Re: infinity

David R Tribble said:
> Matt Gutting said:
> >> I recall you defining the "value range" of a set as the maximum difference
> >> between any two elements of the set. Without regard to whether this
> >> difference is finite or infinite, bounded or unbounded, can you demonstrate
> >> that such a difference exists for any arbitrary set (of natural numbers)?
> >
>
> Tony Orlow wrote:
> > Think of it as the largest POSSIBLE difference in the set. There is no
> > possible infinite difference between finite values. The largest possible
> > difference between finites is finite. Any set of values has some range,
> > some set of differences.
>
> And yet you stated elsewhere in this thread that the maximum range of
> the real points in S = (0,1] was 1, even though the difference between
> any two finite points in S is always less than 1.
And then emended it to say that the range of such an open set is "<1", which
would mean arbitrarily close to but less than 1. The range of the finite
naturals is "<oo".
>
> Quote:
>
> Randy Poe said:
> >> But there is no pair of values separated by 1
> >> [in the set of reciprocals of finite positive naturals],
> >> so this contradicts your own definition of range.
> >
>
> Tony Orlow wrote:
> > Maximum POSSIBLE difference between values in the set. If there are
> > an infinite number of elements int he set, then you will have
> > elements infinitely close to 1, which standard math would consider
> > equal to 1.
>
> So you seem to be willing to grant that a set of reals can have a range
> greater than any possible difference between any two of its members,
> but that a set of naturals can't.
>
> Why is that?
It really can't. That was the first time I was asked about such an interval,
and it wasn't well thought out, since I had been thinking in terms of the
naturals. An open interval' range can be expressed using a "<" to denote the
lack of maximal element. I'd also say that (0,1), [0,1) and (0,1] all have the
same range of <1.
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... > But this list is only a countably infinite number of reals. ... it lists only binary fractions with a finite number ... > the infinite set of all finite naturals. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Tony Orlow wrote: ... Then I say there are infinitely many naturals. ... this an infinite set, ... That identity relationship characterizes all the initial finite subsets ...
    (sci.math)
  • Re: Orlow cardinality question
    ... Tony Orlow wrote: ... > distinct finite naturals, and have pointed out the flaw in the one proof you ... if the set of natural numbers is infinite, ... you give a proof then people don't argue with you, ...
    (sci.math)
  • Re: infinity
    ... >> Tony Orlow says... ... Good way to kludge away the largest finite. ... And of course aleph_0 is just such an infinite n ... so we only have finite naturals. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... > In sci.math, Tony Orlow wrote: ... >> When you declare both the naturals and the even naturals as infinite ...
    (sci.math)
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