Re: Galileo's Paradox

Virgil wrote:
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.
It is a direct consequence of the notion that a proper subset is always smaller, in some sense, than the base set.


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.

Sure it is, Virgle!
.

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