It is well-known that:

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> Given a set $X$ and a collection $\cal S$ of subsets of $X$, there exists a $\sigma$-algebra $\cal B$ containing $\cal S$, such that $\cal B$ is the smallest $\sigma$-algebra satisfying this condition.
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Certain texts, Lieb and Loss, *Analysis*, for instance, state that the proof of this assertion requires transfinite induction. On the other hand, one can define $\mathcal B$ to be the intersection of all $\sigma$-algebras containing $\cal S$. Which statement is correct? Or, is there a hidden transfinite induction contained somewhere?

I must confess here that I have only vague ideaos of the rigorous set-theoretic foundations of mathematics.