Natural numbers
sowa <sowa@turing.pacss.binghamton.edu>
Reply-To: cg@cs.umn.edu
Date: Mon, 15 Nov 93 19:05:10 EST
From: sowa <sowa@turing.pacss.binghamton.edu>
Message-id: <9311160005.AA01473@turing.pacss.binghamton.edu>
To: cg@cs.umn.edu, interlingua@isi.edu
Subject: Natural numbers
Cc: sowa@turing.pacss.binghamton.edu
Fritz,
I'm sorry for the rather abrupt ending to my last note. I received a
message saying that Turing (our computer) was going down, and I typed
"quit", which is not my normal signature.
Many of your examples can be defined rather simply if you take the
natural numbers as given. Cantor & Co. tried to start with set theory
and construct the natural numbers. They gave us such odd definitions
as saying that an integer like 5 is defined as the set of all sets that
have 5 elements. The 19th century mathematician Kronecker found such
definitions rather repulsive, and he made the remark "God gave us
the integers -- all else is man made!"
I sympathize with Kronecker, and I strongly believe that the
natural numbers are far simpler than any of the supposed "explanations"
or "definitions" of them that have been proposed using set theory.
If you grant me the natural numbers, I can define "finite"
"connected" and many other such things in a very simple first-order
formula.
For example, the HOL definition of "finite set" is one that
cannot be put into a 1-to-1 correspondence with any of its
proper subsets. That is very clever, but also stupid.
If I have a sort called Integer, I can just say that a set S
is finite iff (En:integer)count(S,n).
Then I can define connectedness just as easily. I define
path length in the usual way, and then I say that a graph G
is connected iff for any two nodes x and y, there exists a
natural number n where n=pathlength(x,y).
So I'm with Kronecker. Give me the integers, and I can construct
anything I want. You can keep your crazy transfinite sets.
John