Re: [PrimeNumbers] Re: Is 2 a prime
- 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.
----- Original Message -----
From: "Peter Kosinar" <goober@...>
To: "Dr. Michael Paridon" <dr.m.paridon@...>
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 :-)
> [*] 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 hiddenTry: 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]
> ----- Original Message -----