Re: infinity

David R Tribble said:
> Tony Orlow wrote:
> >> Try "number of elements". Stick to basics. Remember Occam's Razor.
> >
>
> David R Tribble said:
> >> You agree that the set of naturals N is infinite.
> >> What do _you_ call the measure of set N? What size is it?
> >
>
> Tony Orlow wrote:
> > Since it is infinite, the size must be putin infinite terms. This set is the
> > natural unit infinity, since it is based on the identity function between
> > element value and position. Its size is N, which means one element per unit,
> > forever.
> >
>
> We call it Aleph_0.
>
>
> >> The set of even integers is infinite.
> >> What size is it?
> >
> > 2N
>
> Huh. I would have thought it was N/2+N/2 = N.
Actually you're right. My mistake.
>
> We call it Aleph_0. Oh well, slight difference there.
Yup.
>
>
> >> The real points in [0,1] is an infinite set.
> >> What size is it?
> >
> > The correlation between the continuum and discrete infinities is a difficult
> > one which may never be solved. A mapping of logN(x) maps the naturals to the
> > reals in [0,1], though not with constant density. It is however fully dense at
> > its least dense point, and can be considered an enumeration of the reals in
> > [0,1]. Given that function and the inverse function rule, there would appear
> > to be N elements in the set, which makes sense since we mapped the N naturals
> > each to one real in [0,1]. However, because of the increasing density in this
> > mapping, over the entire range N, it would appear to have N^N elements, so
> > there is a problem here. It is probably unavoidable to have two unit
> > infinities, as I said a long while back, the discrete unit infinity N, and
> > some continuous unit infinity R, which would be the number of reals in [0,1].
>
>
> >> The set of all reals, R, is infinite.
> >> What size is it?
> >
> > The unit discrete infinity times the unit continuous infinity. N units of R
> > points apiece.
>
> We call it c.
Yup, same as the set of reals in [0,1], eh, or in [39,765)?
>
>
> >> The power set of N contains all the possible subsets of N.
> >> What size is it?
> >
> > 2^N
>
> We call it 2^Aleph_0.
Yup.
>
>
> >> The power set of R contains all the possible subsets of R.
> >> What size is it?
> >
> > 2^(NR)
>
> We call it 2^c.
>
>
> >> The rest of the world has names for all these set sizes (which
> >> we call "cardinalities"). Do you?
> >
> > Avoid the nominative fallacy at all costs.
>
> At first glance, it looks like your N is very much like our Aleph_0,
> and your R (or NR) is very much like our c. Are you sure you're
> not coming around to our points of view?
I do not diagree with every conclusion. There are obviously two types of
infinities, discrete and continuous, or counting and measure related, so the
use of N and R is not unlike the use of aleph_0 and c. There are important
differences, however, in the treatment of infinite sets of quantities, strings,
tree branches, operations, etc.
>
>

--
Smiles,

Tony
.

Relevant Pages

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