Set Theory and the Philosophy of Set Theory

Joel David Hamkins, Professor of Logic, Oxford University

Chapter 8. Set Theory

We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Frege’s formal theory was naive, refuted by the Russell paradox. Zermelo’s set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon a cumulative conception of the set-theoretic universe. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but it also was a new foundational theory with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including in particular, the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.

[Ricardoricar]

I'm a Lawyer and have no formal education on Mathematics. And for some reason I'm hooked into these professor's series of Lectures. Very good! Came here for the philosophy, ended up staying for the Math. I'm definetly going deeper in this amazing field. Thank you, Sir.

[ai_serf]

One of the best lectures I've ever listened too. Not for the total beginner, but if you have an armchair degree from popular youtuber/channels in some set theory, mathematical logic, and real analysis, everything should make sense, even if we don't understand it to any degree beyond the superficial.

[PhilosophicalTrials]

For me this was the first series of lectures on *any* academic subject where I did not miss a single video -- either live or later on YT -- since I first enrolled at University in 2015. I finally got a better picture of how various things fit together in the Philosophy of Mathematics. Great content -- I'll definitely read the book in February!

[georgeconstantinides]

What a great series of lectures. Thanks for making them public.

[Achrononmaster]

@1;23:00 this was lovely, had not heard my affinity with platonism expressed so well before. I am definitely a platonist, but I like this plurality idea of different conceptions of "set", and it follows there could be different conceptions of ℝ. Just like your analogy to non-Euclidean geometry. But it is a fascinating analogy. Non-Euclidean geometries have such clear concrete manifestations (any curved surface or hypersurface). What I've never heard from a set theorist is a similar obvious concrete alternative concept of "set", only different but _very abstract _*_in_*_ their difference_ axiom schemas.

[joeldavidhamkins]

Start at 4:47 for the full-screen speaker view. Apologies for not getting this right from the start.

[javierfernandez]

Thanks for this great lecture! I will go through the book and the lectures simultaneously.
It's great that anyone in the world can have access to this high-quality material.

[rsmt]

Thank you for publishing these lectures. I found them to be very illuminating.

[blargoner]

Great lecture series! Thanks for posting them.

[DrThalesAlexandre]

Thank you for all the enlightening lectures in this series!

[nikitasmarkantes]

I am a lawyer too. Its amazing how this lecture reiterates Socrates' parmenidis. A real and genuine platonist view.

[minch]

Great series overall! You mentioned in the Q&A section about picking a path in an infinite tree and its relationship to a weaker variant of the axiom of choice. I bring it up because I remember going to a conference in my masters year (2014) and seeing a talk by an Oxford PhD student saying that he was working on showing that this graph theoretic principle (or one very similar; it's been a while and my memory's not great!) is equivalent to the axiom of choice. My question is, is what you mentioned (its relation to a weaker version of the axiom) the product of his research? It would be very interesting to hear how the story from so many years ago ended up!

Thanks again for this incredible lecture series. Can't wait for the book to come out next year!

[victormd]

Professor, i read somewhere that you're a pluralist on the philosophy of set theory, i am just not so sure what this view means exactly:
Do you think the cumulative hierarchy ( as defined by the successive application of power sets starting at the empty set) defines multiple universes ( therefore not having an intended model ), each of which is true in some sense or do you believe the cumulative hierarchy defines a single universe, while holding it is not "special" and others - such as the constructible universe - are true in some sense as well?

[radientbeing]

Thanks for a good review of set theory. I think about set theory a lot as well as the philosophy of mathematics.

[MrLove]

Professor Hamkins, hello! I found your channel via your wonderful discussion with Daniel Rubin!

I am curious, could you shed some light on why it is that Cantor's diagonal argument is used to suggest the "uncountability" of certain infinite sets when the same reasoning applies to very small finite sets which can clearly be counted? I am trying to wrap my mind around why having 2^n combinations of numbers with n digits is in any way more or less "countable" than something else, and what is different in this regard for small finite sets and infinite sets? (For example, the set of all 3-digit binary numbers 000, 001, etc.. contains 2^3= 8 elements, there is nothing at all surprising that there are more than three numbers on this list.. and clearly the inverted diagonal number "111" (for example, or whatever you'd like) is on the list. Three and eight cannot be put into one to one correspondence, yet we can all count to 8. If we allow the use of infinity, the same is true for an infinite list.. sure, for numbers with omega digits, the list of all of them will be more than omega, but the inverted diagonal number, omega-digits long will still be on the list, and we can clearly define it's position if the rest of the list is organized in a way that allows communication. Why is one of the numbers considered to be "uncountable"? Is eight an "uncountable" finite number if it is written as 2^3? Etc). Hope that makes sense.

[Achrononmaster]

@1:14:00 no? Not necessarily? Woodin's V=Ultimate-L conjecture would render this conception immune to Cohen forcing. It seemed pretty simple. It is _only_ Cohen forcing that gives rise to models where CH is false. I think this is a terrific possible resolution for CH. It does not imply there is only one unique set theoretic platonic universe, just that there is this nice rigid one!

[Achrononmaster]

@1:33:00 isn't a shorter answer that Peano Arithmetic does not admit power sets. So you do not have ℝ .

[Achrononmaster]

Just a quick note, which I got from Hugh Woodin. Given two doors you can choose one to go through, forbidding forever entry to the other. One leads to an Oracle for Set Theory, the other to an Oracle for Number Theory (and only NT). No one in their right mind is ever going to choose the NT door, since the ST Oracle can already tell you if Number Theory is consistent or not. The number theory Oracle cannot tell you the converse. If the ST Oracle tells you ST is inconsistent, then you can continue on wherever you go developing NT with no worries. But your stupid cousin who took the NT door will never know whether the greater richness of ST could have been a thing.

[draconyster]

Man I am drunk and I have no idea what this is about, but I feel so smart listening to this and kind of picking up on stuff. I don't know why Youtube suggested me this but I am listening for like an hour now, "general comprehension principle" man, you rock!

[ppss]

Regarding the formulation of the cumulative hierarchy, atemporal or not, is it possible to symmetrically reverse it and start(or ongoing without a single starting point) from the unbounded transfinite levels in order to reach ω and ultimately the empty set? In this context, are finite sets inaccessible from above? The true question I believe is what is a step and what really happens at a limit stage.