JT
Posts:
1,388
Registered:
4/7/12


Re: Finitely definable reals.
Posted:
Jan 12, 2013 12:31 AM


On 12 Jan, 06:27, JT <jonas.thornv...@gmail.com> wrote: > On 11 Jan, 23:30, Virgil <vir...@ligriv.com> wrote: > > > > > > > > > > > In article > > <8d0b862ec93542faba29f90eb0ead...@f25g2000vby.googlegroups.com>, > > > WM <mueck...@rz.fhaugsburg.de> wrote: > > > On 11 Jan., 12:36, Zuhair <zaljo...@gmail.com> wrote: > > > > On Jan 11, 12:49 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > Please answer one question: What shall undefinable reals be good for? > > > > > Explaining continuity of space? Possibly? > > > > Something that is underfined and hence unexplained should be able to > > > explain something? > > > Further space is not continuous. > > > The real line is!http://en.wikipedia.org/wiki/Real_number > > Quote: > > The currently standard axiomatic definition is that real numbers form > > the unique Archimedean complete totally ordered field (R,+,·,<), up to > > isomorphism,[1] whereas popular constructive definitions of real numbers > > include declaring them as equivalence classes of Cauchy sequences of > > rational numbers, Dedekind cuts, or certain infinite "decimal > > representations", together with precise interpretations for the > > arithmetic operations and the order relation. These definitions are > > equivalent in the realm of classical mathematics. > > The reals are uncountable, that is, while both the set of all natural > > numbers and the set of all real numbers are infinite sets, there can be > > no onetoone function from the real numbers to the natural numbers > > End quote. > > > > But my question aimed at the application of undefined reals in > > > mathematics. > > > Without them one cannot have a complete infinite Archimedean ordered > > field such as the real number field. > > > > > > They cannot spring off Cantor's argument. > > > > > They do of course, they are a consequence of his arguments. > > > > No, you misunderstand again. Cantor's opinion was (and did not change > > > until he died) that undefined items are nonsense. And ofcourse he was > > > absolutely right. > > > But he still showed that the set of real numbers, i.e., the objects > > forming the unique Archimedean complete totally ordered field was not a > > countable set in the sense that no surjection from N to that set is > > possible. > > > > > Cantor proved that the > > > > > > *definable* reals (those which are definitely different from all reals > > > > > of his list) cannot be put in bijection with N. > > > > > You mean * discernible* reals, there is a difference between > > > > discernible reals and finitely definable reals, two reals might be > > > > discernible (i.e. differ at some finite position of their decimal > > > > expansions) and yet each one of them might be non finitely definable! > > > > Nonsense. If a real number is not finitely definable, then it has no > > > positions. > > > One may know that a real is between 0.1 and 0.2 but still not finitely > > definable. In fact one may know a real accurate to any finite number of > > decimals places but still have it undefineable any further. > > > If you know, say, only the digits of the first three finite > > > > positions, then you have not an undefined real but you have an > > > interval with two rationals as limits, in decimal you have the > > > interval between 0.abc000... and 0.abc999... > > > > You cannot define a real number by increasing step by step the number > > > of known digits. You would never arrive at a point. All you do is > > > shrinking the interval. In order to define a real number you need a > > > finite definition that describes all nested intervals. > > > So if you only know its first n digits, that number is one of those > > undefineables that WM claims do nt exist.. > > > > > YES Cantor proved that his Diagonal real is * discernible* from all > > > > the other members of the list, AS FAR AS THAT LIST IS COUTNABLE, but > > > > that doesn't make out of it *finitely definable*; for it to be > > > > finitely definable it must UNIQUELY satisfy some finite predicate and > > > > proving it discernible doesn't by itself make out of it finitely > > > > definable. Cantor's arguments tells us that we do have MORE > > > > discernible reals than finitely definable ones. We do have UNCOUNTABLY > > > > many discernible reals but we have only COUNTABLY many finitely > > > > definable reals. > > > > There are two cases: > > > 1) If a Cantor list is finitely defined, then you know the entry in > > > every line and you know every digit of the diagonal. > > > So that every list of finitely defined basal numerals, with base >=4, > > is incomplete since its antidiagonal is not listed. > > > > 2) If a Cantor list is undefined and has only, as usual, the first > > > three lines and then an "and so on", then you do neither know the > > > following entries nor the digits of the diagonal. Nothing is > > > "discernible" then except the theorem that two decimals which differ > > > at some place are not identical. But that is not a deep recognition. > > > But any assertion that a list of basal numerals is COMPLETE is > > falsified by the existence of antidiagonals which are provably not in > > the original list. > > > > > > But we know that they> are countable. Undefinable reals are not elements of > > > > > mathematics and > > > > > of Cantorlists. They cannot help to make the defined diagonals belong > > > > > to an uncountable set. > > > > > No some of Non finitely definable reals ARE members of Cantorlists. > > > > Actually for some lists the diagonala is provabley (by Cantor's > > > > arguments) non finitely definable! > > > > Actually some *lists* are not finitely definable (not only the > > > diagonals), and therefore these lists are undefinable. > > > To prove countability of a set, one must be able to prove a surjection > > from N to that set, which is, effectively, proving that one can list > > all its members. > > > So that set which cannot be shown to be listable cannot be shown to be > > countable. > > > And when one can show that any attempted listing is incomplete, one has > > sown that the set cannot be listed. > > > At least that is how things work in the mathematics outside of > > WMytheology > > > > In fact *all* list, that have no finite definition are undefined, > > > i.e., not existing! > > > That is irrelevant when showing that any listing is impossible. > > > > And therefore also their diagonals and anti > > > diagonals are undefined, i.e., not existing. > > > If a list could exist, so must its antidiagonal, so the nonexistence of > > an antidiagonal proves the nonexistence of any list. > > > > Therefore there is > > > nothing "discernable". It is simply not existing. > > > Prove of countability implies listability (A surjection from N to a set > > is a list, possibly with repetitions, of the set's members) > > > Thus disproof of listability proves uncountability. > > > At least everywhere but in WMytheology > > > > > But of course all elements on > > > > Cantor's list and the diagonal (or antidiagonal) all are definitely > > > > discernible (i.e. differ from each OTHER real at some finite position > > > > of their decimal expansions). > > > > But as you don't know the "each" and "other" you don't know anything. > > > You don't have to know them individually to prove unlistability > > > > No. You are confusing intervals and numbers and defined lists and > > > undefined "lists", i.e., not existing "lists". > > > Prove of countability implies listability (A surjection from N to a set > > is a list, possibly with repetitions, of the set's members) > > > Thus disproof of listability proves uncountability. > > > At least everywhere but in WMytheology > > > The standard definition of "countable": > > > A set S is COUNTABLE if and only if there is a surjection from the set > > of natural numbers, N, to the set S. > > Note that for finite sets, S, this cannot be a bijection. > > For countably infinite sets S, it can be but need not be. > > For uncountably infinite sets, S , no mapping from N to S can be > > surjective at all. > > >  > > The real numberline is a joke, the naturals are discrete entities that > have a continum. There is no 0 in the set of naturals, there is no > infinity in the set of naturals. There is no 0 in the set of reals > there is no infinity in the set of reals. They say a real number is a > value that represents a quantity along a continuous line, but the fact > you can make the same cut anywhere on the line representing number 3, > already there tells the truth about natural numbers, they are discrete > entities counted in and so are the reals 3.14 represent a set of 3 > entities and one fractioned entities. The decimal expansion introduce > the zeros because of partition of the imaginary fake numberline. That > is why zeros was introduced in the first place. The idea of zero is > meaningless there is no number zero between discrete elements, this > does not mean null isn't a valuable term of description, but it is not > a mathematical entity. If you line up a series of discrete elements > and cherrypick where the first positive element within the set start, > there is no zero entity between 1.+1 just a void depending upon the > fact that the numberline just isn't there it is created by counting, > there is no least part of the first natural where it is zero just > fractions ***as you count them in***
I think my view of numbers very closely relate to Platos idea of numbers. I have no idea who actually constructed the numberline and introduced zero, but i think it came with the abacus was us arabic or hinduic?

