The Timpson Lyceum · First-Order Predicate Logic (2024)

Axiomatic Set Theory – Lecture 14 (Logic)

Axiomatic Set Theory – Lecture 14

Overview

Set theory is the foundational system that almost every other branch of mathematics presupposes: when you do geometry, algebra, topology, or computation theory, you are quietly relying on sets. This lecture builds set theory axiomatically, starting from just two primitives — the membership symbol $\in$ and the notion of set — and shows why a careful axiomatic approach is needed by walking through Russell's Paradox, the contradiction that sank "naive" set theory. From there we develop the core relations (subset, equality, proper subset), prove the empty set is unique, and introduce the existence axioms that let us actually build sets: pairing, union, replacement, and the power set. The headline result is that set-builder notation $\{y \in M : P(y)\}$ always denotes a genuine set — but only when we filter elements out of an already existing set, never out of "everything."

Key Concepts

Primitives. Everything in the theory is ultimately defined from exactly two undefined notions: the relation $\in$ (membership) and the predicate "is a set." A predicate returns true or false; a relation is a two-place predicate. Quantifiers always range over sets — "for all $x$" means "for every set $x$."

Axiom 1 (Membership is a proposition). For all sets $x, y$, the statement $x \in y$ is a proposition — it is true or false but not both: $\forall x\, \forall y \,\big( x \in y \ \underline{\vee}\ x \notin y \big)$ This looks trivial but it is the axiom that lets us deny sethood to ill-formed collections (see Russell's Paradox).

Russell's Paradox. Define $U$ to be "the set of all sets that do not contain themselves": $x \in U \iff x \notin x$. Ask whether $U$ contains itself:

• If $U \in U$, then by definition $U \notin U$ — contradiction.
• If $U \notin U$, then $U$ qualifies for membership, so $U \in U$ — contradiction.

So "$U \in U$" is neither true nor false, hence not a proposition. By Axiom 1, $U$ is therefore not a set. Naming a collection a "set" no longer forces it to be one — just as calling 4 a prime does not make it prime. This is why we cannot conjure sets into existence merely by describing them.

Basic relations.

• Non-membership: $x \notin y \iff \neg(x \in y)$.
• Subset: $x \subseteq y \iff \forall a\,(a \in x \to a \in y)$ — everything in $x$ is in $y$.
• Set equality (extensionality): $x = y \iff (x \subseteq y \ \wedge\ y \subseteq x)$. Two sets are equal exactly when they have the same members; order and repetition of elements are irrelevant, so $\{1,2\} = \{2,1\}$.
• Proper subset: $x \subsetneq y \iff (x \subseteq y \ \wedge\ x \neq y)$ — $x$ sits inside $y$ but $y$ has at least one extra element. ($\subseteq$ behaves like $\le$; $\subsetneq$ behaves like $<$.)

Sets all the way down. The only things a set can contain are other sets. The empty set $\varnothing$ is the unique set with no members; every other set is built from it. (For example the von Neumann number $4 = \{\,\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\}\}, \{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}\,\}$.)

Axiom 2 (Empty set exists). $\exists x\, \forall y\,(y \notin x)$. After proving uniqueness (below) we name this set $\varnothing$.

Axiom 3 (Pairing). For any sets $x, y$ there is a set whose only members are $x$ and $y$: $\forall x\,\forall y\,\exists m\,\forall a\,\big(a \in m \iff (a = x \ \vee\ a = y)\big),\qquad m = \{x, y\}.$ This is the source of curly-brace notation. Because $\vee$ is commutative, $\{x,y\} = \{y,x\}$. The singleton is defined by $\{x\} := \{x, x\}$, which is why duplicates collapse.

Ordered pairs. Defined so that order is recoverable: $(x, y) := \{\{x\}, \{x, y\}\}$. This gives the key property $(x,y) = (a,b) \iff (x=a \ \wedge\ y=b)$; the left coordinate is identifiable as the element that is a member of the other. Tuples nest: $(x,y,z) := ((x,y), z)$, and in general $(a_1, \dots, a_n) := ((a_1,\dots,a_{n-1}),\, a_n)$.

Axiom 4 (Union). For a set $x$, there is a set $\bigcup x$ whose members are the members of the members of $x$: $\forall y\,\Big( y \in \textstyle\bigcup x \iff \exists u\,(u \in x \ \wedge\ y \in u)\Big).$ Ordinary binary union is then defined by $a \cup b := \bigcup \{a, b\}$, equivalently $a \cup b = \{t : t \in a \ \vee\ t \in b\}$. This is our first operation, and three-or-more-element sets are built from it: $\{a,b,c\} := \bigcup\{\{a,b\},\{c\}\}$.

Axiom 5 (Replacement / image). A functional relation $R$ pairs every input with a unique output ($\forall x\, \exists! y\, (xRy)$, written $R(x) = y$). The image of $R$ on a set $M$ is $\mathrm{Im}_R(M) = \{\,y : \exists x \in M,\ xRy\,\}$. Axiom 5 states: if $M$ is a set and $R$ a functional relation, then $\mathrm{Im}_R(M)$ is a set — applying a function to a set yields a set.

Set-builder notation (theorem from Axiom 5). For any predicate $P$ and any set $M$, the collection $\{\,y \in M : P(y)\,\}$ is a set. Crucially we always filter from an existing set $M$ — never "$\{x : P(x)\}$" over everything, which would resurrect Russell's $U$.

Operations defined from the above.

• Intersection: $a \cap b := \{\,t \in a \cup b : t \in a \ \wedge\ t \in b\,\}$ (naively, $\{t : t \in a \wedge t \in b\}$).
• Set difference: $x \setminus y := \{\,a \in x : a \notin y\,\}$. Not commutative: $a \setminus b \neq b \setminus a$.
• Cartesian product: $x \times y = \{\,(a,b) : a \in x,\ b \in y\,\}$. Properly, the pairs are drawn from $\mathcal{P}\big(\mathcal{P}(x \cup y)\big)$ — which blows up fast, which is why everyday math uses the naive form.

Axiom 6 (Power set). For a set $M$ there is a set $\mathcal{P}(M)$ whose members are exactly the subsets of $M$: $\forall a\,\big(a \in \mathcal{P}(M) \iff a \subseteq M\big).$ E.g. $\mathcal{P}(\{a,b\}) = \{\,\varnothing,\ \{a\},\ \{b\},\ \{a,b\}\,\}$. (The power set is also the discrete topology on $M$.) A separate axiom is needed because finite power sets are constructible from earlier axioms, but infinite ones are not.

Worked Example

Proof that the empty set is unique. First state the standard uniqueness strategy: assume two objects with the property, prove they must be equal. Suppose $A$ and $B$ are both empty, i.e. $\forall y\,(y \notin A)$ and $\forall y\,(y \notin B)$. Show $A = B$ via mutual subset inclusion, using that a false hypothesis makes an implication vacuously true.

Since any two empty sets coincide, the empty set is unique, and we may legitimately name it $\varnothing$.

As a second, smaller illustration, build the ordered triple $(1,2,1)$ purely from sets:

The outer set has two distinct members, and no spurious duplicate appears because the two copies of $(1,2)$ live in different sets.

Summary

• Set theory rests on two primitives, $\in$ and "is a set"; every other notion is defined from them, and all quantifiers range over sets.
• Russell's Paradox ("the set of all sets that don't contain themselves") shows that describing a collection does not make it a set — Axiom 1 lets us deny sethood to such collections.
• Core relations: $x \subseteq y$ (containment), $x = y$ (same members — order and repetition don't matter), $x \subsetneq y$ (proper containment).
• Existence axioms build sets in stages: empty set (Axiom 2, proven unique), pairing (Axiom 3), union (Axiom 4, the first operation), replacement (Axiom 5), and power set (Axiom 6).
• Axiom 5 guarantees that set-builder notation $\{y \in M : P(y)\}$ denotes a set — provided you always filter out of an existing set $M$, which is exactly the safeguard naive set theory lacked.
• Intersection, set difference, and Cartesian product are all defined from union and selection; ordered pairs $(x,y) = \{\{x\},\{x,y\}\}$ let unordered sets encode order and build arbitrary finite tuples.