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I'm currently self studying Analysis I by Tao. He's sprinkled a little set theory into this book, and I'm really struggling to apply it. After painstakingly proving Zorn's Lemma via another lemma, we are given this very much non-obvious exercise:

Let $A$ and $B$ be two non-empty sets such that $A$ does not have lesser or equal cardinality to $B$. Using the principle of transfinite indcuction, prove that $B$ has lesser or equal cardinality to $A$. (Hint: for every subset $X \subseteq B$, let $P(X)$ denote the property that there exists an injective map from $X$ to $A$. This exercise (combined with Exercise 8.3.3 [Schroeder-Bernstein]) shows that cardinality of any two sets is comparable, as long as one assumes the axiom of choice.

His definition of transfinite induction is as follows:

We give some applications of Zorn's lemma (also called the principle of transfinite induction) in the exercises below.

That's it. I feel like I've been thrown not even into the deep end, but pretty much into the ocean. How could I even begin to solve that using Zorn's lemma? I don't see any partial order that could in any way be useful to solve this, and I don't see any way to maximise. Am I missing something obvious, or is this just really hard?

It'sNotALie.
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1 Answers1

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Without wanting to give too much away, here's an idea of how you might use Zorn's Lemma in general:

  • Pick a partial order Q that's relevant for your problem
  • Show it's nonempty
  • Show that every total order in Q has an upper bound in Q
  • You now know Q contains a maximal element: use it to solve your problem

In this case, you're given the hint that you're to consider subsets $X$ of $B$...

Other vague hints: to prove $B$ has lesser or equal cardinality to $A$, you can give an injection from $B$ to $A$.

B. Mehta
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