jan 8, 2017 #6

    fresh_42

    Staff: Mentor

    Yes, only that it can't be another answer. Assuming CH were decidable in some axiomatic system, then it would have to be different from ZFC. However, information theory isn't.
     
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