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The Formal Proof of The Characteristic Property of Ordered Pair

  1. Sep 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove the statement that (a,b) = (c,d) iff a = c and b = d with only using the first order logic (rules), the axiom of extensionality, the axiom of pair, and the definition that (a,b) = {{a},{a,b}}. Any intuitive approach should be avoided.

    The example of intuitive proof:
    http://plato.stanford.edu/entries/set-theory/primer.html [Broken]
    Proof. If a = a′ and b = b′, then, of course, (a, b) = {{a}, {a, b}} = {{a′}, {a′, b′}} = (a′,b′). The other implication is more intricate. Let us assume that {{a}, {a, b}} = {{a′}, {a′, b′}}. If a ≠ b, {a} = {a′} and {a, b} = {a′, b′}. So, first, a = a′ and then {a, b} = {a, b′} implies b = b′. If a = b, {{a}, {a, a}} = {{a}}. So {a} = {a′}, {a} = {a′,b ′}, and we get a = a′ = b′, so a = a ′ and b = b′ holds in this case, too. □

    In the proof above, the author of the proof directly derives from a≠ b that {a} = {a′} and {a, b} = {a′, b′}. This procedure relies on the notion of cardinality of set. But this is not first order logic rule or any other notion of first order logic. Such an approach should be avoided. The demonstrating steps should rely on the first order logic rules and the set theory axioms mentioned above only.

    As for the formal definition and the formal axioms, (that is, the definition and the axioms in first order language) refer to Jech T., Set Theory.

    2. Relevant equations
    none.


    3. The attempt at a solution
    none.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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