🚀 go-pugleaf

On Mar 24, 7:57 pm, David R Tribble <da...@tribble.com> wrote:
> Richard Tobin wrote:
> > One third has to be the limit of the series
> >  3/10, 33/100, 333/1000, ...
> > If you claim the existence of one third you claim the existence of the
> > number
> >  33333.../100000...
> > with infinite numerator and denominator.
> Sounds like an alternative theory to ZFC, which plainly limits
> one's acceptance of perfectly reasonable ideas such as
> Albrecht's infinite ratios (eg, 31415.../10000...). Just because
> these ratios have absolutely no meaning in standard math is
> no reason to reject them outright. Just ask Transfer Principle;
> he'll set you straight on the underlying, but mostly hidden, logic.

Let's look at Robinson's hyperreals here. Do you (Tribble) consider
the hyperreals to be "hidden logic"?

We know that hyperreals are defined as certain equivalence classes
of sequences of reals. So let's define 31415... to be the class
containing the sequence:

{3, 31, 314, 3141, 31415, 314159, ...}

and 10000... to be the class containing:

{1, 10, 100, 1000, 10000, 100000, ...}

Then 31415.../10000... is a certain hyperreal. Its standard part is
pi, yet it's not equal to pi. Why not? In classical analysis, we
know that pi is irrational -- i.e., for every pair of integers a,b
we have that pi does not equal a/b. This Transfers, so that there's
no pair of hyperintegers A,B such that pi equals A/B either. The
hyperrational 31415.../10000... differs from pi by an infinitesimal
while no hyperrational equals pi exactly.

What's the difference between this and Tobin's example? Similarly,
we see that the ratio of:

{0, 3, 33, 333, 3333, 33333, ...}

and the hyperreal labeled 10000... above is infinitesimally 1/3,
yet isn't 1/3 itself. But that doesn't prove that 1/3 is not a
hyperrational -- not by a longshot. Think about quantifiers -- to
show that 1/3 isn't hyperrational, we must show that _every_ pair
A,B of hyperintegers does A/B fail to equal 1/3, but so far we've
only shown it for _one_ particular pair. Indeed, we see that if we
subtract one from 10000... above to obtain:

{0, 9, 99, 999, 9999, 99999, ...}

we see that 33333.../99999... really does equal 1/3 exactly. Thus
1/3 is hyperrational. Of course, there are more trivial examples
of hyperintegers A,B such that A/B=1/3, such as:

{1, 1, 1, 1, 1, 1, ...}
{3, 3, 3, 3, 3, 3, ...}

> Just because
> these ratios have absolutely no meaning in standard math is
> no reason to reject them outright.

Exactly -- and I didn't even have to come up with a brand new
theory to give them meaning. They already have meaning in an
already known theory, Robinson's hyperreals.

(Since I'm responding to Tribble here, I was hoping to use his
suprareals instead of the hyperreals. But I doubt that one can
assign digits to infinite suprareals like eta_1, in the same
manner I assigned digits to infinite hyperreals above. Sorry!)