
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 6, 2017 7:42 PM


On 10/6/2017 6:03 AM, netzweltler wrote: > Am Freitag, 6. Oktober 2017 02:54:40 UTC+2 > schrieb Jim Burns: >> On 10/5/2017 3:12 PM, netzweltler wrote: >>> Am Donnerstag, 5. Oktober 2017 17:59:25 UTC+2 >>> schrieb Jim Burns: >>>> On 10/5/2017 10:00 AM, netzweltler wrote: >>>>> Am Donnerstag, 5. Oktober 2017 15:22:35 UTC+2 >>>>> schrieb Jim Burns:
>>>> [...] >>>>>> _We don't do what you're describing_ >>>>> >>>>> Nevertheless, >>>> >>>> "Nevertheless"? >>>> Do you agree that what you're describing >>>> is not what we're doing? >> >> *NETZWELTLER* >> DO YOU AGREE THAT WHAT YOU'RE DOING >> IS NOT WHAT WE'RE DOING? > > Let's say I agree. Doesn't mean that it is obvious to me > what *you* are doing.
Great. Let's say you agree. Will you stop saying that "0.999... means infinitely many commands"?
> All I've seen so far is, that you define 0.999... to be > the LUB or limit (I guess you use these expressions > interchangeably in this context) of the sequence > (11/10^n)n?N. That's it.
LUB and limit are not interchangeable, but they are closely related. It happens that, in this case, that LUB{ 1  1/10^k  k e N } is the same number as lim_n>oo 1  1/10^n
In this context, I prefer LUB because the argument is very short: That set of real numbers is bounded and nonempty. For every bounded and nonempty set of real numbers, there is a least upper bound. (This is what we mean by "real numbers".) That set has a least upper bound, which is 1. This is why we say 0.999... = 1.
Also, conceptually, LUB "happens all at once". There is no progression from point to point. Limits also "happen all at once" but some people don't get that. I like LUB in this context because there should be no way to mistake it as happening progressively in some infinite way.
 (i) Let us define the set P of decimal places as a set (which we can think of as { 1, 2, 3, ... } ) such that 1 e P (Ax)( x e P > x1 e P ) (all B sub P) ( 1 e B ) & (Ax)( x e B > x1 e B ) > ( B = P )
(What I _did not_ do in this definition is enumerate explicitly what the elements of P are. What we have instead is a way to prove in some cases whether some set we are interested in is P. This does not require infinite many statements to do.)
(ii) Let us define the set D of decimal digits D = {0,1,2,3,4,5,6,7,8,9}
(iii) Let us define an _infinite decimal fraction_ f as a map from places P to digits D.
We can represent that map as a subset f of the Cartesian product PxD such that (all x e P)(exists unique y e D)( (x,y) e f )
(In order to construct the particular map which represents 0.999... we would just say y = 9 for all x.)
(iv) Define D^P as the set of all functions P > D. We're using D^P to represent all the decimal fractions of real numbers  that is, [0,1].
(v) Define, for each map to digits f: P > D (an element of D^P) a map to finite sums S[f]: P > Q such that S[f](1) = f(1)*0.1 (all n e P) S[f](n1) = S[f](n) + f(n1)*10^n
(Remember, n is negative. The addition and multiplication is addition and multiplication for rationals.)
The image S[f](P) of the map to finite sums is set of the values of all the finite, terminating initial parts of the infinite decimal represented by f. If f were "infinitely many 9s following" S[f](P) = { 0.9, 0.99, 0.999, ... }
(vi) Define the _value_ of f in D^P (a real number Val(f)) as the least upper bound of the set S[f](P).
(all f e D^P) (all x e S[f](P))( x =< Val(f) ) & (all y e R) (all x e S[f](P))( x =< y ) > ( Val(f) =< y )
That's it. As you say, that's it.
We can define a function Val from the set of infinite decimals D^P to the real fractions [0,1], Val: D^P > [0,1] as above. A map P > D determines infinitely many digits (our infinite decimal), which determines infinitely many finite sums, which determine a set with a unique real number as its least upper bound. This LUB is the value of the infinitedecimal/map.
Every map to digits has a unique real that is its value, every real in [0,1] has at least one map to digits in D^P which is mapped to it.
(For some reals, there are more than one. 0.0999... and 0.1000... for example. This isn't actually any sort of problem. It's just a conceptual problem for some people who, through long familiarity with decimals, think of them as what real numbers "really" are.)
None of this has been defined using '...' It's possible I left some stuff out, but that can be defined without '...' too.
> Do we agree that a LUB or a limit is a point on the number line >  or to simplify it in case of a positive number  a point on > a geometric line from 0 to +infinity?
I'm going to continue to call these real numbers. Real numbers are a wellknown commodity, with axioms that we should be able to agree upon. Or we could call them the complete ordered field.
> I haven't seen any proof > for that yet other than the claim that it is. For some reason > coincident with point 1.
It is an axiom of the real numbers that every bounded nonempty set of reals has a least upper bound. The closest that I think we can come to a "proof" of an axiom is proving that something is a model of the formal theory with that axiom. This is something I've done for you in the past, large parts of that proof, at least. You could also look it up yourself.
If you "haven't seen any proof", it's possible that you've been ignoring large swaths of the explanations directed at you. Have you considered reading those?

