Re: irrational number continuum

On Jun 23, 6:28 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jun 23, 6:58 am, slartibartfast <tomokane2...@xxxxxxxx> wrote:

I'm not objecting to thenumber1/3; 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

denumerable sequences and infinite sets cannot be fully represented
in the sense that you cannot completely write down "all" the natural
numbers.

even though you can refer to the sqrt(2), you cannot say what it is
precisely, you can only estimate it.

you're saying that even though "you cannot say what it is precisely"
nevertheless it does exist and has an exact value (and which we refer
to as the sqrt(2) )


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 thenumber1/3
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.

example 0+2i is two normal integer units of an imaginary entity.
the "number" applies to the "two normal integer units" part of that
description
i is one normal integer unit of that imaginary entity
but what is that imaginary entity itself? its not a number, its
represented as an "axis" or direction in the complex plane.

yet we use it
usefully.
we don't have to accept the actual existence of such anumberas 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.

I know you have a penchant for saying "we prove it" and "it can be
proven" and "in any introductory text " etc etc

please the proof here, of the existence of irrationals, it couldn't be
that long surely.


twould save a lot of to-ing and fro-ing.







here we come to it.
if something "doesn't have a decimal expansion that can be expressed
in a finitenumberof 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.

we cannot visualise an infinity of digits either.

I had this discussion previously with regard to infinite sets like the
natural numbers.
to me, the point was that a statement about "all the natural numbers"
could be rephrased as a statement " for each single natural number".
Although the set of natural numbers is not finite, it does not make
any sense to say it exists as a completed infinity, since we can only
vizualize a finity.


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 acontinuumbetween 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 meannumberneed be.

And that wasn't the argument for simplicity.

it was argued with respect to rectangles and the unit square.

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

your calculation of the diagonal of a unit square as the square root
of 2 says nothing more than "My calclulation of the diagonal of a
unit square is the square root of 2"
if you wish to only express your answer "in terms of the square root
of 2" then there is no problem. But you certainly did not calculate
what that value "the square root of 2" actually exactly is.

.

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