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THE AXIOMS OF KP

  • Axiom Of Extensionality : Two sets are the same if and only if they have the same elements.

  • , and if for all sets ''x'' it follows from the fact that φ(''y'') is true for all elements ''y'' of ''x'' that φ(''x'') holds, then φ(''x'') holds for all sets ''x''.

  • and denoted {}.

  • Axiom Of Pairing : If ''x'', ''y'' are sets, then so is {''x'', ''y''}, a set containing ''x'' and ''y'' as its only elements.

  • Axiom Of Union : For any set ''x'', there is a set ''y'' such that the elements of ''y'' are precisely the elements of the elements of ''x''.

  • φ(''x''), there is a Subset of the original set containing precisely those elements ''x'' for which φ(''x'') holds. (This is an axiom schema.)

    • Here, a Σ0, or Π0, or Δ0 proposition is one whose quantifiers are all bound, that is, are of the form orall u \in v or \exist u \in v. (More generally, we would say that a formula is Σ''n''+1 when it is obtained by adding existential quantifiers in front of a Π''n'' formula, and that it is Π''n''+1 when it is obtained by adding universal quantifiers in front of a Σ''n'' formula: this is related to the Arithmetical Hierarchy but in the context of set theory.)

  • φ(''x'', ''y''), if for every set ''x'' there exists a set ''y'' such that φ(''x'', ''y'') holds, then for all sets ''u'' there exists a set ''v'' such that for every ''x''ε''u'' there is a ''y''ε''v'' such that φ(''x'', ''y'') holds.


If one knew that some set existed, then the axiom of separation would give us the empty set.

These axioms differ from ZFC in as much as they exclude the axioms of: infinity, powerset, and choice. Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the predicates φ used in these are limited to bound quantifiers only.

The axiom of induction here is stronger than the usual axiom of regularity (which amounts to applying induction to the complement of a set (the class of all sets not in the given set)).


PROOF THAT CARTESIAN PRODUCTS EXIST


Theorem: If ''A'' and ''B'' are sets, then there is a set ''A''×''B'' which consists of all Ordered Pair s (''a'', ''b'') of elements ''a'' of ''A'' and ''b'' of ''B''.

{a} = {a, a} exists by the axiom of pairing. {a, b} exists by the axiom of pairing. Thus (a, b) = { {a}, {a, b} } exists by the axiom of pairing.

If p is intended to stand for (a, b), then a Δ0 formula expressing that is:
\exist r \in p (a \in r \and orall x \in r (x = a)) \and \exist s \in p (a \in s \and b \in s \and orall x \in s (x = a \or x = b)) and orall t \in p ((a \in t \and orall x \in t (x = a)) \or (a \in t \and b \in t \and orall x \in t (x = a \or x = b))).

  Finally, A&timesB <math>\cup</math>{A&times{b} b&epsilonB} exists by the axiom of union This is what was to be proved