[FOM] squaring a set
M. Randall Holmes
holmes at diamond.boisestate.edu
Thu Sep 25 13:39:20 EDT 2003
In response to Slater on squaring sets:
The short answer is, sure you can square a set -- if that set happens
to be a number. Subtyping does occur...
The long answer is that it's fairly easy to counter this kind of
objection in the case of the Frege natural numbers. Arithmetic
operations on Frege natural numbers are fairly simple set-theoretical
operations, and can often be extended to general sets (for example,
they extend very smoothly to general cardinal numbers).
Note that in the development of the definition of the set of Frege
natural numbers (given in my first post) the successor operation had
to be defined for all sets before it could be used to define the set
of all natural numbers.
For example, the square of a Frege natural number n is the set of all
sets which are equinumerous with the cartesian product A x A for some
A belonging to n. This is an operation proper to sets, and it is also
clearly naturally related to what we do when we square a number.
The analogous definition for the von Neumann natural numbers is not
quite as satisfactory -- n^2 is the von Neumann natural number
equinumerous with the cartesian product of n with itself; here the
conventional choice of the von Neumann natural numbers as numerals has
to be explicitly referenced, which is not the case with the definition
for Frege natural numbers.
You can't take the square root of a natural number per se (unless it
is a perfect square) -- that's a category mistake in _mathematics_
(nothing to do with set theory!) The real number 3 does have a square
root, of course. I haven't offered a natural definition for the real
numbers as sets -- it is clear that there are nice implementations,
but there doesn't seem to be a canonical one. Whatever the real
number 3 is, it probably isn't the same as the natural number 3
(though it is convenient to use the same name for it -- a phenomenon
known to computer scientists as "overloading").
I do understand and often agree with the kind of objection that
Slater is raising (notice that I do agree in the case of the usual
definitions of the reals -- these are very sensitive to obviously
conventional details of the definitions).
If one believes, as Frege did, that sets are extensions of properties
(the paradoxes can be understood as telling you that you can't expect
just _any_ property to have an extension, but doesn't show this idea
to be incoherent; the consistency of NFU tells us that this kind of
property can consistently be taken to have an extension), and that
finite cardinal numbers can be understood as properties of finite
sets, then it falls out: 3 is reasonably to be identified with the set
of all sets with 3 elements. Further, _all_ talk about 3 as a set
translates into talk about the actual number 3 under this
interpretation (it has no extraneous artificial properties). The only
interesting facts about a set are what elements it has: the elements
of 3 (understood as a Frege natural number) are exactly the sets with
three elements; there is no extraneous information of a conventional
nature in the structure of the set (as there is in a set constructed
to code a real number).
We try this out on a strange sentence which is true in the Frege
natural numbers and not in the von Neumann natural numbers: "0 is an
element of 1". This translates to "0 has one element", and further
translates to "there is only one set with 0 elements", which is
actually a fact about zero as a property of finite sets.
One can come close to this kind of analysis for the von Neumann
natural numbers. The von Neumann natural number n as a set contains
the elements 0,...,n-1. These do have a relation to n which can be
expressed entirely in terms proper to natural numbers: they are the
natural numbers less than n. So if we understand the relationship of
elementhood on sets restricted to natural numbers to coincide with
"less than", the von Neumann definition of the natural numbers fall
out, and all information in the set theoretical structure of a natural
number translates directly to information about the natural number
_qua_ natural number. The suspect statement "1 is an element of 3" is
simply an odd way to say "1 is less than 3". But there is no natural
reason to identify "less than" as a case of "is an element of": we
only see that it is convenient after the fact. The corresponding move
in the Frege definition of the natural numbers is to identify "has the
property..." as a case of (or as synonymous with) "is an element
of...", which is a _much_ more defensible move from a philosophical
standpoint.
Most definitions of mathematical objects as sets are clearly
conventional in nature: this is _not_ altogether clear in the
case of the Frege definition of the natural numbers.
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes
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