Zermelo-fraenkel set theory

**Zermelo-fraenkel set theory** (ZF) is a form of set theory invented by Ernst Zermelo and Abraham Fraenkel. It has eight axioms, with the Axiom of Choice (C) included to form the variation known as ZFC, used most frequently today.

## Axioms

ZF has six first order logical axioms and two second order logic axioms. These are as follows:

### Axiom of Extensionality

* Two sets are equal if and only if they have the same elements.

The membership predicate is made more clear by this axiom. After all two people can have the same friends but not be the same person. On the other hand, if two sets have the same members they *are* the same set by definition. Set theory is not about friendship, although you can certainly make friends learning set theory.

Repititon of an element can be ignored, so {A,A} is exactly the same as {A}.

### Axiom of Regularity

* In every non-empty set, there is a set which shares *no* members with that set.

This implies that no set can contain itself. If we had the set {X} then by the Axiom of Regularity, {X} can share no members with an element of {X}, but the only element is X, thus X can not be a member of X. This is also called the Axiom of Foundation, because it lets us know that if you keep finding the elements of the elements of the elements of a set you'll eventually reach an end. Every set has a foundation, like a building.

### Axiom of Pairing

* For every two sets A and B, a set containing both A and B exist.

This axiom assures that sets with at least two different elements exist. We can join any two sets to make another set, like joining any two cookies with cream to make new cookies. [See fig. 1]

### Axiom of Union

* For any set, the set of elements of elements of that set exists.

This axiom allows us to form sets with more than two elements. For sets like {} or {{}} where there are no "elements of elements" the set of elements of elements would be the empty set, or {}.

### Axiom of Powerset

* For any set, the set of all subsets of that set exists.

A subset of a set is a collection of elements *exclusively* chosen from that set. For example, {{},{{}},{{{}}}} would be a subset of {{},{{}},{{},{{}}},{{{}}}}, but {{},{{{{}}}}} would not be. The powerset of {} is {{}}, the powerset of {{}} is {{},{{}}}, the powerset of {{},{{}}} is {{},{{}},{{{}}},{{},{{}}}}, as some examples. Sets can never contain themselves element-wise but can contain themselves as their own subsets.

### Axiom of Infinity

* There exists an infinite set.

To form a bunch of other infinite sets we need to form at least one infinite set. This is done by saying there exists a set I containing {} and for every set A within I, the set {a,b,c,d,. . .,A} also exists in I, where a,b,c,d,. . .are the elements of A. These distinct sets would be The infinite set would be {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},. . .}. You can use the Axiom of Powerset on this to create even bigger sets.

### Axiom of Specification

* For any property and any set A, you can make a set B with just elements from A that obey that property.

For example, if we had a set of sets which represented the natural numbers 1 to 1,000 and we had the property of "being prime" we could form another set holding only the sets which represented the prime numbers from 1 to 1,000. It is called the Axiom of Specification because we specify what parts of the set we want to keep to make the new set.

### Axiom of Replacement

* For any function and set A, you can make a set B made of the outputs of the function when the inputs are the elements of A.

For example, if we had a set of sets which represented the counting numbers {1,2,3,4,. . .} and our function was f(x) = xx we could form the set of natural numbered double exponentiations, or 2x. {1,4,27,256,. . .}. The Axiom of Specification and Replacement also work on infinite sets. It is called the Axiom of Replacement because we replace every element of one set with an element of another through the function. Keep in mind a function only has one output for every input.

These axioms are sufficient for answering most problems in mathematics. Sometimes though a ninth is needed for special results like the Banach-Tarski paradox:

### Axiom of Choice

* For any set of non-empty sets A there exists a way to pick one and only one set from every set in A.

This set says there exists a way to pick a set from each set in A but *doesn't* tell us how to do it! For this reason it is called a non-constructive axiom. There are sets like the set of subsets of the reals for which there is no known decision procedure telling us how to pick a member from each subset. We could try the lowest real in each subset, but some subsets like {0.1, 0.01, 0.001, 0.0001,. . .} have no lowest member so that would be useless.

## Extensions

ZF was extended to create ZFC. Likewise, ZFC has been extended for more expressive power, with TG set theory being one such extension.

**Set theory**

**Theories**
Pocket set theory ★ Morse-Kelley set theory ★ Neumann-Gödel-Bernays set theory ★ Tarski-Grothendieck set theory ★ **Zermelo-fraenkel set theory**

**Axioms**
Axiom of Choice ★ Axiom of Extensionality ★ Axiom of Infinity ★ Axiom of Pairing ★ Axiom of Powerset ★ Axiom of Regularity ★ Axiom of Replacement ★ Axiom of Specification ★ Axiom of Union

**Objects**
Zero sharp ★ Von Neumann hierarchy

**Pioneers**
Abraham Fraenkel ★ Ernst Zermelo ★ George Cantor ★ John von Neumann ★ Kurt Gödel