Re: Galileo's Paradox

In article <4579d233@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

stephen@xxxxxxxxxx wrote:
Six wrote:
On 6 Dec 2006 07:08:46 -0800, "Mike Kelly" <mk4284@xxxxxxxxxx> wrote:

Six wrote:
I am very grateful to you for expanding on this. While I'm almost
certain I'm missing something, I'm afraid I still don't get it.

How exactly does claiming that a 1:1 C is not necessarily
indicative of equality of size with infinite sets presuppose an
inability
to map (eg) the binary and decimal representations of integers?

There is still a 1:1 C between the two sets. It is still true that
for any finite sets a 1:1C implies equality of size. Moreover it's still
reasonable to suppose that a 1:1C implies equality of size in the
infinite
case unless there are other, 'functional' reasons to the contrary.
(Vague,
I know. Roughly, 1:1 C is a necessary but not sufficient condition for
equality of size.)

The idea is that the naturals (in any base) form a paradigm or
norm, a standard against which other sets can be measured.
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).

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.

How can the size be base dependent? The natural numbers are not base
dependent.
Any natural number can be expressed in any base. There is no natural
number
expressible in base 16 that is not expressible in base 10, or base 9, or
base 2.

I suppose you could claim that there is a set of decimal numbers, and a set
of base 2 numbers, and a set of hexadecimal numbers, and that they are all
different, and all have different sizes. But it is a strange notion of
"different size" given that all the sets represent the same thing.

Stephen

Measure is something different from the language needed to express it.

Which does not exculpate TO from the wrong he has so recently
promulgated.
.

Relevant Pages

  • Re: Galileos Paradox
    ... more decimal strings representing integers than binary strings, ... How should a map between decimal and binary representations of positive ... for any finite sets a 1:1C implies equality of size. ... The idea is that the naturals form a paradigm or ...
    (sci.math)
  • Re: Galileos Paradox
    ... for any finite sets a 1:1C implies equality of size. ... The idea is that the naturals form a paradigm or ... The set of finite binary strings is a subset of the set of finite ... The natural numbers are not base dependent. ...
    (sci.math)
  • Re: Galileos Paradox
    ... I'm afraid I still don't get it. ... for any finite sets a 1:1C implies equality of size. ... The idea is that the naturals form a paradigm or ... I agree it would be a strange notion of size. ...
    (sci.math)
  • Re: Galileos Paradox
    ... I'm afraid I still don't get it. ... for any finite sets a 1:1C implies equality of size. ... The idea is that the naturals form a paradigm or ... and all have different sizes. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... What is that constraint? ... Depends on your exact usage of bit strings to represent naturals. ... The second is as "normal" as any other representation scheme. ...
    (sci.math)