Cardinality continuum is aleph_1

Discussion in 'Recreational Math' started by William Elliot, Apr 30, 2006.

  1. By Cohen's work CH is independent of ZFC.
    Thus we're free to chose CH or not.

    Thus by Occam's razor CH, the
    cardinality of the continuum is aleph_1

    Is GCH also independent of ZFC ?
    Then yah, GCH by Occam's razor.
    Thus we can finish up some theorems, prove others directly and
    go on to other notions lacking that immense bother.

    Why clutter your abacus with myraids of uselessly colored
    beads on an hallucinatorial rod between the 1's and the 10's?

    --
    Occam also allows the convenient disappearance of
    inaccessible cardinals and gods.

    IIRC

    The denial of inaccessible cardinals is consistence with ZFC. There cannot
    be a proof that the existence of inaccessible cardinals is consistence
    with set theory. Thus I and Occam, on the firm ground of ZFC consistence,
    affirm down with inaccessibles!
     
    William Elliot, Apr 30, 2006
    #1
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  2. [.snip sophistry.]
    Yes. But C is not independent of ZF+GCH.


    --
    ======================================================================
    "It's not denial. I'm just very selective about
    what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
    ======================================================================

    Arturo Magidin
     
    Arturo Magidin, Apr 30, 2006
    #2
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  3. No problem, I'm a prochoice mathematician.
     
    William Elliot, Apr 30, 2006
    #3
  4. Only if you exclude aleph_0, which seems prejudicial. Since aleph_0 is
    strongly inaccessible unless you deliberately exclude it, and since the
    class of all sets behaves like a strongly inaccessible cardinal,
    excluding them seems dubious. This is particularly true since we want
    there to be strongly inaccessible cardinals, as we want to do things
    like work inside of Grothendieck universes for purposes such as doing
    algebraic geometry, providing models of set theory, or making the
    surreal numbers into a field.

    Moreover, unless there is something to prevent large cardinals from
    existing, they ought to exist. What stops them? Why should there not be
    cardinals which behave like proper classes for many purposes?
     
    Gene Ward Smith, Apr 30, 2006
    #4
  5. Oh yes, just like we want fairy tales, are why we crave elves, gnomes,
    batmen, etc. Oh come on, congressional and UN investigation of White
    House war crimes, are much more preferable tales to be told the world.

    BTW, have you noticed how big those gaps in the surreal numbers are?
    Because, as explained above, they come with no warranty and just like gods
    and goddesses, are too big for their britches. Just imagine what a world
    this would be without big inaccessible gods to argue and fight about.
     
    William Elliot, Apr 30, 2006
    #5
  6. I have always seen Occam's razor as a helpful guide rather than a hard
    and fast principle.
     
    Stephen Montgomery-Smith, Apr 30, 2006
    #6
  7. I think applying it to what is postulated by mathematics is a
    misapplication of the idea. Mathematical structures are possibilities,
    and you can't wave a magical ontological wand and make possibilities
    impossible.
     
    Gene Ward Smith, Apr 30, 2006
    #7
  8. Enuf is enough, it's time for a shave, for an end to those incessantly
    itchy long hairs. Speak ye of faerie tails? I'll tell you a tale or two.
    Once upon a when within a neighborhood of then, there was a mathematician
    who had grown long hair ideas longer than the long line, Conway IIRC,
    who's very long long line was thicker and denser than any ever before.
    Yet upon examination by Dr. Occam IIRC, was found tiny tiny gaps
    everywhere which, upon the stresses of flights into fancy, would proof too
    fragile and brittle for safe trips. Instead, Rev. Occam IIRC, recommended
    Conway's long long line with tiny fine gaps everywhere be used in the
    kitchen as a strainer or sieve, to filter out infinitesimals for
    non-standard analysis preparations. Anyway, anyway, just a nanosec...
    Occam! Occam! Will you sweep up those hairs you shaved off?
     
    William Elliot, May 1, 2006
    #8
  9. William Elliot

    Rupert Guest

    Why does CH follow from Occam's razor rather than ~CH?
     
    Rupert, May 1, 2006
    #9
  10. William Elliot

    dkfjdklj Guest

    But why do you suppose "they" are possible? Is it some kind of magical
    ontological possibility wand? Merely proving an assertion is
    consistent with other axioms does not mean that it is possible (in the
    apparently absolute sense you are using it).
     
    dkfjdklj, May 1, 2006
    #10
  11. Because CH mops up a mess of cardinals between aleph_0 and 2^aleph_0, rubs
    them out in fact so they'll go away and not bother us. Bother us, for
    example, requiring longer proofs to show cardinality of a set is >= c when
    it's quit apparent the set is uncountable.

    GCH is even nicer. aleph_xi = beth_xi settling the nether lands with nice
    orderly guys instead of scads of unrulely sorts arguing who's bigger or
    smaller and what's there in between that nobody can show yet believe there
    are, can or will be some.

    Same with CH, quit talking about life on Mars and show us
    your unconstructive wonder between aleph_0 and 2^aleph_0.
     
    William Elliot, May 1, 2006
    #11
  12. William Elliot

    Rupert Guest

    ~CH requires postulating a cardinal between aleph_0 and 2^aleph_0. But
    CH requires postulating a bijection between 2^aleph_0 and aleph_1. It's
    not clear to me why one violates Occam's razor more than the other.
     
    Rupert, May 1, 2006
    #12
  13. William Elliot

    Dave Seaman Guest

    Because CH mops up a mess of cardinals between aleph_0 and 2^aleph_0, rubs
    them out in fact so they'll go away and not bother us. Bother us, for
    example, requiring longer proofs to show cardinality of a set is >= c when
    it's quit apparent the set is uncountable.[/QUOTE]
    According to Occam's razor, it is supposed to be assumptions, not
    consequences, that we are to avoid multiplying without necessity.
     
    Dave Seaman, May 1, 2006
    #13
  14. They're possible, all you need is axiom and lo and behold, there they are.
    They're there if you want them and for those who do, they're fun. Like
    pack rats for whom lots of stuff is fun. Others with a less lavish sense
    of intellectual fun, prefer thoughts Amish style.

    So I put it to ya'll, aleph_omega0 practically has no practical use except
    as jump off point to greater fanstasies aleph_omega1, aleph_epsilon_0
    for ever and ever, aleph_omega_epsilon_omega_epsilon_omega_epsilon_...
     
    William Elliot, May 1, 2006
    #14
  15. ~CH puts more members on the board of directors.
    CH revokes the applications of additional board members.

    Already most board members have nothing to do but
    to look big and aleph_(epsilon_omega) impressive.
     
    William Elliot, May 1, 2006
    #15
  16. CH is an example where the statistical direct proportion of axioms to
    results is locally an inverse proportion. It's important not to multiply
    and fill the mind with cluttering consequences as for example, we could cut
    down on number of assumptions to just one simple axiom that proves almost
    everything we want and even perhaps more, more than we want.
     
    William Elliot, May 1, 2006
    #16
  17. William Elliot

    Nathan Guest

    CH has lots of interesting mathematical consequences. But so do many
    axioms that are inconsistent with CH. All of these are still
    mathematics. Math is a big discipline; there's room for all sort of
    interests.
     
    Nathan, May 1, 2006
    #17
  18. It's supposed to be entities according to Occam, but that gets extended
    to assumptions of a particular kind, which have nothing to do with
    mathematics. Trying to apply Occam's razor to mathematics is a category
    mistake, in that mathematics is not an aspect of the natural world we
    are attempting to explain via some model.
     
    Gene Ward Smith, May 2, 2006
    #18
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