- So I thank everybody very much for explanation. Mathematics has more hidden

beauties I will ever learn, and as I am not in a professional way dealing

with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

Best regrads

Michael

----- Original Message -----

From: "Peter Kosinar" <goober@...>

To: "Dr. Michael Paridon" <dr.m.paridon@...>

Cc: <primenumbers@yahoogroups.com>

Sent: Monday, August 07, 2006 4:19 PM

Subject: Re: [PrimeNumbers] Re: Is 2 a prime

> > Sorry, if I did not post right. I was not about to cause any

> > inconvenience.

> >

> > I did not say Jack's definition is not possible. I think is not the

> > definition mostly used, as usally division by zero is not defined. As

> > well as 0^0, if I remember right. Jack's definition leads to an

> > agreeable result, I admit.

>

> It depends on the area people work in. For a combinatorian, the equality

> 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number

> of functions from empty set to empty set [*]. Moreover, things like

> binomial theorem also work nice with this extension; it allows you to

> evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or

> simply 0^n (ok, I admit, this is just a contrived academic example).

>

> Likewise, if you work in the area of foundations of mathematics, defining

> divisibility using the operation of division is a bit more complicated

> than using the straight existential-quantifer with multiplication (just

> like Jack did); for the division is only a derived operation in e.g. Peano

> Arithmetics and one needs to prove its well-definedness (and possibly some

> other properties) first.

>

> On the other hand, an analyst would probably bop you over the head

> if he saw 0^0 :-)

>

> Peter

>

> [*] This works even in the much more general framework of set-theory --

> If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined

> to be the cardinality of the set A^B which is the set of all functions

> from B to A. If the sets A and B are finite, the cardinal exponentation

> agrees with the usual exponentation of natural numbers.

>

> --

> [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278

>

>

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>

> - Dr. Michael Paridon wrote:
> So I thank everybody very much for explanation. Mathematics has more hidden

Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction.

> beauties I will ever learn, and as I am not in a professional way dealing

> with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

> Best regrads

[snip]

>

> Michael

>

>

> ----- Original Message -----

--

Alan