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

An actual first-order formulation of ZFC?

  1. May 27, 2012 #1
    Can someone point me to a first-order axiomatization of ZFC?

    As I've mostly seen ZFC expressed in higher-order logics.
  2. jcsd
  3. May 27, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    ZFC is a first order theory, where have you been seeing second order formulations? I'm sure every one of the top Google results for "ZFC axioms" will give you a first order formulation. In particular, Wikipedia.
  4. May 27, 2012 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The higher-order axioms are replaced with axiom schema. e.g. the axiom schema of subsets is collection of statements
    {x in A | P(x)} is a set,​
    one for every unary predicate P in the language of first-order set theory.

    First-order ZFC requires infinitely many axioms to specify. First-order NBG, however, is an 'equivalent' set theory in an important sense, but only requires finitely many axioms.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook