Re: abundance of irrationals!)


aeo6 Tony Orlow wrote:
> Dave Rusin said:
> > In article <MPG.1ce175f0d96da894989b7e@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >
> > >Infinite means "without end", and takes several forms.
> >
> > Mathematicians use it to mean "not finite", although since it is an
> > adjective it only appears in a context like "This is an infinite X"
where
> > X is a noun; mathematicians might prefer to treat "infinite X" as
a
> > single noun phrase to be defined, thus never reall defining
"infinite"
> > at all!
> Well you didn't define finite either, so I'm not surprised. I'm
trying to help
> here. If we can't even agree on our definitions for terms, then how
can we use
> them in any precise manner?

There are two separate conventions in common usage. You've
got to define one or the other, but people make different
choices about which is defined.

Convention 1: A finite set is one which can be put into
bijection with {1,2,3,...,n} for some natural number n.
Infinite is defined as "not finite".

Convention 2: An infinite set is one which can be put
into bijection with a proper subset of itself. A finite
set is defined as "not infinite".

They're equivalent, but they differ in which is taken as
definition and which as derived.

> Is there an agreed on definition of "infinite" that enables us to
know what
> each other is talking about?

For sets or numbers?

> > For example, the (correct) statement
> > the limit of f(x) = x sin( 1/x ) as x tends to 0 is 0.
> > may be expressed as a cumbersome but precise statement about the
> > behaviour of the function f near zero. The words "end" or "stop"
> > never occur in the translated statement.
>
> near zero, or at zero?

The limit is a number. It's exactly 0.

What that means is that f(x) gets close to 0 as x gets large.

> Most limits give pretty precise values for what is
> happening at that precise point.

What precise point? There's no precise point in the above
statement of lim(f(x)). As x gets large, f(x) gets
small. Limits are precisely defined, but in terms of
sequences of values (in this case, for any arbitrary
fixed value epsilon you pick, I can find x0 such that
|f(x)|<epsilon for all x > x0). That definition of limit
is precise, but the values epsilon and x0 are variables.

- Randy

.

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