Ordinals and axioms · Definition

Zermelo–Fraenkel axioms

set theoryordinals and axioms

Before you start you need to know

Zermelo proposed a first axiomatization in 1908 to avoid the paradoxes of naive set theory. Fraenkel and Skolem completed it, and the resulting system ZF is now used as the standard foundation.

Definition

The Zermelo-Fraenkel axioms (ZF) are the standard axiom system of set theory, formulated in first-order logic. They allow the usual mathematics to be built from the primitive notions of set and membership (\( \in \)).

\[\forall x\,\forall y\,(\forall z\,(z\in x \leftrightarrow z\in y)\rightarrow x=y)\]

The axioms include extensionality, empty set, pairing, union, power set, infinity, replacement and regularity. With them one defines ordered pairs, functions and the natural numbers as finite ordinals, as in Enderton and Jech. Extensionality states that two sets with the same elements are equal, that is, \( x=y \leftrightarrow \forall z\,(z\in x \leftrightarrow z\in y) \).

Guiding example: the pairing axiom guarantees that for any \( a \) and \( b \) the set \( \{a,b\} \) exists, and from it the ordered pair \( (a,b)=\{\{a\},\{a,b\}\} \) is built. This yields the Cartesian product \( A\times B \) as a set within the theory.

Typical error: believing the axiom of choice is included in ZF. It is usually added as an independent axiom (ZFC), and its independence from ZF was proved by Cohen, as explained in Jech.

Think about it

Think about it: why can the axiom of choice not be deduced from ZF?

Gödel and Cohen proved AC is independent of ZF: if ZF is consistent, neither AC nor its negation follows from ZF.

Exercises 10

  1. Which of these statements is the axiom of extensionality?
  2. What is the name of the axiom that guarantees the existence of \( \{a,b\} \) for any \( a \) and \( b \)?
  3. Which axiom guarantees the existence of a set containing the empty set that is closed under successor?
  4. Which axiom allows forming the set of all subsets of a given set?
  5. If \( A=\{1,2\} \) and \( B=\{3\} \), how many elements does \( A\times B \) have?
  6. What is the union of the family \( \{\{1\},\{1,2\},\{2,3\}\} \)?
  7. Which of these statements about regularity is correct?
  8. Which axiom allows defining an image set by applying a formula to a given set?
  9. Let \( A=\{\varnothing,\{\varnothing\}\} \). How many elements does \( \mathcal{P}(A) \) have, and what is correct about \( A \in A \) under regularity?
  10. Let \( f:A\to B \) with \( A=\{0,1,2\} \) and \( B=\{a,b\} \). How many injective functions from \( A \) to \( B \) exist, and which statement about replacement is correct?

References

  • Herbert B. Enderton, Elements of Set Theory, 1st edition, Academic Press, 1977.
  • Thomas Jech, Set Theory (The Third Millennium Edition), 3rd millennium edition, Springer, 2003.