not<\/em> a set.<\/li>\n<\/ul>\n<\/p>\n
Back in the early days of set theory, people looked at what we now call “naive set theory”. Naive set theory is useful and interesting – but when you try to do complicated things in it, you encounter a problem which makes things fall apart. Naive set theory fails to distinguish between different kinds of things, and that makes it all to easy to use it create paradoxical statements and structures. <\/p>\n
For example – take that old classic, the liar’s paradox. “This statement is not true.” Everyone who’s ever watched Star Trek has encountered that, right? If it’s true, then that means it must not be true. But if it’s not true, then it’s true. <\/p>\n
The structural problem of that statement can be translated into set theory. It’s a statement that’s really talking about the set<\/em> of truth-bindings of statements: the statement operates as a function<\/em> from statements to true or false. So it’s saying is that it<\/em> is a statement whose meaning is a function which maps some set of statements to either true or false, and that in its mapping of statements to true and false, it maps itself<\/em> to false.<\/p>\n In terms of set theory, what’s a function F that maps from a value v to true or false? It’s a set, SF<\/sub>, where F(v) is true if and only if v∈SF<\/sub>. So a statement like the Liars paradox is really a set of true statement – and what it’s doing is asserting that it is not in its set.<\/p>\n To see the problem with that, we need to look at the complement<\/em> of SF<\/sub>, SF<\/sub>-1<\/sup>. SF<\/sub>-1<\/sup> is the set of values that are not<\/em> members of SF<\/sub>: SF<\/sub>-1<\/sup> = {x : x∉SF<\/sub>}. So when F says that a statement M is not true, what it’s saying is M¬it;SF<\/sub>, and therefore M∈SF<\/sub>-1<\/sup>!<\/p>\n Now, finally, here’s the problem. If M is the liars paradox statement, and F is the functional-meaning of M, then SF<\/sub> is the set of values v where F(v)=true. Is M a member of SF<\/sub>? No – because it says that it’s not true, so it necessarily map F(M) to false. . So then SF<\/sub>-1<\/sup> must<\/em> include M. But if SF<\/sub>-1<\/sup> includes M, then that means that M was true, since what M said was that M would be found in SF<\/sub>-1<\/sup>.<\/p>\n If you keep pushing through the semantics, it comes down to a simple paradoxical construct: the liars set: the set of sets that do not<\/em> include themselves as members, L = { s : s∉s }. The paradox is that if L∈L, then L∉L, but in L∉L then L∈L. <\/p>\n Since a big part of the point of set theory is to provide a simple, general toolkit for building well-founded<\/em> mathematical structures, the ability to break it so easily was a serious problem. But no one wanted to give up set theory: it’s too powerful, and too easy to use, to just give up on it because of this problem. So people set out to find a solution. The easiest one which avoids the problem was proposed by Alan Turing, and refined by Kurt Gödel. The idea is that instead of just having one kind of collection of things called a set, we’ll create two different kinds of collections. Any<\/em> collection of things is called a class<\/em>. A collection of things which is also a member<\/em> of some class is called a set<\/em>. A class which is not<\/em> a set is called a proper class<\/em>.<\/p>\n This makes the liar’s paradox disappear. The reason it’s gone is that in this new formulation, the Liar’s set isn’t a set<\/em>. It’s a proper class. So the class<\/em> of sets that don’t include themselves as members is well-defined, and it doesn’t include itself – in fact, can’t include itself – because it’s not a set.<\/p>\n The way that this connects back to the surreal numbers and nimbers is that the nimbers form a proper class, not a set. And the traditional formulation of a field requires that it be defined in terms of a set<\/em> of values. But the nimbers don’t form a set – they’re a proper class<\/p>\n As it happens, we don’t actually worry about that. In his book on the surreals, Conway handwaves his way past it by pointing out that for fields of numbers, there’s no particular reason to not<\/em> allow the definition of class-fields, so long as we’re clear that they are class-fields and not set-fields. So – the nimbers are a class-field, and for most things that we want to do with them, the fact that they’re a class-field and not a set-field just doesn’t actually make a difference.<\/p>\n","protected":false},"excerpt":{"rendered":"This is something that came up in some of the comments on the recent “nimbers” post, and I thought it was worth promoting to the front, and getting up under an easy-to-find title in the “basics” series. 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