Isn't Russel's "Paradox" really a proof that the "set of all sets that are not members of themselves" does not exist?
No, it's much deeper than that.

First, define S as the set of all sets that are members of themselves. That is, self-referential. The S set is well defined, and easy to see that it exists.

Next is the hard part. Define R as the set of all sets not in S. Clearly this is where the trouble comes in. Whether R contains itself is undecidable, which leads back to what it means to be in a set.

Can something be both in the set and not in the set? Can someone be half-pregnant?

So to follow your logic, we can't even define a set properly, at least not completely. We can only say what is in certain kinds of sets, and not what isn't in them. This is starting to sound like the halting problem!? We know if a program halts after the fact, but we don't necessarily know if it doesn't, because maybe we just haven't waited long enough.

All of these are related. And they won't go away, even if someone comes up with a new idea of "set" or "computable" or "decidable". Some other concept in the new system would become undecidable, and we'll have just traded in one problem for another.

-QM
--
Quantum Mechanics: The dreams stuff is made of

In reply to Re^3: BlooP and FlooP and GlooP: Turing Equivalence, Lazy Evaluation, and Perl6 by QM
in thread BlooP and FlooP and GlooP: Turing Equivalence, Lazy Evaluation, and Perl6 by jonadab

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.