Elementary Theory of the Category of Sets explained

In mathematics, the Elementary Theory of the Category of Sets or ETCS is a set of axioms for set theory proposed by William Lawvere.^[1] Although it was originally stated in the language of category theory, as Leinster pointed out, the axioms can be stated without references to category theory.

Axioms

Informally, the axioms are as follows: (here, set, function and composition of functions are primitives)

1. Composition of functions is associative and has identities.
2. There is a set with exactly one element.
3. There is an empty set.
4. A function is determined by its effect on elements.
5. A Cartesian product exists for a pair of sets.
6. Given sets

X and Y , there is a set of all functions from X to Y .

1. Given

f : X \to Y and an element y \in Y , the pre-image f ^{-1} (y) is defined.

1. The subsets of a set

X correspond to the functions X \to \{ 0 , 1 \} .

1. The natural numbers form a set.
2. (weak axiom of choice) Every surjection has a right inverse (i.e., a section).

The resulting theory is weaker than ZFC. If the axiom schema of replacement is added as another axiom, the resulting theory is equivalent to ZFC.

References

• Leinster . Tom . Rethinking Set Theory . The American Mathematical Monthly . 1 May 2014 . 10.4169/amer.math.monthly.121.05.403 . EN. 10.4169/amer.math.monthly.121.05.403.

Further reading

• https://ncatlab.org/nlab/show/ETCS
• https://topologicalmusings.wordpress.com/2008/09/01/zfc-and-etcs-elementary-theory-of-the-category-of-sets/
• https://golem.ph.utexas.edu/category/2024/09/axiomatic_set_theory_1_introdu.html

Notes and References

1. William Lawvere, An elementary theory of the category of sets, Proceedings of the National Academy of Science of the U.S.A 52 pp.1506-1511 (1964).