Re: irrational number continuum

On Jun 23, 6:58 am, slartibartfast <tomokane2...@xxxxxxxx> wrote:
I'm not objecting to the number 1/3; but
firstly to the possibility to represent it as a finite decimal AND
ALSO secondly to the notion of a completed infinity (of 3's after the
decimal point).

If you disallow denumerable sequences, then fine, you're welcome to
adopt whatever alternative axioms or set of priniciples you like that
do not prove the existence of denumerable sequences.

the fact that the differs less and less from 1/3 the more digits we
use is NOT (in combination with the observation that the number 1/3
actually exists) any justification whatsoever for concluding that
"therefore a completed infinity must be notionally possible even if
not actually representable"

We don't argue by any such justification.

That is like arguing that little pixies built the eiffel tower, then
pointing to the existence of the eiffel tower as evidence for the
existence of little pixies!

But that is not an analogy with what we do.

we don't accept the existence of the square root of -1,

Sure we do.

yet we use it
usefully.
we don't have to accept the actual existence of such a number as the
square root od 2 in order to do useful calculations with that "notion"
either. I've never once had to write it out in full!

Then you're welcome to devise a mathematical notion of "actual
existence" and prove what you think "actually exists".

We'd be a lot better off to say "2 doesn't have a decimal
expansion that can be expressed in a finitenumberof
digits" because that's a lot simpler than trying to
exclude certain numbers.

that already presupposes that there even ARE "such (exact) numbers"
even though they "cannot be represented".

No, we prove it.

here we come to it.
if something "doesn't have a decimal expansion that can be expressed
in a finite number of digits" (either in the case of dec. rep. of 1/3
or worse the square root of 2)

there is no justification for assuming that there is ANY definite
entity that that (partial) decimal representation  partially
represents.

We don't assume. We prove.

in the case of 1/3, 1/3 definitely does exist and the quibble is only
with the use of an "infinity" of 3's to "represent" it in the decimal
system. (since all of its digits are known)

See my earlier remark.

in the case of the square root of 2 however, we don''t even know what
those digits would be, and there still "an infinity" of them.

We can't finitely list all the digits. That's not a contradiction.

in all computations however, we only EVER use "approximations" to
them, and these "approxmations" are always rational.

No, I don't always compute with only approximations. In computing the
diagonal of a unit square, I arrive at the square root of 2, not at
any approximation.

why do we assume a continuum between 0 and 1?

Because it's simpler if we do.

no it isn't; its completely unnessary! just because space seems a
continuum, doesn't mean number need be.

And that wasn't the argument for simplicity.

since when we finish manipulating the symbols for such "irrational
numbers" and progress to actually doing the computations with values,
we only ever use "rational number" approximations" ever.

Nope. I just gave you an example. My calclulation of the diagonal of a
unit square is the square root of 2, and not some approximation.

MoeBlee

.

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