Re: Relative Cardinality

"Jiri Lebl" <jirka@xxxxxx> writes:

> David Kastrup wrote:
>> mueckenh@xxxxxxxxxxxxxxxxx writes:
>>
>> > Jiri Lebl wrote:
>> >> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>> >> > > I also thought of another irrational number that escapes
>> >> > > your logic. Take the number: 0.9909000900000009000... That
>> >> > > is the number whose digital representation has 9s in the
>> >> > > places that are powers of 2. It is the "sum(n=0 to oo)
>> >> > > 9*10^(-(2^n))". So then I can say what the 10^10^100's
>> >> > > digit is, it is 0. Also the
>> >> > > 2^100^100^100^100^100^100^100^100's digit is 9. And the
>> >> > > next digit after that is 0. I wonder why this number
>> >> > > doesn't exist and why it cannot be on any list.
>> >> >
>> >> > Interesting number. Thank you for posting this question which
>> >> > is one of the few non-polemic questions I received. Can you
>> >> > say which digit is on the position floor(pi*10^10^100)?
>> >>
>> >> 0, and it is not hard to see.
>> >
>> > Therefore, this number does exist and can be on a list.
>>
>> I count this "Therefore" among one of the most audacious uses of
>> this word that I have yet encountered from our venerable WM.
>
> Yes this is an interesting one. WM argued that only numbers with
> "simple rules" for generating digits are on the list, so if I
> present an irrational number for which it is easy to say what the
> nth digit is for any n that WM is prepared to give it suddenly
> ceases to exist?

No, you should reread what WM wrote again. He acknowledges the
existence of that particular irrational number. He seemingly has
fewer problems with reconciling this with his contention that
irrational numbers as such do not exist than you have.

Now why does he acknowledge the existence of this irrational number?
Because you were able to specify a _single_ digit of this number which
he specified to you by a natural number which, according to his
reckoning, does not exist.

It would be somewhat daring to talk about a "line of reasoning" here.

> Let's follow this reasoning for a different number. Say 1/3 which
> is in decimal: 0.3333... So WM would ask what is the digit in the
> position floor(pi*10^10^100), I would answer "3 and it is not hard
> to see", and WM would conclude that 1/3 does not exist.

No, he'd conclude that 1/3 _does_ exist. Using this particular line
of reasoning, on the other hand, would mean that 1/7 would likely not
exist. At least if the jury is able to establish non-existence, like
existence, by virtue of a single check at a non-existent position.

> Although by some other posts, for WM, n=floor(pi*10^10^100) does not
> exist and so anything you want is true for it.

Well, in this case pretty much the opposite of what you would have
expected. Small wonder you were confused about what he wrote, though
probably less so than himself.

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

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