Re: abundance of irrationals!)

"W. Mueckenheim" <mueckenh@xxxxxxxxxxxxxxxxx> ... 
"r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote ...

The first step of *your* putative definition of sum[k in N] 2^-k leads to the infinite string 0.111..., as you've quoted above.

But you now refuse to follow through with the next step -- that of defining exactly what is *meant* by 0.111... -- without admitting that it is just the binary representation of unity (i.e. the limit of an infinite sequence of partial sums) as used standardly in
diagonal arguments. 

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.

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 ...

*First* you agreed that the infinite string 0.111... is the result of *your* first step in a putative definition of SUM[k in N]1/2^k that
doesn't resort to a limit of a sequence of partial sums.

*Now* you say that this very same infinite string 0.111... *does* represent the limit of a sequence of partial sums! *That* is the
agreed-upon definition that allows writing 0.111... (binary) = 1.
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!


But I do not accept
that 0, as a term, belongs to the sequence (1/n). 

.... which has nothing to do with the issue at hand, namely,
WHAT IS YOUR DEFINITION OF  SUM[k in N]1/2^k  THAT DOES NOT INVOLVE
THE LIMIT OF A SEQUENCE OF PARTIAL SUMS?

And it is similarly wrong to believe that all the digits in 0.111... can be enumerated by natural numbers. 

What a bizarre thing to say -- you don't recognise that it
contradicts your own previous statements!  You said:

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...


Hence, from the following list,

0.1
0.11
0.111
...

we obtain two theorems:

A) the nth partial sum of the diagonal can always be found in the nth
line.
B) The binary representation of 1 is not in a line. Conclusion:
C) The binary representation of 1 is not in the diagonal.
Set theory denies C. So set theory is incompatible with {limit-mathematics} u {logic}. 

As usual, your "conclusion" does not follow from the premises.
.

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