### By Estefania Olaiz

A notable mathematician Georg Bernhard Riemann once said:

"If only I had the theorems! Then I should find the proofs easily enough.”

It is with this sentiment that Riemann expressed the inductive dependency of theorems on mathematical proofs.

To clarify, a proof is a mathematical argument used to support a conclusion inferentially. Former theorems may be used to deduce the new theorems; one may also refer to axioms, which are self-evident principles regarded as the basis of logically sound assumptions.

For instance, take the following axiom of philosophy: “Everything that is, is.” Now one of science: “Past evidence can be trusted or developed into new theories.” Lastly, take an axiom of mathematics: “If there is a contradiction, one may question the logic.”

This leads us to Russel’s Paradox, postulated by polymath Bertrand Russell in 1901. More specifically, a logical inconsistency called “The Barber’s Paradox,” which illustrates the following:

If there is a barber who lives on an island, and the barber solely shaves all men who live on the island who do not shave themselves, does the barber shave himself?

Moreover, if the barber shaves himself, then the barber is shaving someone who shaves themselves. This contradicts the fact that he only shaves those who don’t shave themselves. On the other hand, if the barber doesn't shave himself, then he would, by principle, shave himself.

In terms of set theory, this illustrates that, because a property can define a set, then its inability for self-membership defines it. Unfortunately, a set can’t be or fail to be, a member of itself.

This created a need for an axiomatic system, with Zermelo-Fraenkel set theory as the most highly regarded. First proposed in 1908 by Ernst Zermelo, it sought to develop set theory without the presence of paradoxes like the Barber’s Paradox.

Now, let’s discuss the system of axioms that constitute the Zermelo-Fraenkel Set Theory, denoted “ZF.”

In the following conditional statements, ∃ stands for “there exists”, ∀ for “for all”, ∈ for "is an element of," ø for “the empty set”, ⇒ for “implies”, ∧ for AND, ∨ for OR, and ≡ for "is equivalent to." An axiom such as ∀ x ∃ p ∀ s . s ⊆ x iff s ∈ p would be read like this:

For every x, there is a p, which is going to be the power set effects, such that for every set s, s is a subset of x, if and only s is a member of p.

Axiom of Extensionality: Establishes that a set is completely characterized by its elements alone.

∀ x (x ∈ a ≡ x ∈ b) ⇒ a = b

Empty Set Axiom: Ensures that there is at least one set.

∃ x ∀ y (! y ∈ x)

Axiom of Subset Selection: Declares subsets of a given set as sets themselves.

∀ X ∀ p ∃ Y ∀ u (u ∈ Y ≡ (u ∈ X ∧ φ (u, p)))

Power Set Axiom: Allows us to construct a bigger set from a given set.

∀ X ∃ Y ∀ u (u ∈ Y ≡ u ⊆ X)

Axiom of Replacement: Allows us, given a set, to construct other sets of the same size.

∀ x ∀ y ∀ z [φ (x, y, p) ∧ φ (x, z, p) ⇒ y = z]
⇒ ∀ X ∃ Y ∀ y [y ∈ Y ≡ (∃ x ∈ X) φ (x, y, p)]

Axiom of Union: Allows us to take unions of two or more sets.

∀ X ∃ Y ∀ u (u ∈ Y ≡ ∃ z (z ∈ X ∧ u ∈ z))

Axiom of Infinity: Gives us at least one infinite set.

∃ S [ø ∈ S ∧ (∀ x ∈ S) [x U {x} ∈ S]]

Axiom of Foundation: Guarantees a set doesn't contain itself, thus avoiding certain paradoxical situations.

∀ S [S ≠ ø ⇒ (∃ x ∈ S) S ∩ x = ø]

Axiom of Choice: Allows us to find a choice set for any arbitrary collection of sets.

∀ x ∈ a ∃ A (x, y) ⇒ ∃ y ∀ x ∈ a A (x, y (x))

Ultimately, these axioms helped restore the ongoing foundational crisis that resulted from the circularity of self-reference in logic. It is with the Zermelo-Fraenkel axiomatization of set theory that we began to formalize the foundations from which the rest of mathematics is derived.

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