A finite family $(A_i)_{i \in I}$ of subsets of a finite set $X$, the intersections $A_J$ for $J \subseteq I$, and the convention $A_\varnothing = X$

A sieve family consists of a finite set $X$, called the ambient set, a finite set $I$, called the index set, and a family $(A_i)_{i \in I}$ of subsets of $X$, that is a function $I \to \mathcal{P}(X)$ (Finite, countably infinite, countable, uncountable, The cardinality $\lvert A\rvert$ of a finite set). For $J \subseteq I$ set

$A_J \;:=\; \begin{cases} \displaystyle\bigcap_{i \in J} A_i, & J \ne \varnothing,\\[4pt] X, & J = \varnothing,\end{cases}$

and write $U := \bigcup_{i \in I} A_i$ for the union of the family.

Why the ambient set has to be named, and why $A_\varnothing := X$ is a stipulation. For $J \ne \varnothing$ the intersection $\bigcap_{i \in J}A_i$ is the set of elements belonging to every $A_i$ with $i \in J$, and it is determined by the family alone. For $J = \varnothing$ that description is satisfied by every set whatsoever, so it determines nothing; an intersection of no subsets of $X$ is $X$ only relative to $X$. Naming $X$ as part of the data and stipulating $A_\varnothing = X$ is what makes the symbol $A_J$ defined for all $J \subseteq I$, which is what the complementary form of the sieve identity requires.

(a) Every $A_J$ is a finite subset of $X$. For $J \ne \varnothing$ pick $i \in J$; then $A_J \subseteq A_i \subseteq X$. For $J = \varnothing$, $A_J = X$. In both cases $A_J \subseteq X$ is finite by clause 1 of A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$, and so is $U \subseteq X$; hence $\lvert A_J\rvert$ and $\lvert U\rvert$ are natural numbers (The cardinality $\lvert A\rvert$ of a finite set).

(b) The index sets of the sieve's sums are finite. $\mathcal{P}(I)$ is finite with $\lvert\mathcal{P}(I)\rvert = 2^{\lvert I\rvert}$ ($\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$); the set $\mathcal{P}(I) \setminus \{\varnothing\}$ of nonempty subsets of $I$ and the set $[I]^{j}$ of $j$-element subsets of $I$ are subsets of $\mathcal{P}(I)$, hence finite (A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$), and $\lvert [I]^{j}\rvert = \binom{\lvert I\rvert}{j}$ (The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$).

(c) Monotonicity. If $J \subseteq J' \subseteq I$ then $A_{J'} \subseteq A_J$. For $J = \varnothing$ this is clause (a); otherwise an element lying in every $A_i$ with $i \in J'$ lies in every $A_i$ with $i \in J$.

(d) The trace of a point. For $x \in X$ put

$T(x) \;:=\; \{\, i \in I \ :\ x \in A_i \,\} \subseteq I ,$

a finite set. For every nonempty $J \subseteq I$,

$x \in A_J \iff J \subseteq T(x) ,$

both sides saying that $x \in A_i$ for every $i \in J$. And $x \in U$ if and only if $T(x) \ne \varnothing$. Writing $t(x) := \lvert T(x)\rvert$, clause (b) applied to $T(x)$ gives $\lvert [T(x)]^{j}\rvert = \binom{t(x)}{j}$, and for nonempty $J$ the condition $x \in A_J$ with $\lvert J\rvert = j$ says exactly that $J \in [T(x)]^{j}$.

• The counts stay in $\mathbb{N}$; the identities do not. Each $\lvert A_J\rvert$ is a natural number. Every identity that sieves them carries a minus sign, and $\mathbb{N}$ has no subtraction, so those identities are stated in $\mathbb{R}$ through the canonical natural and read back by its injectivity. That is a property of the identities, not of this definition, which introduces no arithmetic at all.

• $I$ is an arbitrary finite index set, not a natural number. Nothing below numbers the sets $A_0, A_1, \dots$; the subsets $J \subseteq I$ are the objects the sums run over, and $\lvert J\rvert$ rather than any position is what carries the sign.

• The clause $A_\varnothing = X$ supplies every empty-subfamily term. In the complementary form at $J = \varnothing$ it contributes $\lvert X\rvert$; later sieve instances also use it when identifying the intersection at the empty subfamily. Removing the stipulation would leave those terms undefined.