7. Write an official WFF expressing the Empty Set Axiom

Unofficial WFF:

∃y(∀z zЄy)

There exists a y such that for all z, z is not an element of y.

Possible official WFF:

∃y∀z¬(z Є y)

8. Intersection WFF

Statement: Given a set, there is a set ∩x consisting of all elements that belong to x.

∩x = ∀z(zЄy ↔ z belongs to every w in x)

∀z(zЄy ↔ (∀w(wЄx → zЄw)))

Statement: ∩∅ exists.

Suppose towards contradiction that x=∅. For the formula above, inputting the empty set for x gives ∩∅=∀z(zЄy ↔ (∀w(wЄ∅ → zЄw))). The right side of the equation is always true as it reads that for all w, w is the empty set. Thus, the left side of the equation must also be true resulting in ∩∅=(zЄy), which is obviously incorrect. To account for this, we can rewrite the formula as:

∩x = ∀z(zЄy ↔ (x≠∅) ∧ (∀w(wЄx → zЄw)))

Statement: ∩$\mathbb{N}$

Suppose towards contradiction that x=$\mathbb{N}$.

∩$\mathbb{N}$ = ∀z(zЄy ↔ ($\mathbb{N}$≠∅) ∧ (∀w(wЄ$\mathbb{N}$ → zЄw))).

For every z that exists in w, w must exist in x. For this to be true, z must be the empty set, but z is then not in x. w as an element of x is true, but z as an element of w is false and thus the equation returns the empty set. Hence, ∩$\mathbb{N}$ does not exist because $\mathbb{N}$ itself is not empty, only its elements are.

Comments - Andrew Furash

Your statements are correct. Nice work.

Writing ∩x = ∀z(zЄy ↔ (x≠∅) ∧ (∀w(wЄx → zЄw))) doesn't make sense. The intersection of a set is not equal to a wff. What you want to say is that for any set x, ∩x=y is given by ∀z(zЄy ↔ (x≠∅) ∧ (∀w(wЄx → zЄw))).