If you can remove any agency on his part the logical structure can continue to eat itself; because, lets face it, the proposition is fucking nuts.

Ah yes, what *does* it mean to have shaved ones self? ]]>

My hunch is that if we could look directly into our own blind spot and *see* something, this is the kind of work that would get us there.

]]>Second, I should mention that I intended no real criticism with my comments. I mostly took the thread as a chance to give a little bit of background information about foundational stuff that most people who are just getting into higher math don’t usually hear too much about. I knew about set theory for a pretty long time before I learned anything about category theory, and the latter is what really got me into higher math. Exposing future math geeks to interesting topics is worth going a little off topic.

]]>My very favourite thing about math is that in the UK they say “maths” but here in the US we just say “math”. I like “maths” better. Also, shrimps.

]]>You think that’s bad? Try mapping the real numbers into the points on a plane.

]]>But I don’t think their objection is based on anything that deep. I went to elementary school in the 1960s, so I got to learn arithmetic the old-fashioned way and then learn it over again through New Math, including set theory, and my guess is that *somebody’s kid brought home their New Math homework and mom&dad didn’t have a clue what it meant,* and they decided that New Math in general and Set Theory in particular teaches rebellious children to challenge their parents’ authority by thinking they know things their parents don’t*.

It’s possible that they also blame Bertrand Russell for some of it, or would if they could understand him. And set theory gets into paradoxes about whether the barber is a member of the set of people that don’t shave themselves or not, and paradox bothers people like that.

(*My parents didn’t have that kind of problem; my mom borrowed my 7th grade math book to prep for the GREs when she went back to college, since set theory hadn’t been something she’d learned in high school or as an undergrad French major.)

]]>Or both!

]]>Mathematicians go mad, and cashiers; but creative artists very seldom. I

am not, as will be seen, in any sense attacking logic: I only say that

this danger does lie in logic, not in imagination.” David Foster Wallace replied that stories of mad, suicidal mathematicians are overemphasized, especially in regards to Cantor and the other set theorists – and then, after writing a truly brilliant book on the subject, went on to kill himself. ]]>

Correspondences can be used to compare the sizes of much larger collections than six goats—including infinite collections. The rule is that, if a correspondence exists between two collections, then they have the same size. If not, then one must be bigger. For example, the collection of all natural numbers {1,2,3,4,…} contains the collection of all multiples of five {5,10,15,20,…}. At first glance, this seems to indicate that the collection of natural numbers is larger than the collection of multiples of five. But in fact they are equal in size: every natural number can be paired uniquely with a multiple of five such that no number in either collection remains unpaired. One such correspondence would involve the number 1 pairing with 5, 2 with 10, and so on.

I’m sure I’d have gotten this question out of my system if I’d taken a few more math courses, but I have a problem completely getting my head around this.

I get if we pair 1 with 5, 2 with 10, etc. that we never run out of numbers in category A (goats) to tie to category B (trees). But what if I choose another kind of pairing? What if I choose to pair 5 with 5, 10 with 10, etc. I also tie a goat to every tree, but now for every tree I also have 4 extra goats that got passed over. It seems like the equivalence only works for an arbitrary method of correspondence.

]]>Hail Eris! All hail Discordia! ]]>

that new, as yet unknown, axioms will show the Hypothesis to be true or false.” (It also contradicts the statement that CH has been shown to be undeterminable from the usual bases of Set Theory.) If we add axioms to any of the foundations, we will *create* a mathematics in which CH is true, or is false–we won’t be showing the hypothesis to be true or false. Essentially, adding different axioms could split mathematics into one field where CH is true and another where it is false. This says nothing about the real world or the naturalness of the conjecture: what Godel and Erdos proved was that neither the assertion or the negation of CH can be shown to be inconsistent with current foundational axioms (and hence, neither is inconsistent with anything that follows from those axioms.) ]]>

But existence is damn sure part of our normal set theory talk, and insofar as we expect our sets to be determined by the axioms, we expect there to really be an answer as to which sets exist.

It’s funny – the way we talk about mathematics makes it easy to slip into a kind of Platonism. We imagine our theories and their axioms as simply describing a domain of pre-existent objects and allowing us to prove things about them. We like to think of them as having this independent existence and that all the facts about them are set in stone in some way or another. But there’s a sense in which what we’ve learned is that you can never quite set everything in stone; some facts are just never determined. It’s strange to find out that there’s just no fact of the matter when it comes to a system you’re working with – you thought set theory gave you a relatively concrete domain of objects, but your set theory hasn’t decided if certain perfectly reasonable sets exist or not.

For those that take the sort of Platonistic talk of mathematics seriously, there might really be an answer about whether CH holds, but this just makes our axiomatization look too weak to capture important features of the theory of sets – in fact, it makes *every* axiomatization look weak. It’s getting used to this other viewpoint where things are only definite up to proof that takes a bit of effort for a lot of people to wrap their heads around.

]]>Is not surprising, really, that axiom systems are incomplete. People for ages tried to prove Euclid’s fifth postulate from the other four, since it was seen as complex, but it was shown in the 18th or 19th century that the fifth postulate was undecidable from the other four. (Euclid’s fifth, also, was about infinity, too, btw – what does it mean that two lines never intersect, after all?) ]]>

But neither brings any resolution to the Continuum Hypothesis, so why did you bring them up?

]]>However such maths don’t seem to be a magic bullet.

]]>In all seriousness, there’s lots of stuff going on in set theory, but more than being an interesting domain in its own right, it’s mostly just a necessary tool for doing a lot of higher math. It’s a kind of basic language out of which mathematical objects can be constructed.

Beyond this role, set theory was intended to be a kind of unifying framework that aided interdisciplinary work – since the 60s, category theory (and now higher category theory) has sort of slowly filling that role. There are certain things that are much more natural from a category theoretic perspective. The best one sentence description of the difference that I’ve heard is this: when it comes to mathematical structures, set theory is good for answering the question “How is it made?” while category theory is good for answering the question “How does it work?”

Some mathematicians find set theory a bit annoying as a foundational theory in certain regards. There’s ‘too much structure’ in a certain sense that can allow stupid distinctions and hide important ones. There have been some recent and very interesting attempts to replace it. The most promising (from my perspective) is the whole univalent foundations program. The idea is to get rid of all the excess structure and have mathematical objects and domains be only describable in terms of distinctions that actually matter (rather than the too-fine-grained distinctions that come up in set theory). It’s all rather fascinating, but I haven’t been totally convinced they’ve created a workable meta-language yet. Time will tell.

]]>