Re: infinity

In article <MPG.1d73c03d71c5e13098a117@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1d6d63de88f1507198a0f8@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >
> >
> > > When you map [0,1] to [0,2] and getting the same number of
> > > elements, the second has only half the density of the continuum.
> > > It's equivalent to only having even numbers on the infinitesimal
> > > scale: 000...000, 000...002, 000...004, etc.
> >
> >
> > By this method of measurement, one can easily show that any finite
> > closed interval is strictly less dense than itself.
> >
> > Biject [0,1] to [0,2] and then [0,1] as a subset of the new [0,2]
> > will have to be less dense than the the original [0,1], even though
> > is still bijects with the original in the obvious way.
> >
> > And TO proposes this idiocy to ELIMINATE contradictions?

> You can certainly talk about sets with different densities in the
> real numbers, and of course this needs to be taken into account when
> comparing sets of numbers to the continuum. If you assume equal
> density between sets, then relative size is the same as relative
> value range.

Then either a single set may have different densities in different
applications or it can have different reanges in different applications.

Such differences mean that the measure is subjective rather than
objective, and unsussitable for any mathematical applications.
> >
> >
> > > > Prove it. What values of b or d, presumably residing in that
> > > > second half, do you have in mind? There should be an infinite
> > > > number of them to choose from, right?
> >
> > > For a range n, the number of whole numbers in n is twice the
> > > number of even numbers in n, even if n=N.
> >
> > Only for finite ranges, so that TO must be claiming that the
> > cardinality of N is finite.

> No, I am saying the fact holds true for infinite sets as well.

Since the only method of comparison of such sets is bijection, at least
for standard sets of naturals , all members of which must be finite,
TO's measure does not work in standard arithmetic, only in TO's
pseudo-arithmetic.
> >
> > > To claim they are the same number, one has to give the evens
> > > twice the range. This is why I am advocating the use of unit
> > > infinities, because otherwise comparison is impossible.
> >
> > Comparisons are quite possible using bijections/injections, but
> > with TO's additions to Cantor one can for some sets prove that they
> > are strictly smaller than themselves, as shown above.

> No, you were being sloppy. Sorry.

I was merely using TO's sloppy definitions
.

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