Re: Galileo's Paradox

From: "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx>
The density of infinite sets, within each other or supersets, has
definite meaning and certainly corresponds to some notion of
relative size.

Only in an absolutist sense that the subset or the complement may
be finite, or both may be infinite. With only two sets, no metric
of any kind, that's all you can meaningfully say. Consider for
example the set of squiggles and the set of zigzags, and their
union. There are an infinite number of each of the two subsets (of
the union). Can you tell me what fraction of the union are squiggles?

Half the integers are even.

Treating the integers *only* as a set, without any metric or ordering:
! For any F such that 0 < F < 1, F of the integers are even.
Picking some particular F and acting like that's the only F that
works, is absurd.

A proper subset is smaller than the set.

Before that can be meaningful, you need to say what the word
"smaller" means as you use it. If you are simply defining "smaller"
to mean "proper subset of", then you have no way to compare
virtually any two sets as to whether one is "smaller" than the
other or not, except in the rare case where one happens to be a
subset of the other.

The size of the proper subset is less.

Before that can have any meaning, you need to say what you mean by
"size". It is certainly *not* true per the usual (Cantor)
definition of "size" (cardinality). If you have some alternate
definition of "size", put up or shut up.

Consider the notion of some probability distribution over the
naturals of which you know not its form, except each integer has
non-zero probability in selection, that selects a (natural)
integer. In a wager for something you value, would you be
willing to accept 10:1 odds in your favor that it was a multiple
of 12? 1200?

If nobody is willing to tell me what probability distribution is
going to be used for the betting game, then I have no way to
compute/estimate any of the odds, and I would be unwilling to place
a bet.

Is the density of a subset of the natural integers in the
naturals integers a meaningful comparison of its size compared to
the density of other subsets of the natural integers?

"density" requires stating at the outset what measure is to be used.
There is no such thing as "the density" of a subset in the absense
of an a priori definition of the measure you're going to use.
For one metric, the even numbers can have density one half, while
for another metric the even numbers can have density one quarter,
while for another metric the even numbers can have density 0.995,
etc. etc. (Note that it's *impossible* for every integer to have
the same weight. I.e. it's impossible to define a metric that
assigns each integer the same weight. The best you can do is
artificially balance the odd and even numbers while decaying their
weight as you go along. For example w(0) = w(1) = 1/4, total 1/2
so-far, w(2) = w(3) = 1/8, subtotal 1/4, total 3/4 so-far, etc.)

Would you think a random integer is more likely to be a multiple
of two, or a multiple of two billion?

Since multiples of two billion are a subset of multiples of two, no
matter what measure you pick a priori, w(2E9*n) <= w(2*n). But if
the question was multiple of two vs. multiple of 2,000,000,001,
then for some metrics one will be larger and for other metrics the
other will be larger and there are some metrics where the two
subsets of integers are exactly the same size. Consider this
weighting (measure), where K=2,000,000,001:
w(0) = w(1) = 1/4 (subtotal 1/2, 1/2 so-far)
w(2) = w(K) = w(3) = 1/12 (subtotal 1/4, 3/4 so-far)
w(2*K) = w(5) = 1/16 (subtotal 1/8, 7/8 so-far)
w(4) = w(3*K) = w(7) = 1/48 (subtotal 1/16, 15/16 so-far)
w(4*K) = w(9) = 1/64 (subtotal 1/32, 31/32 so-far)
w(6) = w(5*K) = w(11) = 1/192 (subtotal 1/64, 63/64 so-far)
...
i.e. alternate between a two-way split between the next multiple of
2*K and the next unused odd number, and a three-way split between
the next unused multiple of 2 but not K, the next unused multiple
of K but not 2, and the next unused odd number.

Now if you want the probability of an even number to be exactly
1/2, and also the probability of a multiple of K = two billion and
one to also be exactly 1/2, that can be arranged with some
additional effort. One way to do it is like this:
w(0) = w(2) = w(K) = w(3) = 1/8
w(2*K) = w(4) = w(3*K) = w(5) = 1/16
w(4*K) = w(6) = w(5*K) = w(7) = 1/32
...
i.e. always have a four-way split equally: Next multiple of 2*K,
next unused multiple of 2, next unused multiple of K, and next
unused odd number.

Exercise for the reader: Let S be a finite set of natural numbers,
no two of which have any factor in common. Define a weighting
(measure) such that multiples of each single number of S have
density (total probability) exactly 1/2.

only half the integers are even.

Any fraction you want (strictly between 0 and 1) can be even.
You want 99.44% even, you can get it your way!

The density of a set of the natural integers, which is a very
particular set of numbers where all the number-theoretic
statements about them apply, is a number between zero and one, ...

Please specify the complete set of what you call "number-theoretic
statements, and then give me an example of a weighting (measure)
that satisfies all of those statements.

The universe is infinite, then infinite sets are equivalent,
because the universe is an example that infinite set and
powerset, itself, are equivalent.

That's nonsense.
.

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