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Trying to understand an expression in Peano's Principia Arithmetices

  1. Dec 8, 2013 #1
    In the attached image, I am having trouble with point 66. I realize the notation is a bit archaic, so I'll explain as much as I can.

    [(x,y)∈]a means "the set of solutions of the proposition a(x,y)." For example, the set of solutions to the proposition x < y.

    -= means "is not equivalent to", and refers to two sets in each case here.

    Λ means "the empty set".

    The period is part of dot-parentheses notation. I'll tie everything together in parentheses in a little bit rather than explain it.

    = means "if and only if" and links two propositions.

    The next part is what bothers me.

    I think [y∈]a is the set of solutions of a(x,y) in terms of x alone. But, what does this really mean? What would the resultant set be if a were the proposition x<y? What if it were x⇒y?

    I suspect that this has some ties to lambda calculus, but that doesn't help me much since I have no real knowledge of it.

    Now, all together, the tautology is:

    ([(x,y)∈]a -= Λ) = (([x∈]([y∈]a -= Λ)) -= Λ)

    Can anyone help me understand the partial solution [y∈]a?

    Attached Files:

    Last edited: Dec 8, 2013
  2. jcsd
  3. Dec 16, 2013 #2
    Okay, I have been flipping through the pages of the discrete math book I'll be using next semester, and found a more intuitive (and probably more modern) way to understand this whole expression. It is simply this proposition:

    ∃x∃y(a(x,y)) ↔ ∃x(∃y(a(x,y)))
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