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

Set Theory & Foundations of Arithmetic – Lecture 15 (Logic)

Overview

This lecture develops axiomatic (ZFC) set theory and shows how all of arithmetic can be built on top of it. Starting from just two undefined notions — set and membership ($\in$) — we recap the axioms and definitions that let us build sets, then prove the theorem that justifies set-builder notation. From there we define ordered pairs, functions, the standard set operations, and the infinite set, and use these to construct the natural numbers, addition, multiplication, and the order relation. The payoff is conceptual: every familiar number and operation is "sets all the way down," and arithmetic is a consequence of set theory rather than a separate starting point.

Key Concepts

Undefined terms. Set theory takes exactly two primitives: set and the membership relation $\in$. Everything else is defined from them.

Membership axiom (A1). "$x \in y$" is a proposition (true or false, never both) if and only if $x$ and $y$ are sets. This blocks Russell's paradox: the "set" $U = \{x : x \notin x\}$ leads to $U \in U \iff U \notin U$, which is neither true nor false — so $U$ is not a set. You don't get to call something a set just by naming it.

Basic definitions.

• $x \notin y$ means $\lnot(x \in y)$.
• Subset: $x \subseteq y \iff \forall a\,(a \in x \to a \in y)$.
• Equality (extensionality): $x = y \iff x \subseteq y \wedge y \subseteq x$. The logical notion of equality is extended here: two sets are equal exactly when they have the same members.
• Proper subset: $x \subsetneq y \iff x \subseteq y \wedge x \neq y$.

Empty set (A2) + uniqueness. There exists an empty set, and a theorem shows it is unique, so we may name it $\varnothing$.

Pairing (A3). For sets $x, y$ there exists $\{x, y\}$. Consequences: $\{x, y\} = \{y, x\}$ (order doesn't matter), and the singleton $\{x\} := \{x, x\}$ (duplicates don't matter).

Ordered pairs. Order does matter, so we encode it with sets: $(a, b) := \{\{a\}, \{a, b\}\}$. From the set you can always recover which element came first. Tuples are recursive: $(a_1, \dots, a_n) := ((a_1, \dots, a_{n-1}), a_n)$.

Union axiom (A4). If $x$ is a set of sets, then $\bigcup x$ exists, with members being the members of the members of $x$. This lets us define finite sets beyond pairs: $\{a, b, c\} := \{a, b\} \cup \{c\}$, and in general an $n$-element set is an $(n-1)$-element set unioned with a singleton.

Relations and functional relations. A relation is a two-term predicate (takes two terms, returns true/false). A relation $R$ is functional iff every $x$ is related to exactly one $y$ — i.e. $\forall x\,\exists! y\,(xRy)$. When $R$ is functional we may write $y = R(x)$. (This is a more restricted notion than the later set-theoretic function, but everything true of the set-theoretic function is true here.)

Replacement / image axiom (A5). If $M$ is a set and $R$ a functional relation, then the image of $M$ under $R$ (the set of all outputs) is a set. ("Image" and "range" mean the same thing.)

Set-builder notation (Theorem). For any predicate $P$ and any set $M$, the collection $\{y \in M : P(y)\}$ is a set. The predicate acts as a filter, and crucially we always filter from a set $M$ we already know exists — this is what keeps us safe from Russell-style "too big" collections (proper classes, e.g. the collection of all ordinals).

Derived definitions (via set-builder).

• Big intersection: $\bigcap x = \{a \in \bigcup x : \forall b \in x\,(a \in b)\}$.
• Power set (A6): for a set $M$, $\mathcal{P}(M)$ exists, with $\mathcal{P}(M) = \{X : X \subseteq M\}$.
• Cross product: $A \times B = \{(a, b) : a \in A,\ b \in B\}$, formally drawn from $\mathcal{P}\big(\mathcal{P}(A \cup B)\big)$.
• Binary union/intersection: $A \cup B = \{x : x \in A \vee x \in B\}$, $A \cap B = \{x : x \in A \wedge x \in B\}$.

Axiom of infinity (A7). There exists a set $X$ with $\varnothing \in X$ and, for every $y \in X$, $\,y \cup \{y\} \in X$. This guarantees an infinite set from which the naturals are built.

Successor and the natural numbers. Define $0 := \varnothing$, and the successor $S(x) := x \cup \{x\}$. Then $1 = \{0\},\quad 2 = \{0, 1\},\quad 3 = \{0, 1, 2\},\ \dots$ so each natural number is the set of all smaller naturals, and the cardinality of $n$ is $n$. The naturals are the set of all finite ordinals.

Arithmetic by recursion.

• Addition: $m + 0 = m$ and $m + S(n) = S(m + n)$.
• Multiplication: $m \cdot 0 = 0$ and $m \cdot S(n) = (m \cdot n) + m$.
• Order: $m < n \iff m \in n$. Membership is the less-than relation.

Number systems sketch. Integers, rationals, and reals follow (via equivalence classes, not fully developed here): an integer is encoded by an ordered pair of naturals $(a, b)$ standing for $a - b$; a rational by an ordered pair of integers; and the reals are realized as $\mathcal{P}(\mathbb{N})$ (equivalently $2^{\mathbb{N}}$), reflecting that the reals are a strictly larger infinity than the naturals.

Foundation / regularity (A8). Every non-empty set $X$ contains an element $y$ with $X \cap y = \varnothing$. This forbids pathologies like $X = \{X\}$ (no set is a member of itself).

Axiom of choice (A9). If $M$ is a collection of non-empty, mutually disjoint sets, there exists a set $X$ containing exactly one element from each set in $M$. ($X, Y$ are disjoint iff $X \cap Y = \varnothing$; a family is mutually disjoint iff any two distinct members are disjoint.) Choice guarantees you can select one element from each of infinitely many sets even when no rule tells you which — its famous consequence is that every set (even $\mathbb{R}$) can be well-ordered, though it never tells you the order explicitly. Adding choice gives ZFC; omitting it gives ZF.

Worked Example

Goal: compute $2 + 2$ using only the recursive definition of addition, $m + 0 = m$ and $m + S(n) = S(m+n)$, together with $2 = S(1)$ and the previously established fact $2 + 1 = 3$.

Apply $2 = S(1)$, then unfold with the successor rule:

The same machinery drives multiplication. For example $3 \cdot 1$, using $m \cdot 0 = 0$, $m \cdot S(n) = (m \cdot n) + m$, and $1 = S(0)$:

No one computes arithmetic this way in practice; the point is that it can be done, so arithmetic is fully recoverable from set theory.

Summary

• Set theory rests on two undefined terms, set and $\in$; the membership axiom restricts what counts as a set and thereby dissolves Russell's paradox.
• Equality is extensional, ordered pairs are encoded as $(a,b) = \{\{a\},\{a,b\}\}$, and the union axiom extends $\{\dots\}$ notation to any finite set.
• The image axiom (replacement) proves the set-builder theorem: $\{y \in M : P(y)\}$ is a set — but you must always filter from an existing set to avoid proper classes.
• The axiom of infinity plus the successor $S(x) = x \cup \{x\}$ builds the naturals as $0 = \varnothing$, $1 = \{0\}$, $2 = \{0,1\}, \dots$, so $n$ has exactly $n$ elements and $m < n$ just means $m \in n$.
• Addition and multiplication are defined recursively through the successor function, recovering ordinary arithmetic entirely within set theory.
• Foundation forbids self-membership; the axiom of choice (giving ZFC) guarantees selections and well-orderings of infinite collections without specifying them.