[FOM] what numbers can be
Randall Holmes
holmes at diamond.boisestate.edu
Fri Sep 19 12:40:04 EDT 2003
Responding to Slater's recent post:
It is quite clear that it is artificial to identify the natural numbers
with the particular sets \emptyset = 0, {0}=1, {0,1} =2, etc.
These are pretty clearly conventional representations of numbers,
"numerals" if you will. (the convention is a good one, though -- it
extends very nicely to infinite ordinals).
There is a set theoretical definition of the natural numbers which is
arguably not artificial at all. This is Frege's definition which has
the effect of defining each concrete natural number n as the set of
all sets with n elements (this is of course not the formal definition
-- it would be circular -- but a formal definition with this effect is
straightforward).
The difficulty is that there are no such sets in ZFC -- but ZFC is not
the only set theory. In NFU (Quine's New Foundations with urelements,
shown to be consistent by Jensen) the Frege natural numbers do exist
individually as sets, and so does the set of Frege natural numbers,
and in fact all the machinery needed for the usual mathematical
development of numbers.
To understand "the set A has three elements" as meaning "A belongs to
3" is much more natural than to understand it as meaning "there is a
bijection between A and {0,1,2}". It is natural if one supposes that
sets are associated with (selected) properties, and observes that
(finite) cardinal number is a property of (finite) sets. (Of course this
implies an extensional view of properties, and one also has to consider
the reasons why only selected properties can have extensions...)
I'm not saying that natural numbers "are" the Frege natural numbers in
a Platonically true set theory which admits them; but the possibility
of defining numbers in this way suggests that it is not so very
obvious that numbers are not sets.
Conclusion: there is no knockdown argument for Slater's assertion that
numbers are not sets.
I append a brief summary of the definition of the Frege natural numbers
as sets:
0 is defined as {{}} (the set whose only element is the empty set).
for _any_ set A (remember that we don't know what a number is yet)
we define A+1 as {a U {y} | a \in A and y \not\in a} (the set of disjoint
unions of elements of A with singletons.
Note that 0+1 =[def] 1 will be the set of all singletons (one-element
sets), 1+1 =[def] 2 will be the set of all 2-element sets, and so
forth.
Define the set N of natural numbers as the intersection of all
sets which contain 0 and are closed under the successor operation.
All of these set constructions work in NFU. Note that we succeed in
defining each concrete natural number n as the set of all sets with
n elements without danger of circularity.
I don't actually maintain that the natural numbers "are" the Frege
natural numbers thus defined -- but I do think that if the natural
numbers are "really" sets, then these are the sets that they should
be.
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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