On 13.02.2013 08:28, Norbert Thiebaud wrote:
On Tue, Feb 12, 2013 at 10:09 PM, Rob Weir <rabas...@gmail.com> wrote:
On Feb 12, 2013, at 10:39 PM, Pedro Giffuni <p...@apache.org> wrote:
(OK, I guess it's better to re-subscribe to the list).
In reply to Norbert Thiebaud*:
In the Power rule, which *is* commonly used for differentiation, we take a
series
of polinomials where n !=0. n is not only different than zero, most importantly,
it is a constant.
[...]
For those interested in the actual Math... in Math words have meaning
and that meaning have often context. let me develop a bit the notion
of 'form' mentioned earlier:
for instance in the expression 'in an indeterminate form', there is
'form' and it matter because in the context of determining extension
by continuity of a function, there are certain case where you can
transform you equation into another 'form' but if these transformation
lead you to an 'indeterminate form', you have to find another
transformation to continue...
hence h = f^g with f(x)->0 x->inf and g(x)->0 x->inf then -- once it
is establish that h actually converge in the operating set, and that
is another topic altogether -- lim h(x) x->0 = (lim f)^(lim g).
passing 'to the limit' in each term would yield 0^0 with is a
indeterminable 'form' (not a value, not a number, not claimed to be
the result of a calculation of power(0,0), but a 'form' of the
equation that is indeterminate...) at which point you cannot conclude,
what the limit is. What a mathematician is to do is to 'trans-form'
the original h in such a way that it lead him to a path to an actual
value.
in other words that is a very specific meaning for a very specific
subset of mathematics, that does not conflict with defining power(0,0)
= 1.
wrt to the 'context' of the quote I gave earlier:
"Proposition 9: : Let X and Y be two sets, a and b their respective
cardinals, then the set X{superscript Y} has cardinal a {superscript
b}. "
( I will use x^y here from now on to note x {superscript y} for readability )
"Porposition 11: Let a be a cardinal. then a^0 = 1, a^1 = a, 1^a = 1;
and 0^a = 0 if a != 0
For there exist a unique mapping of 'empty-set' into any given set
(namely, the mapping whose graph is the empty set); the set of
mappings of a set consisting of a single element into an arbitrary set
X is equipotent to X (Chapter II, pragraph 5.3); there exist a unique
mapping of an arbitrary set into a set consisting of a single element;
and finally there is not mapping of a non-empty set into the
empty-set;
* Note in particular that 0^0 = 1
"
Here is the full context of the quote. And if you think you have a
proof that there is a mathematical error there, by all means, rush to
your local university, as surely proving that half-way to the first
volume, on set theory, of a body of work that is acclaimed for it's
rigor and aim at grounding the entire field of mathematics soundly in
the rigor of set theory, there is an 'error', will surely promptly get
you a PhD in math... since that has escaped the attentive scrutiny and
peer review of the entire world of mathematicians for decades...
Thanks for providing some real math into this thread. I don't claim to
have understood everything you write, but still, I learned something new.
And I would never have expected to hear the name of Bourbaki on the dev
list. Thanks for that, too.
-Andre
