fom
Posts:
1,033
Registered:
12/4/12
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 5, 2013 10:02 PM
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On 1/5/2013 4:22 PM, Virgil wrote:
> In article
> <bb7be2f4-a0c3-4538-baf6-527a064d1771@b8g2000yqh.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>
>> On 4 Jan., 19:47, fom <fomJ...@nyms.net> wrote:
>>> The rationals
>>> are not complete. So much so, in fact, that they are a set
>>> of measure zero.
>>>
>>> But, wait.
>>
>> No reasonable reason to wait. "Measure zero" is a nonsense expression
>> because it presupposes aleph_0 as a meaningful notion. There are
>> convergent sequences, but the limit is never assumed, not "after
>> aleph_0 steps".
>
> So WM would throw out all of measure theory with his bathwater.
It is much worse than that.
What is a number? Between the ancient Greeks and
the nineteenth century came Vieta. His "logic of
species" (algebra) effectively treats arithmetical
monads and geometric magnitudes homogeneously. That
it works is why we use it.
But, the underlying invariance is something found
with Lebesgue measure. The two posts about how
WM is cheating document the issue (although I had
one particular mistake by attributing a property
of the Baire space to the Cantor space).
Because of the peculiar nature of Lebesgue measure,
its product measures do not require the full theory
of product measures. What comes of it is the relation
between measurable sets and Borel sets. If
a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_10, a_11, a_2
are a sequence of zeros and ones associated with a
real number, then
a_1, a_2, a_4, a_7, ...
a_3, a_5, a_8, ...
a_6, a_9, ...
a_10, ...
and so on
give you real numbers in infinite dimension. Eventually
constant sequences give a sequence of eventually constant
sequences and, by virtue of modulo arithmetic on the
subscript sequences, repeating sequences give a sequence
of repeating sequences. So, every rational yields
a rational sequence and every irrational yields an
irrational sequence. Moreover, the Lebesgue measure
is unchanged by the transformation.
Measure is preserved because of the relationship of
Lebesgue measure to the Borel hierarchy. Although
you would not find it discussed anywhere, Lebesgue
measure is the invariant property that justifies
certain aspects of Vieta's algebra.
So, WM would have us back in Macedonia.
I do not criticize the finitism. But, there
are responsible ways of addressing these matters
and merely learning about foundations for the
sole purpose of promoting one's religious belief
system is reprehensible.
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