- From: "William Hughes" <wpihughes@xxxxxxxxxxx>
- Date: 2 Dec 2006 18:50:40 -0800
Stephen Montgomery-Smith wrote:
It only shows that subtraction depends of the properness of subsethood,
while addition do not?
I think there is a mistake in infinite cardinal arthmetic,
one day will be revealed.
You are not being quite fair here. You are OK with the notion of odd or
even being undefined for aleph-null (this was your response to Doug).
So what is then wrong with the notion of substraction being ill-defined
for infinite cardinals?
there is certainly week point about cantor's cardinality, I have not
doubt in this at all. First : it is too counter-intuitive to have any
usefullness other than academic luxury. As I say always Cantor's
cardinality is too counter-intuitive to be true.
The concept of infinity itself is questionable? even its definition by
dedekind is not so strong one ( an infinite set is a set injectable to
some proper subset of it ) this is not a good definition, it is biased
from the start.
In what way is this "biased". The charaterization "there is no
to a proper subset" is a simple characterization of a finite set.
It will not help to try to find another characterization. Either it
be equivalent to this, or it will not define what we intuitively
to be a finite set. The Dedekind definition of infinite follows from
"an infinite set is one that is not finite". Note it does not
depend on ordering.
By AC, every set is well ordered. So it is easier to define " infinite
set" as: any set is said to be infinite iff when its members are
arranged in a well ordered manner, the successor of every member of it
is in it.
while a finite set is any set if its members are arranged in a well
ordered manner then there exists a member of it that do not have a
sucessor in it and that member is called the last member .
No, well orderings are not unique. Even if we assume that a set
can be well ordered (as you point out we need AC for this to hold
for every set) a set (e.g. the natural numbers) can have two
well orderings, one with a last element, and one without a last
These are easier approaches to define a finite and an infinite set.
You need to characterize a finite set. You then
negate this characterization to get a characterization of infinite set.
The only way to
get a simpler definition is to find a simpler characterization of
finite set than "a set which does not admit a bijection to a proper
Then we go and define bijection, injection and serjection. and from
these definitions and the above definition of a finite set and an
infinite set , we might derive Dedekind definition as a theorum.
anyhow, even then I am not quite satisfied with that Dedekindian
definition of an infinite set. It leads to all counter-intuitiveness
that makes cantor's cardinality something hard to beleive that it is
Learn to live with it. If you accept that changing the name of every
element of a set does not change its size, then you have to
accept that if there is a bijection from A to B, then
A and B have the same size.
- William Hughes