Re: Existence of mathematical entities (Re: Successor Axiom: on what grounds TF?)

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 02/13/05

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    Date: 12 Feb 2005 19:55:33 -0800

    robert j. kolker wrote:
    > Ross A. Finlayson wrote:
    > >
    > > Hi Bob,
    > >
    > > I've heard of that before, the antidiagonal argument. Because I
    think
    > > that infinite sets are equivalent, to some extent due to the
    > > well-ordering and transfer principles,
    >
    > There does not exist a one to one from the integers onto the reals.
    Period.
    >
    > Bob Kolker

    So I have heard.

    Ah ha ha ha ha, that's pretty funny.

    "Everything exists"? When you say everything, do you mean, for
    example, the collection of every set, in a set-theoretic universe where
    there exists only sets, every thing, and such notions as category
    theory and Chu spaces are representable therein?

    Bob, you must agree that there is a well-ordering of the reals in ZFC
    because everybody says so. How can that be formed when it is subject
    to Cantor's first even in transfinite induction because the reals are
    complete? If you happen to describe a particular well-ordering of the
    reals, that might be of interest to others in this group as it is one
    of Hilbert's problems.

    I forgot explicitly the principle of induction. Ah, transfinite
    induction: plain induction is plenty, thank you, for, for example,
    that a process with infinitely many steps can trisect the arbitrary
    angle with edge and compass. Then again, a circle is approximated by
    the regular polygon as the number of sides goes to infinity.

    Bob, regardless, we've discussed the antidiagonal argument and Cantor's
    first, and Cantor-Bernstein and the powerset result in thousands of
    posts. Here, I want your opinion on one thing:

    Can you consider a _function_ from a set of physical objects to other
    physical objects to itself be a physical object?

    If so, the universe has infinitely many physical objects, and, infinite
    sets are equivalent.

    Then, you might want a theory to explain those real things where that
    is so.

    Ross

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