Since Tilemachos Vassias asked (in a comment to Mariano Suárez-Alvarez's answer) about "the appearance of set theory in seemingly random and completely unexpected places in mathematics," I'd like to give a more general answer than the one given by Juris Steprans'sSteprans, who concentrated on dimension. I've come to expect set theory to appear in any area of mathematics that gets beyond the consideration of countable or "essentially countable" (e.g. separable, in topology) structures. This expectation is not so much based on the foundational role of set theory, mentioned in Fernando Muro's comment, but on its role as the study of combinatorial structures on infinite sets. It is not surprising (to me) that when one analyzes problems or structures in depth, combinatorial issues arise. Indeed, it happens surprisingly often that analysis of a problem reduces it entirely to a combinatorial question, and then one expects to use set-theoretic tools. In some cases, that leads to independence results; in other cases, the tools produce solutions in ZFC. One difference between these two sorts of outcomes is that we know how to prove independence results only when uncountability is involved in an essential way (unless you count applications of Gödel's incompleteness theorems, but they're not really set theory), whereas the use of infinite combinatorics to prove results in ZFC can also arise in essentially countable situations.
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Since Tilemachos Vassias asked (in a comment to Mariano Suárez-Alvarez's answer) about "the appearance of set theory in seemingly random and completely unexpected places in mathematics," I'd like to give a more general answer than the one given by Juris Steprans's, who concentrated on dimension. I've come to expect set theory to appear in any area of mathematics that gets beyond the consideration of countable or "essentially countable" (e.g. separable, in topology) structures. This expectation is not so much based on the foundational role of set theory, mentioned in Fernando Muro's comment, but on its role as the study of combinatorial structures on infinite sets. It is not surprising (to me) that when one analyzes problems or structures in depth, combinatorial issues arise. Indeed, it happens surprisingly often that analysis of a problem reduces it entirely to a combinatorial question, and then one expects to use set-theoretic tools. In some cases, that leads to independence results; in other cases, the tools produce solutions in ZFC. One difference between these two sorts of outcomes is that we know how to prove independence results only when uncountability is involved in an essential way (unless you count applications of Gödel's incompleteness theorems, but they're not really set theory), whereas the use of infinite combinatorics to prove results in ZFC can also arise in essentially countable situations. |
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