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Statement

Setting (§3, p. 340, and §4, p. 341). $M$ is a set and $r$ is a rank function in $M$: an integer $r(A)$ for every finite $A\subseteq M$, with $r(\varnothing)=0$, $r(A)\le r(A\cup\{x\})\le r(A)+1$, and $r(A\cup\{x\})=r(A\cup\{y\})=r(A)$ implying $r(A\cup\{x,y\})=r(A)$, for all finite $A$ and all $x,y\in M$ (the paper's axioms (4)–(6)). A subset $L\subseteq M$ of any cardinal is independent, written $f(L)=1$, when $r(A)=|A|$ for every finite $A\subseteq L$; a base of $L$ is a maximal independent subset of $L$, that is, an independent $L^*\subseteq L$ with $L^*\cup\{x\}$ dependent for every $x\in L\setminus L^*$. The definitions page gives these in full.

Theorem (p. 341, unnumbered, quoted in the paper's notation, where $+$ is union, juxtaposition is intersection, $-$ is set difference, $\theta$ is the empty set and $\subset$ is inclusive):

"(i) If $|L| < |L'|$; $f(L) = f(L') = 1$, then there exists $x' \in L' - LL'$ satisfying $f(L + \{x'\}) = 1$.

(ii) If $L_1 \subset L$; $f(L_1) = 1$, then there exists a base of $L$ which contains $L_1$. In particular ($L_1$ the empty set) every set $L$ possesses at least one base.

(iii) If $L'$ and $L''$ are bases of $L$, then $|L'| = |L''|$."

In the corpus's words: for subsets $L,L',L_1$ of $M$ of arbitrary cardinal,

• (i) if $L$ and $L'$ are independent and $|L|<|L'|$, some $x'\in L'\setminus L$ leaves $L\cup\{x'\}$ independent;
• (ii) every independent $L_1\subseteq L$ is contained in a base of $L$, so every subset of $M$ has a base;
• (iii) any two bases of the same set $L$ have the same cardinal.

Rank cardinal (p. 341, the paragraph after the Theorem). By (ii) and (iii) the paper defines the rank cardinal $r(L)$ of any $L\subseteq M$ as the largest cardinal of an independent subset of $L$, equivalently the common cardinal of all bases of $L$, and notes that for finite $L$ this agrees with the given rank function.

Source. R. Rado, Axiomatic treatment of rank in infinite sets, Canadian Journal of Mathematics 1 (1949), 337–343: the axioms on p. 340, the definition of a base and the Theorem on p. 341, the proof on pp. 342–343. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the three parts and the rank-cardinal paragraph were read clause by clause on the printed pages. The arguments on the linked part pages are the corpus's own; nothing here is independently reviewed.

Proof pointer

Pages 342–343. Part (i) is proved by contradiction: if no element of $L'\setminus L$ augments $L$, each $x'\in L'$ has a finite set $A(x')\subseteq L$ whose rank it does not raise, Whitney's exchange inequality (the paper's (11)) gives the Hall-type condition $|A(x'_1)\cup\dots\cup A(x'_k)|\ge k$, and Lemma 2, applied with cardinality as the rank function, yields an injection of $L'$ into $L$. Part (ii) applies Zorn's lemma to the independent sets between $L_1$ and $L$, which have finite character. Part (iii) follows from (i). The corpus's arguments for the three parts are on part (i), part (ii) (which also records the slip on p. 343, where the union of the chain is printed as a member of the chain $\Lambda'$ rather than of $\Lambda$) and part (iii).

Dependencies

Lemma 2 of the same paper, and through it Lemma 1 and R. Rado, A theorem on independence relations, Quarterly Journal of Mathematics 13 (1942), 83–89, Theorem 3; H. Whitney, On the abstract properties of linear dependence, American Journal of Mathematics 57 (1935), 509–533; and Zorn's lemma, cited to M. Zorn, A remark on method in transfinite algebra, Bulletin of the American Mathematical Society 41 (1935), 667. The external-input page states the forms used.

Bears on

No Erdős problem directly. The Theorem is the infinite-rank part of the paper; the paper's link to the problems runs through Lemma 1.