Re: New countable infiniity logic

From: Barb Knox (see_at_sig.below)
Date: 11/27/04 

Date: Sat, 27 Nov 2004 15:19:54 +1300

In article <b453b903.0411201121.6f4ad3ac@posting.google.com>,
 whit0911@umn.edu wrote:

>Barb Knox <see@sig.below> wrote in message
>news:<cmu9kq$qs3$1@lust.ihug.co.nz>...
>> In article <b453b903.0411101156.25208b52@posting.google.com>,
>> whit0911@umn.edu wrote:
>
><snip>
>
>> >
>> >If (0.3, 0.33, ...) is an infinite sequence where each specified
>> >element in this representation of the sequence has a finite number of
>> >digits, isn't (0.3, 0.33, ..., 0.3333...) a valid representation of
>> >the infinite nature of this sequence showing that as the number of
>> >elements in the infinite sequence approaches infinity, the finite
>> >number of digits associated with each element also approaches
>> >infinity.
>>
>> It looks like you've been tripped up by the notation:
>> "0.333..." is an infinite sequence of digits.
>> "0.3, 0.33, ..." is an infinite sequence of finite sequences of digits.
>> This is the sort of thing your function generates.
>
>I am also saying that the set Dec3 = { 0.3, 0.33, ... } is an
>infinite set

Correct.

>and therefore *must* have an infinite number of digits

It looks like you're inappropriately mixing levels again: the top-level set
has elements which are sequences, each of which has elements which are
digits. So the set can be viewed as "having" digits only indirectly, by
virtue of the fact that its elements have digits.

So in that sense the set can "have" an infinite sequence of digits in one of
two ways (or both together):
1. Some particular element is an infinite sequence of digits.
2. Each particular element is a non-zero sequence of digits, and there is
an infinite sequence of such elements.

Clearly (2) is true here; you are arguing that (1) is also true. But note
that if (2) is true but (1) is false then the set still "has" an infinite
sequence of digits. That is the case here.

>since it is formed from an infinite sequence (albeit of finite
>sequences of digits).

Right, that's (2).

>For the set not to be finite, there must be an infinite number of digits,

Yes, in the sense that (1) OR (2) (or both) must be true.

>and therefore 0.333... must be present in
>set Dec3 or else Dec3 is an not infinite set.

No. You're arguing "Since (1) or (2) must be true then (1) must be true".
That is clearly fallacious.

[snip] 

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