Set Theory (0)

The axiom of the empty set ∃y{∀x[¬(x ∈ y)]} asserts there exists a set that contains no elements. Ever seen an empty bag? The empty set in mathematics is represented by either { } or Ø.

The axiom of extensionality ∀A{∀B[∀x(x ∈ A ⇔ x ∈ B) ⇔ ∀y(A ∈ y ⇔ B ∈ y)]} claims that if you find two sets each with the same elements, then you will find each set contained by precisely the same sets. Basically, ∀x(x ∈ A ⇔ x ∈ B) and ∀y(A ∈ y ⇔ B ∈ y) are long winded versions of A = B. With the axiom of extensionality it can be proved that every set equals itself. In other words, ⊨ ∀S(S = S).

ST1.) ⊨ ∀S(S = S)

1.) ⊢ P ⇔ P                                       Th54

2.) ⊨ x ∈ S ⇔ x ∈ S                          Substitution

3.) ⊨ ∀S[∀x( x ∈ S ⇔ x ∈ S )]        Gen

4.) ⊨ ∀S(S = S)                                 Definition of =

Definition of ⊆

⊨ ∀A{∀B[(A ⊆ B) ⇔ {∀x[(x ∈ A) ⇒ (x ∈ B)]}]}

ST2.) ⊨ ∀S(Ø ⊆ S)

1.) ⊢ F ⇒ P                                       tautology with F false

2.) ⊨ x ∈ Ø ⇒ x ∈ S                         Substitution

3.) ⊨ ∀S[∀x( x ∈ Ø ⇒ x ∈ S )]        Gen

4.) ⊨ ∀S(Ø ⊆ S)                              Definition of ⊆