Jun 3, 2026 · 1 hr 25 min · 16 segments
aboutlogic #13 | In this episode of aboutlogic, we’re joined by Joel David Hamkins, professor at the University of Notre Dame and a leading figure in set theory, mathematical logic, and the philosophy…
Joel David HamkinsGuestTorstenHostCo-HostHostUh, so may I ask you to explain for our audience, maybe assuming that they already know that CH is independent, and they heard that there is this forcing construction, um, what is the multiverse view? Uh, how much does it entail? Are all universes mo- still classical? And maybe also, is it about the models themself, or is it about the whole multiverse as one thing, or how to imagine that?
So the way I think about it is, uh, um, one wants to get straight on what mathematics is about.
So what are we doing when we're doing mathematics, and what, what is the subject matter? And, uh, in set theory, I think, uh, the situation maybe, uh, for a long time is quite different from in other parts of mathematics.
But, uh, there was, uh, when I was a graduate student, for example, the kind of pervasive view in set theory was that, look, set theory is about the set-theoretic universe.
There is this one true set-theoretic universe that we're trying to understand the nature of the truths of that, uh, of that universe.
Maybe it's something like a attitude that people have about the natural numbers.
It's quite common that there is this thing, the natural numbers, zero, one, two, three, and so on, and it has a certain arithmetic structure, and there are truth values, so any arithmetic statement has a definitive truth value in that structure.
Okay, this would be a kind of arithmetic version of the universe view, that there's a kind of, uh, singular reality to arithmetic truth.
And it was quite common, uh, years ago in set theory that a similar situation was holding in the set-theoretic universe.
So every set-theoretic statement, according to this universe view, um, would have, uh, its ultimate truth value, and we were trying to figure out what those truth values were and what the theory was.
And of course, uh, Zermelo-Fraenkel set theory is part of that picture, but, but, uh, large cardinals and so on, uh, uh, were probably also part of this picture.
And so the idea is that on the universe view, we're sort of converging to the one true set theory.
Set-theoretic pluralism is the view that, that there isn't just one true set-theoretic reality, but rather a, a kind of, uh, plurality of different set-theoretic concepts, each giving rise to their own kind of set-theoretic truth.
Uh, it's just kind of, um, I mean, the way it's often described, set-theoretic pluralism, it's not about the theory and our descriptions of the theory.
There are different concepts of set that give rise to their own independent set-theoretic worlds, so to speak, and those worlds come to different, uh, truth values for set-theoretic assertions, such as the continuum hypothesis or maybe even the axiom of choice or large cardinals or any of the other, uh, statements that we know are independent.
Um, I mean, the-- I f- find it helpful to think about the case of geometry, uh, the analogy with geometry.
Uh, so may I ask you to explain for our audience, maybe assuming that they already know that CH is independent, and they heard that there is this forcing construction, um, what is the multiverse view? Uh, how much does it entail? Are all universes mo- still classical? And maybe also, is it about the models themself, or is it about the whole multiverse as one thing, or how to imagine that?
So the way I think about it is, uh, um, one wants to get straight on what mathematics is about.
So what are we doing when we're doing mathematics, and what, what is the subject matter? And, uh, in set theory, I think, uh, the situation maybe, uh, for a long time is quite different from in other parts of mathematics.
But, uh, there was, uh, when I was a graduate student, for example, the kind of pervasive view in set theory was that, look, set theory is about the set-theoretic universe.
There is this one true set-theoretic universe that we're trying to understand the nature of the truths of that, uh, of that universe.
Maybe it's something like a attitude that people have about the natural numbers.
It's quite common that there is this thing, the natural numbers, zero, one, two, three, and so on, and it has a certain arithmetic structure, and there are truth values, so any arithmetic statement has a definitive truth value in that structure.
Okay, this would be a kind of arithmetic version of the universe view, that there's a kind of, uh, singular reality to arithmetic truth.
And it was quite common, uh, years ago in set theory that a similar situation was holding in the set-theoretic universe.
So every set-theoretic statement, according to this universe view, um, would have, uh, its ultimate truth value, and we were trying to figure out what those truth values were and what the theory was.
And of course, uh, Zermelo-Fraenkel set theory is part of that picture, but, but, uh, large cardinals and so on, uh, uh, were probably also part of this picture.
And so the idea is that on the universe view, we're sort of converging to the one true set theory.
Set-theoretic pluralism is the view that, that there isn't just one true set-theoretic reality, but rather a, a kind of, uh, plurality of different set-theoretic concepts, each giving rise to their own kind of set-theoretic truth.
Uh, it's just kind of, um, I mean, the way it's often described, set-theoretic pluralism, it's not about the theory and our descriptions of the theory.
There are different concepts of set that give rise to their own independent set-theoretic worlds, so to speak, and those worlds come to different, uh, truth values for set-theoretic assertions, such as the continuum hypothesis or maybe even the axiom of choice or large cardinals or any of the other, uh, statements that we know are independent.
Um, I mean, the-- I f- find it helpful to think about the case of geometry, uh, the analogy with geometry.
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