The Zermelo-Frankel Axioms

The ZF-Axioms are kind of a bed rock in Mathematics. A place where everything can begin. Now this is not the only way to formalise mathematics. There many theories around. I haven't got there yet. Category theory and Homeotopic type theory come to mind here.

Now, let the caveat be stated here clearly.

I am not a mathematician, I am not formally trained in the topic and I do understand there may be errors here. If you are so disposed and would like to free the Internet of error. Please let me know! I will be happy to be schooled in this manner. What appears below is how I have understood the subject and it may be helpful to you.

Axiom of Extensionality

Means that any two sets with the same members is the very same set.

∀x∀y (∀z z ∈ x ≡ z ∈ y) ≡ (x = y)

Lets hold this true that for every set x & y, where for all z such that z is member of x if & only if it is also a member of y, this is the same as saying x = y

Axiom of the Empty Set

There is a set with no members.

∃x∀y y ∉ x

There exists a set x that for all y, y is not a member of x.

Axiom of Unordered Pairs

From any two sets x & y, we can construct a set that contains both x & y.

The notation for that set is {x,y}.

∀x∀y ∃z ∀w w ∈ z ≡ (w = x ∨ w = y)

This says for every pair of sets x & y there exists a set z (for all w) that contains x and y and no other members. Here w act like a dummy variable, it cannot equal anything but x or y.

Axiom of Union

∀x∃y ∀z z ∈ y ≡ (∃t z ∈ t ∧ t ∈ x)

Given any set x, there is a set y such that, for any element z, z is a member of y if and only if there is a set t such that z is a member of t and t is a member of x.

Axiom of Infinity

∃x∅ ∈ x ∧ [∀y (y ∈ x) → (y ∪ {y} ∈ x)]

Axiom Scheme of Replacement

B(u, v) ≡ [∀r(r ∈ v ≡ ∃s[s ∈ u ∧ A n (s, r)])]

[∀x∃!yA n (x, y)] → ∀u∃v(B(u, v))

Power Set Axiom

∀x∃y∀z[z ∈ y ≡ z ⊆ x]

Axiom of Choice - An add on Axiom

For every,

e ∈ C f (e) ∈ e.

∀C∃f∀e[(e ∈ C ∧ e = ∅) → f (e) ∈ e]

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