Re: infinity

Virgil said:
> In article <MPG.1dc052f49831462098a4eb@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > > The rationals are not a continuum, but that is not the same as being
> > > "continuous" versus "discrete". What is TO's definition of "discrete" if
> > > the naturals and rationals are but the reals are not?
> > >
> > That they are generated element-by-element by a recursive definition, whereas
> > the reals are defined, most appropriately, as representing the real distance
> > between any two points on a continuous line.
>
> The naturals are not "generated" recursively, they are only related to
> each other recursively. They are "generated" by the axioms which say
> that colectively they exist.
>
> The rationals are not even "generated recursively" in that sense.
>
> Does TO claim that in his TOmatic wonderland there is some "successor
> operation" on rationals by which each rational recursively generates a
> next one. If so, it only happens in TOmatics and nowhere else.
>
In TOmatics, the set of rationals is generated via a two-child recursive
definition.

1. 1/1 is a rational
2. rational(a/b) -> rational(a/succ(b))
3. rational(a/b) -> rational(succ(a)/b)

So, each element has two successors. Each element is also the successor to two
other elements. It's not a well-ordering, but a recursive construction
nonetheless.
--
Smiles,

Tony
.

Relevant Pages

  • Re: Cantor Confusion
    ... The "number" pi is definitely a merely fictitious element of continuum. ... naturals, integers, rationals, irrationals, or reals. ... intergers and naturals are genuine. ... genuine numbers to the reals is tempting but not justified. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... continuum is something every part of which has parts ... the so called 4th logical possibility is valid: The reals must ... not quantitatively compared with the rationals. ... Cantor ascribed two different infinite numbers, so called cardinalities, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... continuum is something every part of which has parts ... the so called 4th logical possibility is valid: The reals must ... not quantitatively compared with the rationals. ... Cantor ascribed two different infinite numbers, so called cardinalities, ...
    (sci.math)
  • Re: Should oo+a=oo be abandoned?
    ... Fictive elements of the continuum cannot be reached ... so you've decided togo for the two-point compactification ... of the reals. ... DA1 is used to prove that the rationals are countable. ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... > between naturals and rationals in terms of their density in the reals, ... under a linear ordering. ... number and its successor. ...
    (sci.math)