Joel David Hamkins: prediction

Transcript context

…So what does that mean? So the continuum hypothesis was shown to be independent from the ZFC axioms of mathematics? Right. So the ZFC axioms were the axioms that were put forth first by Zermelo in 1908 in regard to his proof of the well-order theorem using the axiom of choice. That wasn’t fully ZFC. At that time, it was just Zermelo theory because he sort of… There was a kind of missing axiom, the replacement axiom and the foundation axiom were added later, and that’s what makes the Zermelo-Fraenkel axiomatization, which became sort of standard. Actually, there’s another aspect, which is Zermelo’s original theory allowed for the existence of ur-elements, or these atoms, mathematical objects that are not sets but out of which we build the set theoretic universe, whereas set theorists today generally don’t use ur-elements at all. I argue that it’s really the philosophy of structuralism that leads them to omit the ur-elements because it turns out that if you adopt ZFC axioms with ur-elements, ZFCU it’s called, or ZFA, then any structure that exists, any mathematical structure that exists in that set theoretic universe with the atoms is isomorphic to a structure that doesn’t use the atoms at all. And you don’t need the atoms if you’re a structuralist because you only care about the structures up to isomorphism anyway, and the theory is simply more elegant and clear without the atoms. They’re just not needed. And so that’s why today when we talk about set theory, generally we talk about the atom-free version, and ZFC has no ur-elements. Okay. So we formulate the ZFC axioms of set theory. These are expressing the main principle ideas that we have about the nature of sets and set existence. And Cantor had asked about the continuum hypothesis in the late 19th century, and it remained open, totally open until 1938. We should mention, I apologize, that it was the number one problem in the Hilbert’s 23 set of problems formulated at the beginning of the century.…

Stored transcript either side of the excerpt. The highlighted words are the published quote; the surrounding text is unedited source, never generated.