From jcowan@reutershealth.com Fri Mar 3 09:45:05 2000
X-Digest-Num: 382
Message-ID: <44114.382.2166.959273826@eGroups.com>
Date: Fri, 03 Mar 2000 12:45:05 -0500
From: John Cowan
Subject: Re: jimc on MEX
And Rosta wrote:
> I've never had any trouble with defining positive integers in this way,
> but feasibility of "the obvious extension" is not at all apparent to my
> unmathematical mind. As far as I can see, only positive integers are
> valid set cardinalities, while other numbers can be derived by operations
> on integers but not on sets.
Okay, here we go. Notation: {x, y, z} means the set with members x, y, z.
As explained before, the natural numbers (positive integers) are identified with
as the set of all sets with appropriate cardinality. Thus 5 is the set of all
fivesomes, and 3 is the set of all triplets. This particular convention is called
"Frege natural numbers" after its inventor. It is extended to other numbers
as follows:
0 is just {{}}, the set of all null sets (there is only one).
Negative numbers are construed by a trick such as -5 = {0, 5}.
To get the rational numbers, you also need the notion of an ordered pair, notated
(a, b), which is identified with {a, {a, b}}: the set containing a set of the two
members and also the first member. Given this set, you can reconstruct the
pair: the set has two members, one of which is a sub-member of the other.
That's the first element of the pair. The second element is then easy to discover.
This convention extends to ordered n-tuples by construing (a, b, c, ...) as
(a, (b, (c, ...))).
Now we identify 5/3 with the ordered pair (5, 3) which is just the set {5, {5, 3}}.
Real numbers are represented as infinite sets: \pi is the infinite
ordered sequence (3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ...),
representing a sequence of rationals the limit of which is \pi. By 3.1
here I mean 31/10, which is {31, {31, 10}}. We have to put some simple
restrictions on the sequences to prevent getting a representation for
infinity, and to avoid multiple representations for 2.0000000... as (2, 2, 2, 2, ...)
and (1, 1.9, 1.99, 1.999, 1.9999, ...). I omit the details here.
Complex numbers are just ordered pairs of real numbers: 3+4i =
(3.0000...., 4.0000...).
Many other schemes are possible, with different representation tricks at
different levels. Von Neumann's positive integers, for example, use the
conventions:
0 = {},
1 = {0} = {{}},
2 = {0, 1} = {{}, {{}}},
3 = {0, 1, 2} = {{}, {{}}, {{}, {{}}}}
etc. etc. They are better for some purposes and worse for others: see below.
One nice property of the von Neumann integers is that the cardinality of the
set just *is* the integer: the number N has N members. The Frege integers
discussed above are a little more complex: every member of the number N
has N members.
> I don't understand this. That is, I don't understand how classes can
> be summed, especially not in a way that gives the results you describe.
First we define + on numbers recursively, by Peano's axioms (invented by
a man named Dedekind, curiously):
if X = 0, X + Y = Y;
else X + succ(Y) = succ(X + Y)
where "succ" is a function meaning "successor of". We can see that this
is true by following the recursion:
2 + 2 is 2 + succ(1),
which is succ(2 + 1),
which is succ(succ(2 + 0)),
which is succ(succ(2)),
which is succ(3),
which is 4.
How do we interpret the "succ" function in terms of sets? Well, let us use the von
Neumann interpretation of numbers instead of the Frege one: 0 is {}, and the
successor of a number has as its members, (a) of the members of that number
and (b) the number itself.
We can generalize this definition to any set: the successor of {Frank, George,
Peter} is {Frank, George, Peter, {Frank, George, Peter}}. Unfortunately,
{Frank, George, Peter} is not the successor of any set, so it is no use
asking what the sum of {Frank, George, Peter} and {Mary, Louise, Catherine}
might be. But that doesn't matter, because we only care about + as applied
to numbers, and our method for doing so always generates the correct (von Neumann)
number. We can also define + for sets that are Frege numbers, but I won't
do so, as the principle is the same and the details are more tedious.
> Can't Lojban already do the equivalent by using the ordinary apparatus
> or integers and variables plus the maths gismu?
Yes, and even integers you can skip, by construing "li pa" as "lo'i te pamoi".
> > Units of measure are defined to multiply the second place argument (a
> > number or other expression) by the unit. They would normally be used in
> > restrictive subordinate clauses. "I weigh 70 kilograms" in Lojban would
> > come out /mi tilju poi kilgrake lo'i mei ze no/.
>
> ze no mei?
A little guaspi leaking in, where the equivalents of mei and moi are
prefixes as in Chinese rather than suffixes as in English. Makes more
sense, really.
> And that poi doesn't seem to make sense.
It isn't grammatical.
I think the right way is to say that *my weight* (a quantity abstraction) is-in-kilos
the number 70: "le ni mi tilju cu kilgra li 70".
> Last time I read the guaspi file (admittedly, some years ago now) I found that
> you seem to share my unfortunate gift for writing expositions that are
> supposed to be, but aren't, intelligible to the general reader ;-)
I guess I must be a special reader, then.
--
Schlingt dreifach einen Kreis vom dies! || John Cowan
Schliesst euer Aug vor heiliger Schau, || http://www.reutershealth.com
Denn er genoss vom Honig-Tau, || http://www.ccil.org/~cowan
Und trank die Milch vom Paradies. -- Coleridge (tr. Politzer)