Set Theory Vs. Real Analysis
Here's the dirt on how these objects relate to one another:
In one direction, set theory is needed to actually construct the real numbers: without set theory, analysis is just a kind of speculation: "*IF* some set of objects exists with all these wonderful properties, *THEN*..." Some interesting set theory is needed to actually prove that "the reals exist" (i.e., that some ordered field with sups and infs exists).
Going into more advanced territory, set theory provides Zorn's Lemma (which is equivalent to the Axiom of Choice), which analysts invoke constantly.
Going the other way, the reals provide a very natural and intuitive playground for one of the deepest issues in set theory, the Continuum Hypothesis.
Going into more advanced territory, set theory provides Zorn's Lemma (which is equivalent to the Axiom of Choice), which analysts invoke constantly.
Going the other way, the reals provide a very natural and intuitive playground for one of the deepest issues in set theory, the Continuum Hypothesis.