Re: An uncountable countable set

Han de Bruijn wrote:
Tony Orlow wrote:

Han de Bruijn wrote:

Precisely! Mathematicians get confused by the idea of a "bijection",
which is an Equivalence Relation, which in turn is a "generalization"
of "common equality" (yes: the one in a = b). But the funny thing is
that EQUALITY HAS NEVER BEEN DEFINED. So there is actually nothing to
"generalize". Equivalence relations are a "generalization" of nothing.

But, fortunately, reality is more simple than this. Every equality is
an equivalence relation. And every equivalence relation is an equality.
[ ... snip ... ]

I agree with that last statement, but would disagree that equality is not definable. It depends on difference, most basically, and where none is detected, two things can be said to be equal.

Panta rhei, ouden menei (= everything flows, nothing remains the same).
There is no such thing as "difference, [...] where none is detected" in
nature (and culture). All equality is "in some sense" and relative. But
a picture says more than a thousand words:

http://hdebruijn.soo.dto.tudelft.nl/fototjes/gezocht.htm

Han de Bruijn


Consider the equally spaced staircase from (0,0) to (1,1), as the number of steps increases from 1 without bound. Is it the same as the diagonal line? Inductively we can prove that the length of the staircase is 2 at every step. Does it really suddenly become sqrt(2) in the infinite case? By the measures of point set topology, all points in the staircase become indistinguishable in location from the those of the diagonal, so by this thinking, all difference has disappeared, and the two objects are equal. However, using a segment-sequence topology, staircase n is the concatenation of n pairs of segments, denoted by x and y offset, described by {0,1/n} {1/n,0}, whereas the corresponding segments of the diagonal, between the points on the diagonal where perpendicular lines pass through the vertices of the staircase, are of the form {sqrt(2)/2n,sqrt(2)/2n}. The fact that the directions of the two curves are different at every point explains the difference in length, but this distinction cannot be detected by looking at pointwise location alone.
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