Re: Galileo's Paradox

In article <456cfd7e@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Bob Kolker wrote:
Tony Orlow wrote:

It also seems reasonable to use measures of set density, and more
sophisticated methods of comparison, such as are employed in the
converse situation, with infinite series. It seems natural to say
that, if half the elements of A are in B, and all elements in B are in
A, then B is half the size of A, as is the case where A=N and B=E. The
proper subset as a smaller set should not be a notion violated by set
theory, in my opinion.

Do you know the difference between cardinality and measure?

I know that cardinality is a purported method of measure of a set.
Otherwise it is is not a quantity of any sort relating to anything.


A straight line segment unit length and a straight line segment twice
unit length have the same cardinality (taken as sets of points). But one
has twice the measure of the other.

That is correct, and that is where cardinality fails as a measure of
such sets.

That would only hold if one insists that there is only one way to
measure a set. In analysis, for example, there are many ways to measure
sets, and restricting things to any single measurement of sets would
require throwing out a great deal of analysis.

Raw bijection determines cardinality, but measure involves a
consideration of the actual mapping function which establishes the
bijection. The two are not incompatible, Bob.

"Measure", in the sense of measure theory, is not preserved merely by
bijection, but neither is it preserved by any of TO's methods.
.

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