Re: OT: Once again Inez delves into the murky waters of philosophy.

Rupert Morrish <rupert@xxxxxxxxxxx> wrote:

J. J. Lodder wrote:
Rupert Morrish <rupert@xxxxxxxxxxx> wrote:

J. J. Lodder wrote:
Rupert Morrish <rupert@xxxxxxxxxxx> wrote:

J. J. Lodder wrote:
John Wilkins <j.wilkins1@xxxxxxxxx> wrote:

J. J. Lodder <nospam@xxxxxxxxxxxxxxxx> wrote:

John Wilkins <j.wilkins1@xxxxxxxxx> wrote:

J. J. Lodder <nospam@xxxxxxxxxxxxxxxx> wrote:

John Wilkins <j.wilkins1@xxxxxxxxx> wrote:
The standard term for what logicians and mathematicians often do
to prove a claim is "reductio ad absurdum": to show that X is
true, prove that not-X will lead you to a contradiction. This is
a bit like what
Inez is after.
Brouwer objects,

Yeah, well I don't trust intuitions.
Ah, you believe in god instead,

Not at all. I'm an agnostic about mathematical objects.
You didn't recognize a reference.
In one of their public confrontations at some conference
there was Brouwer was lecturing, and explaining things
like not all reals having a decimal expansion.
(which was considered most shocking at the time)
I'm still most shocked. Can you give an example of a real number that
does not have a decimal expansion?
One of the ways to arrange it is to construct a real number r
such that for succeeding approximations to it
1 - 10^{-n} < r < 1 + 10^{-n} holds,
with the greater or smaller depending on the outcome
of some Brouwerian question.
So you can know the number r to arbitrary precision,
(with n equal to the number of known decimals of pi for example)
but you still don't know the first decimal,
for you don't know if it starts with
0.9999999.... or 1.0000000.....
and there is no way to find out,

Best,

Jan

I'm not sure that answers the question. If you knew the value of r,
you'd know the decimal expansion.

But you don't.
You only know it to arbitrary precision.
1 - 10^{-n} < r < 1 + 10^{-n}
for as far ijn n as you can/want to calculate.

I still have the feeling that this example is demonstrating something
other than what you claim. The fact that you can't determine the decimal
expansion of a number you haven't specified doesn't mean that there is
an actual real number that doesn't have a decimal expansion.

The algorithm for calculating the numeber is well defined.

Until
there's only 1 real number in the range (1 - 10^{-n}, 1 + 10^{-n}).

There are uncountably many reals in the interval
(1 - 10^{-n}, 1 + 10^{-n})
for all values of n.
For sufficientle large n our number may or may not be outside the
interval, depending on the outcome of a Brouwerian question.
If the question is unsettled the number is in the interval,
if it gets settled at n it is out, for all values > n

This looks more like one of those series where you can prove
convergence, but the series oscillates around the convergent value.

No oscillation involved,

Jan

.

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