On 1/5/2013 4:22 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> 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
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, ...
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.