Ramble, Part 72: Get Ready, Get Set...
Jun. 6th, 2009 02:39 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowBoth the Logicist and Formalist factions had an interest in establishing a set of axioms for set theory; the paradoxes that Cantor's set theory led to were unsatisfying to both. Several sets of axioms have been suggested; I'm going to describe one of the first, devised by Ernst Zermelo in 1908, under the cut.
The first point that needs to be made is that set theory was intended to be foundational; in other words, for these purposes everything mathematical is a set. (What? Even numbers and points? Yes; I'll talk about that a little later on.) The more difficult step is deciding what sets there are. Zermelo put forward eight axioms, to answer this question. (Actually, there were nine, but the ninth is, mm, controversial enough that it will get its own post.) The axioms themselves, in keeping with either school of thought, are simply strings of logical symbols; I will, instead, give approximate verbal translations.
1) Most of the axioms describe how to construct new sets from old, but we need a place to start; the first axiom declares that the empty set exists. (More formally: "there is a set x such that, for all u, u is not an element of x".)
2) The definition of any set is extensional; that is, a set is completely determined by what its elements are. (For example, the set of four-sided triangles is the same as the set of circular squares, both being empty.) (More formally: "for all x and y, if, for all u, u is in x if and only if u is in y, then x equals y".)
3) We need more sets, so we declare that, given any two sets, there is another set whose elements are precisely those two sets. (Note that we do not assume that the two sets are different; in particular, if x is any set, then so is {x}.)
4) We can take unions: if we have a set x, the union of the elements of x is also a set.
5) We have power sets: if x is a set, then there is a set y whose elements are the subsets of x. (Since, as Cantor showed, the power set of x is strictly larger than x, this lets us have arbitrarily large sets.)
6) We want to have sets associated with each predicate F(x), but we can't be indiscriminate about it - that path leads to Russell's Paradox. Instead, we say that, if y is a set, there is a set whose elements are the elements of y which satisfy the predicate F(x). In other words, we don't have a set of all red objects, but we can have a set of all the red objects in the set y.
7) This one, the "regularity" axiom, is tricky. It says this: given any predicate F(x), if there is any set satisfying the predicate, then there is a "smallest" set satisfying it - that is, there is an x which satisfies the predicate, but none of its elements do. This prevents, in particular, any set from having itself as an element. (Let F(x) be the property "x is an element of x". If there were any set satisfying F(x), it would contain an element - itself - which also satisfied F(x); so there must not be any such set, by the axiom.)
8) This one, the "axiom of infinity", is crucial; without it, we can't even get the Peano axioms to work. It says, basically, that there is an infinite set. Specifically, it says that there is a nonempty set with no largest element - if x is an element of that set, there is some y, also an element of that set, which is not x but contains x as a subset.
These axioms were defined with two aims in view: first, they had to be capacious enough to allow for Peano's axioms, and second, they had to avoid the various paradoxes that had been discovered - Russell's, Burali-Forti's, and others which had popped up since. There remained the problem that, even if they avoided the known paradoxes, it was possible that there were other paradoxes to which they were still vulnerable. Hilbert, of course, hoped that his metamathematics would allow a proof that this, or some other set of axioms, would avoid all possible paradoxes.
Oh, yes: if everything is a set, then what are numbers? Sets, of course. We define 0 to mean the empty set. 1 is the set {0}. 2 is the set {0,1}; 3 is {0,1,2}; and so on. In general, once we've defined n, the successor of n is the set whose elements are the elements of n, together with n itself.
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The first point that needs to be made is that set theory was intended to be foundational; in other words, for these purposes everything mathematical is a set. (What? Even numbers and points? Yes; I'll talk about that a little later on.) The more difficult step is deciding what sets there are. Zermelo put forward eight axioms, to answer this question. (Actually, there were nine, but the ninth is, mm, controversial enough that it will get its own post.) The axioms themselves, in keeping with either school of thought, are simply strings of logical symbols; I will, instead, give approximate verbal translations.
1) Most of the axioms describe how to construct new sets from old, but we need a place to start; the first axiom declares that the empty set exists. (More formally: "there is a set x such that, for all u, u is not an element of x".)
2) The definition of any set is extensional; that is, a set is completely determined by what its elements are. (For example, the set of four-sided triangles is the same as the set of circular squares, both being empty.) (More formally: "for all x and y, if, for all u, u is in x if and only if u is in y, then x equals y".)
3) We need more sets, so we declare that, given any two sets, there is another set whose elements are precisely those two sets. (Note that we do not assume that the two sets are different; in particular, if x is any set, then so is {x}.)
4) We can take unions: if we have a set x, the union of the elements of x is also a set.
5) We have power sets: if x is a set, then there is a set y whose elements are the subsets of x. (Since, as Cantor showed, the power set of x is strictly larger than x, this lets us have arbitrarily large sets.)
6) We want to have sets associated with each predicate F(x), but we can't be indiscriminate about it - that path leads to Russell's Paradox. Instead, we say that, if y is a set, there is a set whose elements are the elements of y which satisfy the predicate F(x). In other words, we don't have a set of all red objects, but we can have a set of all the red objects in the set y.
7) This one, the "regularity" axiom, is tricky. It says this: given any predicate F(x), if there is any set satisfying the predicate, then there is a "smallest" set satisfying it - that is, there is an x which satisfies the predicate, but none of its elements do. This prevents, in particular, any set from having itself as an element. (Let F(x) be the property "x is an element of x". If there were any set satisfying F(x), it would contain an element - itself - which also satisfied F(x); so there must not be any such set, by the axiom.)
8) This one, the "axiom of infinity", is crucial; without it, we can't even get the Peano axioms to work. It says, basically, that there is an infinite set. Specifically, it says that there is a nonempty set with no largest element - if x is an element of that set, there is some y, also an element of that set, which is not x but contains x as a subset.
These axioms were defined with two aims in view: first, they had to be capacious enough to allow for Peano's axioms, and second, they had to avoid the various paradoxes that had been discovered - Russell's, Burali-Forti's, and others which had popped up since. There remained the problem that, even if they avoided the known paradoxes, it was possible that there were other paradoxes to which they were still vulnerable. Hilbert, of course, hoped that his metamathematics would allow a proof that this, or some other set of axioms, would avoid all possible paradoxes.
Oh, yes: if everything is a set, then what are numbers? Sets, of course. We define 0 to mean the empty set. 1 is the set {0}. 2 is the set {0,1}; 3 is {0,1,2}; and so on. In general, once we've defined n, the successor of n is the set whose elements are the elements of n, together with n itself.
That this is a set can be proven using Zermelo's axioms. First, we use Axiom 5 to create the power set of n. Next, we define the predicate F(x) to mean "either x is an element of n, or x equals n". An application of Axiom 6, using the power set of n and this predicate, produces the desired successor.
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