Re: Galileo's Paradox


Six wrote:
On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284@xxxxxxxxxx> wrote:
<snip>
The set of finite binary strings is a subset of the set of finite
decimal strings.

I confess I hadn't fully appreciated this simple point, that
together with the fact that the strings just are, so to speak, the natural
numbers (in a given base).

I'd say the strings are a representation of the set of natural numbers.
I'd certainly expect the sets of strings and the set of natural numbers
to be the same size.

Then b) precludes them being the same size.

They are also both the same size as the set of natural numbers.

Thus they are the same size as each other.

Contradiction.

One is driven to the conclusion that there is no base-independent
size for the natural numbers.

One is driven to the conclusion that EITHER a proper subset can be the
same size as its superset OR that there is no base-independent size for
the natural numbers.

Frankly the latter makes no sense whatsoever to me for any reasonable
interpretation of "the natural numbers".

This does not make the discussion of the relative size of, for
example, natural numbers and squares meaningless. It's just that a given
base would have to be understood, and that whatever is said about the
relative size of the two sets is understood to apply mutatis mutandis to
any other base.

So you really believe "the set of natural numbers represented in base
10" is a different set from "the set of natural numbers represented in
base 2"? Hmm.

But for me at least, it has certainly opened my eyes to the
implications of the original argument.

Much appreciated,

Six Letters

Well... you're welcome, I guess.

--
mike.

.

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