Extensionality Axiom

The ax­iom of ex­ten­sion­al­ity is one of the fun­da­men­tal ax­ioms of set the­ory. Ba­si­cally, it pos­tu­lates the con­di­tion, by which two sets can be equal. This con­di­tion can be de­scribed as fol­lows: if any two sets have ex­actly the same mem­bers, then these sets are equal. A for­mal no­ta­tion of the ex­ten­sion­al­ity ax­iom can be writ­ten as:

$$ \forall A \forall B : ( \forall x : (x \in A \iff x \in B) \Rightarrow A=B)$$

Examples

  • \(\{1,2\} = \{2,1\}\), be­cause what­ever ob­ject we choose, it ei­ther be­longs to both of these sets (\(1\) or \(2\)), or to nei­ther of them (e.g. \(5\), \(73\))

com­ment:

  • If \(A = \{x \mid x = 2n \text{ for some integer } n \}\) and \(B = \{x \mid x \text{ is even } \}\), then \(A=B\). The proof goes as fol­lows: \(\forall x : (x \in A \Leftrightarrow (x = 2n \text{ for some integer } n ) \Leftrightarrow (x/2 = n \text{ for some integer } n) \Leftrightarrow (x/2 \text{ is an integer}) \Leftrightarrow (x \text{ is even}) \Leftrightarrow x \in B)\)

that, if sim­plified, gives \(\forall x : (x \in A \iff x \in B)\), which, by ex­ten­sion­al­ity, im­plies \(A=B\)

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Fix the for­mat­ting in the cur­rently commeneted ex­am­ple. Every new state­ment needs to be in a new line, lined up.

Add more ex­am­ples.

Ax­iom’s converse

Note, that the ax­iom it­self only works in one way—it im­plies that two sets are equal if they have the same el­e­ments, but does not provide the con­verse, i.e. any two equal sets have the same el­e­ments. Prov­ing the con­verse re­quires giv­ing a pre­cise defi­ni­tion of equal­ity, which in differ­ent cases can be done differ­ently. noteSome­times the ex­ten­sion­al­ity ax­iom it­self can be used to define equal­ity, in which case the con­verse is sim­ply stated by the ax­iom. How­ever, gen­er­ally, the con­verse fact can always be con­sid­ered true, as the equal­ity of two sets means that they are the same one thing, ob­vi­ously con­sist­ing of a fixed se­lec­tion of ob­jects. The sub­sti­tu­tion prop­erty of equal­ity?

Parents:

  • Set

    An un­ordered col­lec­tion of dis­tinct ob­jects.