FOM: {n: n notin f(n)}
Dean Buckner
Dean.Buckner at btopenworld.com
Wed Aug 28 13:35:39 EDT 2002
Buckner:
>>1. Let f be a function, f: N -> S, from objects to sets of objects.
>>Suppose f is "onto"
>>2. Suppose that there is a set M, of all objects satisfying "x is not in
f(x)".
>>3. For some m, M = f(m) (since f is onto)
>>4. Forall x, x in f(x) iff x not in f(m) (from 2)
>>5. Thus if m is in M, it is not in M, and if it is not in M, it is in M
(instantiating m)
>>Since (5) is a contradiction, it's standard to reject assumption (1): that
there is such a mapping from things to sets of things.
>>But why can't we reject assumption (2), that there really is a set of all
numbers > [actually I meant "things"] satisfying the predicate "x is not in
f(x)"?
Heck:
> Obviously, one could, in principle, reject (2). But the question is what
> else you would have to reject to reject it.
It's always good to hear from Richard as he has an elegant and economical
way of putting his things. Check out also his website which has some new
and interesting additions. Including a section under development promising
a closer look at his record collection, indeed. No clues yet as to what
lies therein, except an F.O.M. snippet some years back where Richard refers
to the cardinality of the concept "disco records in my [Richard's]
collection" (zero). Clearly he would have little interest in my own
collection of dance music from all eras.
Anyway. Richard's reply to my posting uses the time-honoured technique of
re-writing one's position then drawing unsuitable or nasty conclusions from
it. Not that I'm wholly uncomfortable with these, but I'm not happy with
Richard's re-write, that has me proposing a relation R such that, for each
subconcept G of F, there is some x such that Fx where aR__ just is G.
(ER)(G)[(x)(Gx --> Fx) --> (Ey)(Fy & (z)(Gz <--> yRz))].
Let me translate my actual position into English.
>>1. Let f be a function, f: N -> S, from objects to sets of objects.
Suppose f is "onto"
I'm uncomfortable with set-theoretic way of putting things precisely because
it commits one to nasty things like infinite sets of objects. I'd translate
this as: there is a way ("f") of associating things of a certain kind (say
natural numbers) with collections of objects of the same kind. The function
is "1-1 and onto" i.e. we can pair up the things and the collections. I'm
using the word "kind" and collection" deliberately. By a "kind" of thing I
mean roughly any thing or things that satisfy the same predicate, by
collection I mean a "finite set" (but I hold that "finite" is otiose
because, really, there aren't any other kinds of set).
>>2. Suppose that there is a set M, of all objects satisfying "x is not in
f(x)".
I go on to deny this, but let me walk through why I do this, in plain
English. As I said, I don't believe in "infinite sets". But there can of
course be an infinite number of things of a certain kind. Meaning, take any
collection of things of that kind, and you find there is at least one thing,
of that kind, not in that collection. Meaning there is another collection
comprising the first collection + the thing you found, meaning here is at
least one further thing & so on ad infinitum.
For example, try associating things (people), with sets of things as
follows
Armstrong { Armstrong, Basie, Calloway}
Basie {Gillespie, Parker, Miller)
Calloway {Locke, Berkeley, Hume}
Dorsey {Beck, Page, Clapton }
Ellington {Chopin, Schumann, Schubert }
You soon see that the collection of things that are not in the collection
they are associated with, is not one of the collections on the list. If
there are a finite number of things, there must be more collections than
things (obviously). But if there are a infinite number of things, why can't
we just go on matching things with collections "for ever"? I don't see why
not. So long as there isn't a collection of *all* the things satisfying "x
is not in f(x)", we are perfectly all right. That's why I deny (2).
Richard argues that this will be controversial, or at least involves denying
something " not typically taken to be controversial" (Axiom of Separation).
As a matter of fact, I would probably as card-carrying ultra-nominalist want
deny something like this, but not here. Richard has me denying it
as the result of accepting something this
(*) (ER)(G)[(x)(Gx --> Fx) --> (Ey)(Fy & (z)(Gz <--> yRz))]
yet I'm not sure I do accept it. My position is that there are two
radically different kinds of predication, as follows:
Some people were at that bar. Some of them were drunk. Some other
people at the bar were laughing at them.
We have "x is a person at the bar", which can be satisfied by any one of
either of the groups of people at the bar, and we have "x was one of them"
which can only be satisfied by one of the first group mentioned. I'd hold
that the first kind of predicate can be true of an infinite number of things
(and is how we get to the idea of infinity in the first place), and
that the second kind can be true only of a "finite" number of individuals.
To represent my position correctly, we would have to recognise these two
different types of predication, which (*) doesn't. And it's really (*),
isn't it, which allows us to get from my (2) above to the contradictory (5)?
But that's a longer story. Richard defers to some of the issues
involved in his interesting essay on his website (On Singular Terms (?)),
though he says he no longer holds the position argued for.
As an exercise I'd challenge Richard or any other FOM'er to translate the
simple 3-sentence story above into formal set-theoretical terms, in a way
that preserves the logic of the story.
Dean
Dean Buckner
London
ENGLAND
Work 020 7676 1750
Home 020 8788 4273
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