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Source. R. Rado, Axiomatic treatment of rank in infinite sets, Canadian Journal of Mathematics 1 (1949), 337–343, §1, printed p. 337, §3, printed p. 340, and §4, printed p. 341 (canonical PDF).

Let $M$ be a set. A finite-rank function is an integer-valued function $r$ on the finite subsets of $M$ satisfying, for every finite $A\subseteq M$ and every $x,y\in M$,

$$r(\varnothing)=0, \tag{R1}$$ $$r(A)\le r(A\cup\{x\})\le r(A)+1, \tag{R2}$$ $$\begin{split} r(A\cup\{x\})=r(A\cup\{y\})=r(A) \quad\Longrightarrow\quad r(A\cup\{x,y\})=r(A). \end{split} \tag{R3}$$

These are the source's equations (4)–(6). Nonnegativity and the upper bound $r(A)\le |A|$ follow from (R1)–(R2); they are not extra assumptions. The finite-rank deductions also establish monotonicity and the elementary exchange facts used below.

An arbitrary subset $L\subseteq M$ is independent if $r(A)=|A|$ for every finite $A\subseteq L$. This is a finite-character definition, also for uncountable $L$. A base of $L$ is an independent $B\subseteq L$ maximal under inclusion among the independent subsets of $L$. Equivalently, $B\cup\{x\}$ is dependent for every $x\in L\setminus B$. The term does not initially mean a set of maximum cardinality; that consequence is proved in part (iii) of the theorem.

No rank has yet been assigned to an infinite set. The notation $r(L)$ for a possibly infinite cardinal is introduced only after base existence and equal cardinality have been established. In particular, the integer axioms above are never applied directly to an infinite argument.

Set-theoretic scope. This compilation follows the paper in a setting with the axiom of choice. The selection proof uses well-ordering and transfinite recursion; base extension uses Zorn's lemma. Cardinal inequalities compare cardinalities of sets, not order types. All index collections and ambient collections here are sets, not proper classes. See the exact external inputs.

Notation. The source uses $A+B$ for union, $AB$ for intersection, $\theta$ for the empty set, and $\subset$ inclusively. We use $\cup$, $\cap$, $\varnothing$ and $\subseteq$. Its letters $A,B,N$ denote finite sets throughout the relevant sections. We repeat finiteness in each result statement so this convention is not lost.