On 24.06.2016 01:58, H. S. Teoh via Digitalmars-d wrote:
On Fri, Jun 24, 2016 at 01:33:46AM +0200, Timon Gehr via Digitalmars-d wrote:
On 24.06.2016 00:53, H. S. Teoh via Digitalmars-d wrote:
Because 0^^0 = 1, and 1 is representable.
E.g. n^^m counts the number of functions from an m-set to an n-set,
and there is exactly one function from {} to {}.
This argument only works for discrete sets.
No, it works for any cardinals n and m.
It doesn't. What is the meaning of m-set and n-set when m and n are
real numbers? How many elements are in a pi-set? How many functions
are there from a pi-set to an e-set? Your "number of functions"
argument only works if m and n are discrete numbers.
...
Most real numbers are not (identified with) cardinals, so I don't see
how that contradicts what I said, or how what you are saying is the same
thing as what you previously stated.
If n and m are reals, you'd need a different argument.
I don't want to argue this at all. x^^0 is an empty product no matter
what set I choose x and 0 from.
And 0^^x is a non-empty product when x != 0. So why should we choose
the limit of x^^0 as opposed to the limit of 0^^x?
...
I don't care about any of the limits. ^^ has an essential discontinuity
at (0,0), so the limits don't need to have a bearing on how you define
the value, but if they do, consider that there is only one direction in
which the limit is 0, and an uncountably infinite number of directions
in which the limit is 1.
Have a look at this plot: http://www.wolframalpha.com/input/?i=x%5E-y
Can you even spot the discontinuity? (I can't.)
0^^0 = 1 is the only reasonable convention, and this is absolutely
painfully obvious from where I stand. What context are you using 'pow'
in that would suggest otherwise?
When computing the limit of x^y as x->0?
...
That limit is 1 if y=0 and 0 if y!=0. I assume you mean the limit of 0^y
as y->0. Why is that important? Why should that single odd direction
ruin it for everyone else?
Also, Andrei's implementation explicitly works on integers anyway.
Even for integers, the limit of x^y as x->0 is 0.
...
This is wrong. Again, I assume that you mean 0^y as y->0.
In any case, on integers, you cannot define this limit without giving
the value at (0,0) in the first place, and the limit is whatever value
you gave.
My point is that the choice is an arbitrary one. It doesn't arise
directly from the mathematics itself. I understand that 0^0 is chosen
to equal 1 in order for certain theorems to be more "aesthetically
beautiful",
Why do you think that is? Again consider my example where a+b is
actually a+b unless a=b=2, in which case it is 5.
Now, the theorem that states that (the original) addition is associative
gets a lot less "aesthetically beautiful". Do you consider the
definition 2+2=4 an 'arbitrary choice'? If you do, then our disagreement
is founded on different ideas of what it means for notation to be
'arbitrary' as opposed to 'well-designed'.
Notation is almost exclusively about aesthetics!
just like 0! is chosen to equal 1 because it makes the
definition of factorial "nicer". But it's still an arbitrary choice.
...
n! = Γ(n+1).
0! = Γ(1) = 1. Consider it arbitrary if you wish.
(The offset by 1 here is IMHO a real example of an unfortunate and
arbitrary choice of notation, but I hope that does not take away from my
real point.)
Anyway, 2+2=4 because this makes the definition of + "nicer". It is not
an arbitrary choice. There is /a reason/ why it is "nicer".
