On 5/3/2013 11:51 AM, Dan wrote: > On May 3, 6:47 pm, WM <mueck...@rz.fh-augsburg.de> wrote: >> On 3 Mai, 08:45, Dan <dan.ms.ch...@gmail.com> wrote: >> >>> 1/9 = 0.1111 ..... (infinity ones) it's ALSO a valid continuous >>> magnitude . >> >> It is a limit, but it has no decimal representation. >> >> If it had a decimal representation, then it would be in the following >> list which contains all decimal representations with the digit 1 >> having a finite index. >> >> 0.1 >> 0.11 >> 0.111 >> ... >> >> Or can you determine a finite index which is not occupied by a 1? >> >> Being infinite is a valid mathematical notion, but it is not a means >> to distinguish decimal representations. If there is a sequence of 1 >> larger than all sequences of the list (not only larger than every), >> then let me know the difference by naming the indices. Otherwise stop >> asserting unmathematical properties. >> >> Regards, WM > > Every real number ,in decimal representation , is an infinite > sequence . But those on your list are followed by an infinity of > zeroes : > 0.1 ............. > 0.110000000 ..... > 0.111000000 ..... > > that we choose not to write , for convenience .
It is more than "convenience".
Let me qualify that with the provision that I hold certain non-standard views.
There is an issue here with the nature of the identity relation. One does not "know" that a terminated expansion is exact except in relation to the algorithm that generates the expansion.
However, in order for that algorithm to demarcate each terminated expansion from all other terminating expansions, the algorithm carries the presupposition of a completed infinity.
The difference between numerical identity (singular term) and equivalence is understood with respect to bivalent logic and mutually exclusive truth valuation.
Equivalence corresponds with the received paradigm of ontological self-identity:
Numerical identity, however, depends on something different:
(x=t_0) and (not(x=t_1)) and (not(x=t_2)) and (not(x=t_3)) and ...
(not(x=t_0)) and (x=t_1) and (not(x=t_2)) and (not(x=t_3)) and ...
(not(x=t_0)) and (not(x=t_1)) and (x=t_2) and (not(x=t_3)) and ...
(not(x=t_0)) and (not(x=t_1)) and (not(x=t_2)) and (x=t_3) and ...
where each t_i may be thought of as a canonical name. It is not that there is some particular collection of names that is special. Rather, it is that there is a presupposition of denotation by singular terms. The paradigmatic singular term is a name. So, while there need not be a particular canonical naming, the semiotics of naming is presupposed prior to the presupposition of denotation.
Observe that when only the ontological self-identity is taken into account, the list collapes to a Russellian enumeration,
(x=t_0) or (x=t_1) or (x=t_2) or (x=t_3) or ...
t_0 and t_1 and t_2 and t_3 and ...
This construct respects Bolzano's admonition against "doing violence to the language" by adhering to Wittgensteinian identity,
not(i=j) -> not(t_i=t_j)
That is, there is no need of a sign of equality for this presupposed enumeration of names. Each distinct inscription purports to represent an individual and each purport of representation excludes prior inscriptions from further use to prevent representational ambiguity.
In general mathematicians consider the Peano axiom,
AmAn((m+1=n+1) -> (m=n))
as a statement fixing the successor function as a well-defined function. Arguably, however, it expresses the fact that ordinal series implicitly define the identity relation on domains.
Among other things, one of the changes that had been made in the nineteenth century is an understanding of mathematics in terms of systems. The sign of equality becomes relativized to the identity relation of those systems.
If, for the moment, questions such as "What is a number?" or "Is a completed infinity legitimate?" are ignored, then the question of "How does the result of the Euclidean division algorithm designate an individual in relation to the system identity?" can be seen for the difficult problem that it is.
Of course, when one simply works within a system of axioms, the problem is obscured by the presupposition of denotation. It merely hides itself in set theory as the question of a well-ordering for the reals.