Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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)))
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Trying to understand an expression in Peano's Principia Arithmetices
Loading...