Zermelo-Fraenkel Set Theory

Zermelo–Fraenkel set theory, often abbreviated as ZF, is a foundational system for mathematics based on set theory. Set theory provides a way to talk about collections of objects. It is a framework in which all mathematics can be described, providing rigorous definitions and a solid foundation.

Zermelo–Fraenkel set theory is used to avoid paradoxes in classical set theory, such as Russell's paradox. This is accomplished using axioms, which are statements assumed to be true without proof. The theory defines the limits on which sets can be created and how they can be combined, allowing mathematicians to work in a consistent way with infinite collections.

Axioms of Zermelo–Fraenkel set theory

Zermelo-Fraenkel set theory includes a number of axioms. Each axiom is a rule that defines how sets behave. The collection of these axioms is known as an axiom system.

Axiom of extendibility

This axiom states that two sets are equal if they have the same elements. In simple words, the identity of a set is determined only by its members.

A = B if for all x, x ∈ A if and only if x ∈ B.

For example, consider the set:

A = {1, 2, 3}
B = {3, 1, 2}

According to the principle of extensionality, A and B are the same set because they have the same elements although their order is different.

Empty set axiom

This axiom accepts the existence of a set with no elements, called the empty set. It is represented by {} or ∅.

∃A (∀x ¬(x ∈ A))

The empty set is unique because inclusion in this set is a condition that is never true.

Pairing principle

The principle of pairing allows us to form a new set by combining two sets.

For any sets A and B there exists a set C = {A, B}.

Example:

If A = 1 and B = 2, then C = {1, 2}.

Axiom of association

This axiom asserts that for any set, there is a union set that contains all the elements that are members of at least one set in the collection.

For any set A, there exists a set B: x ∈ B if and only if there exists a set C in A such that x ∈ C.

Example:

Let A = {{1, 2}, {2, 3}, {4}};
union(a) = {1, 2, 3, 4}

Axiom of power set

The axiom of power set states that for any set, there is a set of all subsets of the original set.

For any set A, there exists a set B: x ∈ B if and only if x is a subset of A.

Example:

A = {1, 2}
Power set of A = {{}, {1}, {2}, {1, 2}}

Axiom of infinity

This axiom guarantees the existence of infinite sets. In particular, it includes a set that includes the empty set and the property that for every set x that is a member, the set {x} is also a member.

There exists a set A for which: 
∅ ∈ A and (x ∈ A implies x ∪ {x} ∈ A)

Visual representation of an infinite set:

The circle on the left represents the empty set, and the circle on the right represents the infinite set, which shows that you can keep adding more elements.

Axiom of replacement

Given any set and a defined operation, this axiom allows us to construct a set by replacing each element of the original set with another set.

For any set A and any defined function F, there exists a set B: for every x ∈ A, there exists a y ∈ B such that F(x) = y.

Example:

Let A = {1, 2, 3} and F(x) = x + 1
B = {2, 3, 4}

Axiom of regularity (also called basis)

This axiom ensures that every set is well-founded, meaning that no set can contain itself as a member, either directly or indirectly. This prevents infinite descending chains.

In every nonempty set A, there exists a ∈ A such that A and a are disjoint.

Example:

For any set B = {{1, 2}, 3}, no element of B contains all the elements of B.

Axiom of choice

The Axiom of Choice states that for any non-empty set, there is a selection function that selects exactly one element from each set.

For any set X of nonempty sets, there exists a selection function f defined on X.

The Axiom of Choice is widely used in areas of mathematics such as algebra and topology, although it is more controversial than other axioms because it leads to results that are counter-intuitive.

Relations and functions in ZF

In set theory, a relation between sets is defined as any subset of their Cartesian product. A function is a more specific type of relation.

If we have sets A and B, then the Cartesian product , denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

A = {1, 2}, B = {x, y}
A × B = {(1, x), (1, y), (2, x), (2, y)}

A function f from a set A to a set B can be viewed as a subset of A × B such that for every element x in A, there is exactly one element y in B that satisfies the ordered pair (x, y) ∈ f.

Example of the function:

A = {1, 2, 3}, B = {4, 5, 6}
The function f : A → B
f(1) = 4, f(2) = 5, f(3) = 6

The function f can be represented as a set of ordered pairs:

F = {(1, 4), (2, 5), (3, 6)}

In this way, set theory allows us to understand complex logical relationships and define mathematical objects in a sophisticated way.

Zermelo–Fraenkel and equivalence with mathematics

Zermelo-Fraenkel set theory is the cornerstone on which much of modern mathematics is based. By defining mathematical objects as sets and characterizing their relationships through axioms, ZF allows for a better understanding of both finite and infinite processes.

Here's an important thing about ZF: although it seems simple at first glance, it is a very powerful tool for mathematics. When we add the axiom of selection to ZF, we get what is known as Zermelo-Fraenkel set theory with the axiom of selection, denoted by ZFC. It is often considered the default foundation for mathematics.

Potential contradictions avoided by ZF

Before ZF, set theory suffered from problems such as Russell's paradox, which arose when considering self-contained sets.

R = { x | x ∉ x }

If R is a member of itself (R ∉ R), then by definition it must not be in itself (R ∈ R), which is contradictory. The axioms defined in ZF are careful to specify how sets can be constructed and which sets can be meaningfully considered.

Conclusion

Zermelo–Fraenkel set theory provides a strong foundation for mathematics, preventing paradoxes and helping to organize the understanding of infinity and other complex topics. While ZF is able to describe all of standard mathematics, adding the axiom of choice (to create ZFC) allows mathematicians to access a much broader range of mathematical tools and concepts, including those that confront non-constructive proofs.

Adherence to Zermelo-Fraenkel's careful and structured definitions allows it to maintain the consistency of mathematical logic. This represents the great triumph of formal set theories: the ability to provide a mathematically rigorous, naturally logical basis for mathematical thinking, free of paradoxes, suitable even for complex mathematical and philosophical endeavors.

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