On 07/31/2011 09:16 PM, Nate Hayes wrote:
Arnold Neumaier wrote:
On 07/28/2011 10:30 PM, Nate Hayes wrote:
Arnold Neumaier wrote:
I had given an example where Motion 27 gives erroneous results for
decorated intersections:
The expression f(x)= x/((x+1) intersect x^2) is undefined for any x in
[1,3], but Definition 7 claims a safe answer for f([1,3]).
Arnold, this is not any example of erroneous results.
For example, the expression is undefined
for
x=1, for x=2, and for x=3, although all these are in [1,3].
u(1) = (1+1) = 2
v(1) = 1^2 = 1
f(1) = 1/(u(1) intersect v(1))
= 1/(2 intersect 1)
= 1/empty
= empty
The set-theoretic intersection of u(1) and v(1) is empty and is not
undefined (it is safe).
Never before I was told that the set theoretic intersection of two numbers
should be empty.
There are three main traditions for defining numbers.
In Zermelo-Fraenkel set theory, the intersection of two cardinal numbers
is the smaller one.
In Dedekind's definition of real numbers as , the intersection of two real
numbers is also the smaller one.
In Peano's definition of natural number, the intersection of two natural
numbers is undefined.
In no case, the intersection is empty.