Re: abundance of irrationals!)

"W. Mueckenheim" <mueckenh@xxxxxxxxxxxxxxxxx> wrote ...
"r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote ... 
["W. Mueckenheim" <mueckenh@xxxxxxxxxxxxxxxxx> wrote ...]
> It is used standardly and erroneously in diagonal arguments. The limit
> (of the partial sums) is not part of the bijection of the sequence (of
> the partial sums) with N (as realized by the lines of Canor's list).

That's a gross non sequitur -- The standard diagonalization arguments
work *because* the limit, represented by an (anti)diagonal string, is
*not* represented by an item on the list.  BUT THAT'S JUST A DIVERSION
AWAY FROM YOUR FAILURE TO STATE YOUR CLAIMED ALTERNATIVE DEFINITION OF
SUM[k in N]1/2^k  AS DISTINCT FROM A LIMIT OF A SEQUENCE OF PARTIAL SUMS.


Wrong. The diagonal argument should show that any a_nn is different
from b_n. This has nothing at all to do with the limit, the digits of
which cannot all be enumerated by natural numbers (as 0 in the
sequence (1/n) is not enumerated by a natural number).


Arguments can be made about *strings* of digits, but if you want
to relate those strings to the real numbers they *represent*,
you must state your definition of 0.111... as a real number, since
it is not the standard one.  The string 0.111... arose in the first
of your steps to define SUM[k in N]1/2^k  as something *other* than
a limit of partial sums.  You steadfastly refuse to state that
alternative definition.


>> Given that you do not accept the *standard* definition of 0.111...
>> as a binary representation of unity, your "explanation" amounts to
>> another refusal to say what is your non-standard definition.
>
> I accept the agreement that binary 0.111... = 1.

Then you CONTRADICT yourself ... 

Not at all. I accept the agreement. But that has noting to do with my
position.


Your "position" on whether  0.111... (binary) = 1  is *not* the issue.
The issue is your failure to state your claimed alternative definition
of 0.111...  as representing something *other* than a limit of partial
sums.  All this other chatter is a diversion that so far has exposed
your gross confusion about the use of digital representations of real
numbers.

1 - 0,999... = 0 only if the digis "9" are not all on finite
positions, because for finite positions k we have 10^-k =/= 0.


Let's not change the base to decimal, since the issue concerns
your definition SPECIFICALLY of SUM[k in N]1/2^k, for which the
binary representation is 0.111..., and is an expression which
you "accept" -- although you claim to define it nonstandardly
(since for you 0.111... is not defined by the standard limit of
a sequence of partial sums, YOU must state YOUR definition of
0.111... as representing a real number).

Until you define what the string 0.111... means as a number,
the expression 1 - 0.111... remains UNDEFINED BY YOU.

IF YOU ARE "ACCEPTING" SOME IMAGINED AGREED-UPON DEFINITION OF 0.111...
*OTHER* THAN THE STANDARD DEFINITION AS A BINARY REPRESENTATION --
I.E., AS A LIMIT -- THEN YOU ARE OBLIGED TO TELL US WHAT IT IS!


I tell you that intermingling the sum over all n and the sum from 1 to
oo is the reason for the would-be-existence of set theory.


You can "tell" all you wish, but without proper definitions your
telling is nothing but gibberish.


WHAT IS YOUR DEFINITION OF  SUM[k in N]1/2^k  THAT DOES NOT INVOLVE
THE LIMIT OF A SEQUENCE OF PARTIAL SUMS?

It is the diagonal number of the binary sequence (list) given below. 

Yet another dodge.  The diagonal NUMBER requires an interpretation
as a number represented by the diagonal STRING.  You refuse to state
your claimed definition of the real NUMBER represented by 0.111...,
which you say is *other* than a limit of a sequence of partial sums.

>>>Take a number from each line which is indexed by a natural
>>>number (but not by oo). Hence do not perform the limit process.

By your own construction, *every* line is indexed by a natural
number!  By your own prescription, just quoted, the *infinite*
string 0.111... is produced (which you already admitted):
FOR EVERY n IN N, there is a 1 in the nth position of 0.111...

But that is not sufficient to form 1/9 (decimal) or 1 (binary). 

What's "formed" is an infinite string.  You've agreed that the
infinite string 0.111.. *is* constructed by this prescription.
That infinite string is a standard binary representation of 1
as a limit of the sequence of partial sums.

Given your refusals to stick to the issue at hand,
I do not intend to participate further in this subthread.

--r.e.s. 

.

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  • Re: abundance of irrationals!)
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