Analysis--Set Theory

Posted by GwanSiu on December 27, 2017

In this article, ZFC axiom will be discussed.

1. ZFC Axiom

Axiom 1(Sets are objects): If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is also an element of B.

Axiom 2(Equality of sets): Two sets A and B are equal, A=B, iff every element of A is an element of B and vice versa. To put it another way, A=B if and only if every element $x$ of A belongs also to B, and every element $y$ of B belongs also to A.

Axiom 3(Empty set): There exists a set $\varnothing$, known as the empty set, which contains no elements, i.e., fpr every object $x$ we have $x\in \varnothing$

Axiom 4(Singleton sets and pair sets): If $a$ is an object, then there exists a set {$a$} whose only element is $a$, i.e., for every object $y$, we have $y\in {a}$ if and only if $y=a$; we refre to {$a$} as the singleton set whose element is $a$. Furthermore, if $a$ and $b$ are objects, then there exists a set {$a,b$} whose only elements are $a$ and $b$; i.e., for every object $y$, we have $y\in {a,b}$ if and only if $y=a$, or $y=b$; we refere to this set as the pair set formed by $a$ and $b$.

Axiom 5(Pairwise union): Given any two sets $A,B$, there exists a set $A\cup B$, called the union of A and B, which consists of all the elements which belong to A or B or both. In other words, for any object $x$,

Axiom 6(Axiom of specification): Let A be a set, and for each $x\in A$, let $P(x)$ be a property pertaining to $x$ (i.e., $P(x)$ is either a true statement or a false statement). Then there exists a set, called {$x\in A: P(x)\text{ is true}$}(or simply {$x\in A:P(x)$} for short), whose elements are precisely the elements $x$ in A for which $P(x)$ is true. In other words, for any object $y$,

Axiom 7(Replacement). Let A be a set. For any object $x\in A$, and any object $y$, suppose we have a statement $P(x,y)$ pertaining to $x$ and $y$, such that for each $x\in A$ there is at most one $y$ for which $P(x,y)$ is true. Then there exists a set {$y:P(x,y) \text{ is true for some }x\in A$},such that for any object $z$,

Axiom 8(Infinity). There exists a set $\mathbb{N}$, whose elements are called natural numbers, as well as an object 0 in $\mathbb{N}$, and an object $n++$ assigned to every natural number $n\in \mathbb{N}$, such that the Peano axioms hold.

Axiom 9(Universal specification). Suppose for every object x we have property $P(x)$ pertaining to $x$ (so that for every $x$, p(X) is either a true statement or a false statement). Then there exists a set {$x:P(x) \text{ is true}$} such that for every object $y$,

Axiom 10(Regularity): If A is a non-empty set, then there is at least one element $x$ of $A$ which is either not a set, or is disjoint from A.

Axiom 11(Power set axiom): Let $X$ and $Y$ be sets. Then there exists a set, denoted $Y^{X}$, which consists of all the functions from $X$ to $Y$, thus,

Axiom 12 (Union): Let A be a set, all of whose elements are themselves sets. Then there exists a set $\bigcup A$ whose elements are precisely those objects which are elements of the elements of A, thus for all objects as