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Re: [PrimeNumbers] Re: What is infinity?

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  • Phil Carmody
    ... as p ranges over the unbounded set of prime numbers greater than or equal to 2. ? Which certainly could be shortened without any wild ambiguity. ...
    Message 1 of 5 , Nov 30, 2001
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      On Fri, 30 November 2001, jack@... wrote:
      > --- In primenumbers@y..., Chris Caldwell <caldwell@u...> wrote:
      > > >2) Infinity is an odd number.
      > >
      > > No, it is not an integer.
      > >
      > > >3) Infinity is a prime number.
      > >
      > > No, it is not an integer.
      >
      > Both of the statements Chris made are true by definition.
      >
      > However, this leads to an interesting question.
      >
      > What is the "proper" way to say (for instance):
      >
      > 1. Compute the sum of 1/p^2 as p ranges over the prime numbers
      > from 2 to infinity.

      as p ranges over the unbounded set of prime numbers greater than or equal to 2.
      ?

      Which certainly could be shortened without any wild ambiguity.

      > or
      >
      > 2. Compute the product of 1+(1/x^2) as x ranges over the
      > odd numbers from 1 to infinity.

      ... unbounded set of odd numbers greater than or equal to 1.

      > I think these are both proper and it is understood that the
      > "infinity" reference is not to an actual number but is rather
      > a succinct way of stating that the range goes on without bound.

      Infinity as a bound => no finite bound => no bound

      > However, I could understand how someone reading (1) or (2) above
      > might think that infinity is a prime number or an odd number...

      You don't explicitly say whether the set is open or closed. Why should someone assume a closed set, and thus an included upper bound, rather than a half-open set, with an excluded upper bound? The danger is more that of people infering something erroneous from something not said, rather than misinterpreting something actually said. Errors like that are always harder to prevent. You can't exclude every possible inference, I'm sure.

      <JSH> What ring were you working in, again? </JSH>

      Phil

      Don't be fooled, CRC Press are _not_ the good guys.
      They've taken Wolfram's money - _don't_ give them yours.
      http://mathworld.wolfram.com/erics_commentary.html


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