Re: abundance of irrationals!)

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!

>All those forms and understandings must be in accord.

Open a dictionary and count the number of definitions for "set".
They're not all even from the same part of speech, much less being
in accord.

(We do hope the multiple uses of a single word are more or less similar
but this is not always the case for historical reasons. If I have a
closed 1-form on a closed set, there is no connection between the
two uses of the word "closed". Pity, that.)



>> The word limit tells it also we are always on our way but we are never reaching it.
> Last I heard a limit was an end, a stopping point.

Informally, perhaps. I'm sure many mathematicians would say they've
reached their limit with this discussion. But when they're using the
word "limit" in a mathematical sentence, they have a precise meaning
for it, which does not involve the terms "end" or "stopping point"
at all. 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.


dave
.

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