9 votes

The complete idiot’s guide to the independence of the Continuum Hypothesis: part 1

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  1. [2]
    skybrian
    Link
    From the blog post:

    From the blog post:

    But look more carefully. What Cantor really shows is only that, within our mathematical universe, there can’t be an enumeration of all the reals of our universe. For if there were, we could use it to define a new real that was in the universe but not in the enumeration. The proof doesn’t rule out the possibility that God could enumerate the reals of our universe! It only shows that, if so, there would need to be additional, heavenly reals that were missing from even God’s enumeration (for example, the one produced by diagonalizing against that enumeration).

    Which reals could possibly be “missing” from our universe? Every real you can name—42, π, √e, even uncomputable reals like Chaitin’s Ω—has to be there, right? Yes, and there’s the rub: every real you can name. Each name is a finite string of symbols, so whatever your naming system, you can only ever name countably many reals, leaving 100% of the reals nameless.

    Or did you think of only the rationals or algebraic numbers as forming a countable dust of discrete points, with numbers like π and e filling in the solid “continuum” between them? If so, then I hope you’re sitting down for this: every real number you’ve ever heard of belongs to the countable dust! The entire concept of “the continuum” is only needed for reals that don’t have names and never will.

    1 vote
    1. PendingKetchup
      Link Parent
      Official petition to rename the continuum to "The Nameless".

      Official petition to rename the continuum to "The Nameless".

      1 vote