Re: infinity

In article <MPG.1dc6b8f0e5383cfb98a535@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> David Kastrup said:
> > Tony Orlow <aeo6@xxxxxxxxxxx> writes:
> >
> > > 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.
> >
> > Last time I looked, 1/1 and 2/2 were the same rational number.

> They have the same quantitative value. Consider it a dual or multiple
> representation, as you get in the digital reals (discounting
> infinitesimals). The set of rationals is certainly redundant that
> way, but as a symbolic representation of quantity, they're two
> different numbers.

There are many different numerals (representations) for the same
number. For example 8 and VIII are different as numerals but represent
the same number. In the same way 1/2 and 2/4 are different numerals
reresenting the same number. Sensible people understand the distinction
between a number and a numereal representing a number.

> At least, that's the way I see it. There is no unique linear order
> for the elements, since parents are not unique.
>
> >
> > Try generating 3/4 in that manner:
> >
> > 1/1 -> 1/2 -> 1/3 -> 1/4 -> 2/4 = 1/2 Oops.
> >
> > 1/1 -> 1/2 -> 1/3 -> 2/3 -> 3/3 = 1/1 Oops.
> >
> > 1/1 -> 1/2 -> 1/3 -> 2/3 -> 2/4 = 1/2 Oops.
> >
> > 1/1 -> 2/1 Oops (the denominator can't pass the enumerator without
> > falling back onto 1/1).

> I don't really consider this redundancy a problem. It's just an
> inherent property of the mode of representation. Rational numbers are
> sort of an overlaying and nesting of all the different number bases.
> Perhaps rationals should be built from the set of primes in some
> manner, rather than the naturals, if you want quantitative
> uniqueness.

For naturals, successorship is unique and creates a sequencial order
among the naturals.

For rationals, TO-successorship does not even create a partial order, as
there are loops, like 1/1 -> 1/2 -> 2/2 = 1/1.
> >
> > So TOmatics is not even able to reach 3/4 in a straightforward way.
> > How about a crooked way?
> >
> > 1/1 -> 1/2 -> 1/3 -> 1/4 -> 1/5 -> 1/6 -> 1/7 -> 2/7 -> 3/7 -> 4/7
> > -> 5/7 -> 6/7 -> 6/8 = 3/4
> >
> > Now, can you prove that for every rational, there is a TO way to
> > generate it? Sounds almost as much fun as the Collatz conjecture.
> >
> > And yours is supposed to be a _definition_, not a riddle.
> >
> >
.

Relevant Pages

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