• TheMadFool
    13.8k
    First, some definitions and set operations to take care of:

    1. Nothing = Not anything = N

    2. { } = The empty set = { }

    3. A n B = The intersection of sets A and B

    4. A u B = The union of sets A and B

    5. A' = The complement of set A

    6. H is a subset of A iff all the elements of set H are in set A

    Suppose now that there's a universe U with only 4 things viz. v, w, x, and y. Suppose also that v and w are form the set A and x and y form the set B as below,

    7. A = {v, w}

    8. B = {x, y}

    9. T = The universal set = A u B = {v, w, x, y}

    10. In U, Nothing, as the definition asserts, is not v and not w and not x and not y. Basically, in terms of sets, Nothing is not A and not B.

    Now,

    not A = A' = {x, y}

    not B = B' = {v, w}

    Nothing = not A and not B = A' n B' = {x, y} n {v, w} = { } = not v and not w and not x and not y.

    11. A' n B' = { }

    However,

    12. A' n B' yields the set whose elements are those that are common to both A' and B'. We now know that Nothing ( N ) is common to both A' and B'. In other words,

    13. A' = {x, y, N}

    and

    14. B' = {v, w, N} and since N is common to both A' and B',

    15. A' n B' ={N}

    16. { } = {N} The empty set contains Nothing ( N ) [From 11 and 13]

    17. (A u B)' = [{x, y, N} u {v, w, N}] ' = {v, w, x, y, N}'

    and

    18. (A u B)' = A' n B' (DeMorgan's law)

    Therefore,

    19. {v, w, x, y, N}' = {N} (from 15, 17, and 18)

    Statement 19 is a paradox because the complement of a set can't contain an element that's in that set but in this case the element N = Nothing is both in the set (A u B) and also in (A u B)'. The Nothing-Empty Set Paradox.

    P.S. To those who object to line 16. { } = {N}, remember that the empty set, { }, is a subset of every set and that means it must have an element that's in every other set. The only element that fits the bill is Nothing = N
  • khaled
    3.5k
    You have an element that is literally defined as not-any-element. Of course weird things will happen. N is not an element. So this doesn't hold.
    the complement of a set can't contain an element that's in that setTheMadFool
  • Philosophim
    2.2k


    I think you are not including all that is needed for this to make sense.

    13. A' = {x, y, N} should be written as A' = {v', w', N} because the negation of A, is the negation of v and w, with the inclusion of N. Point 14 would need the same adjustment. The negation of A does not mean the inclusion of B in logic. To do this, you would need to include steps that include the universal set as well in your equation, which you do not.

    The compliment to A' would then be retranslated as A = {v, w, N'}. Same with B.

    So with the adjustements of 13 and 14, that leaves 15 still correct, that the negation of both A and B includes N.

    But, if you are unifying A and B, then you would get {v, w, x, y, N' }

    If you negate both A and B, this translates to
    {v, w, x, y, N'}' = {v', w', x', y', N} without any paradox to my mind.
  • jgill
    3.5k
    I think it's really something that you are still in pursuit of nothing! Keep up the good work. To many this means nothing, but they are wrong, it is something! :cool:
  • fishfry
    2.6k
    I got plenty of nothing
    And nothing's plenty for me
    I got no car - got no mule
    I got no misery
    Folks with plenty of plenty
    They've got a lock on the door
    Afraid somebody's gonna rob 'em
    While there out (a) making more - what for
    I got no lock on the door - that's no way to be
    They can steal the rug from the floor - that's OK with me
    'Cause the things that I prize - like the stars in the skies - are all free
    I got plenty of nothing
    And nothing's plenty for me
    I got my gal - got my song
    (I) Got heaven the whole day long
    - Got my gal - got my love - got my song

    https://www.youtube.com/watch?v=bx1YfQF0WNQ
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