netzweltler
Posts:
460
From:
Germany
Registered:
8/6/10


Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 3, 2017 3:15 PM


Am Dienstag, 3. Oktober 2017 14:20:11 UTC+2 schrieb FromTheRafters: > netzweltler brought next idea : > > Am Dienstag, 3. Oktober 2017 03:22:11 UTC+2 schrieb Jim Burns: > >> On 10/2/2017 2:47 PM, netzweltler wrote: > >>> Am Montag, 2. Oktober 2017 20:35:56 UTC+2 > >>> schrieb Jim Burns: > >>>> On 10/2/2017 1:58 PM, netzweltler wrote: > >>>>> Am Montag, 2. Oktober 2017 17:59:21 UTC+2 > >>>>> schrieb Jim Burns: > >>>>>> On 10/1/2017 3:22 AM, netzweltler wrote: > >>>>>>> Do you agree that 0.999... means infinitely many commands > >>>>>>> Add 0.9 + 0.09 > >>>>>>> Add 0.99 + 0.009 > >>>>>>> Add 0.999 + 0.0009 > >>>>>>> ...? > >>>>>> > >>>>>> 0.999... does not mean infinitely many commands. > >>>>> > >>>>> But that's exactly what it means. > >>>> > >>>> That's not the standard meaning. > >>> > >>> So, you disagree that > >>> 0.999... = 0.9 + 0.09 + 0.009 + ... ? > >> > >> Your '...' is not usable. If we say what we _really_ mean, > >> in a manner clear enough to reason about, then the '...' > >> disappears. Also, what we are left with are finitely many > >> statements of finite length. You will not find infinitely > >> many commands in those finitelymany, finitelength > >> statements. > >> > >> We sometimes write the set of natural numbers as > >> { 0, 1, 2, 3, ... } > >> The '...' is informal. We do not use '...' in our reasoning, > >> we use a correct description of what the '...' stands for. > >> > >> Do you see '...' anywhere in the following? > >> > >> The set N contains 0, and for every element x in N, its > >> successor Sx is in N. > >> > >> This is true of N but not true of any _proper_ subset of N. > >> > >> _Therefore_ , if we can prove that B is a subset of N > >> which contains 0 and which, for element x of B, contains Sx, > >> then B is not a _proper_ subset of N. > >> > >> B nonetheless is a subset of N, we just said so. The only subset > >> of N which B can be is N. Therefore, B = N. > >> > >> This is finite reasoning about the infinitely many elements > >> in N. Note that there is no '...' in it. > >> > >> I could continue and derive 0.999... = 1 from our definitions, > >> and nowhere in that derivation will be '...'. There will not be > >> infinitely many commands in it either. > >> > >>>> You give it some other meaning, and then you find a problem > >>>> with the meaning you gave it. Supposing I wanted to sort out > >>>> what that other meaning was, and how to make sense of it, my > >>>> attention to your meaning would not affect the standard meaning. > >>>> > >>>> I am not a math historian, but the impression I have > >>>> is that great care was taken in choosing the standard meaning > >>>> in order to avoid problems like the ones you are finding. > >>>> > >>>> You have the ability to create and then wallow in whatever > >>>> problems you choose. No one is able to take that power away > >>>> from you. But you can't "choose" by an act of your will to > >>>> make your created problem relevant to what everyone else > >>>> is doing. You are not the boss of us. > >>>> > >>>>> Infinitely many commands. Infinitely many additions. > >>>>> Infinitely many steps trying to reach a point on the number line. > >>>>> > >>>>>> There is a set of results of certain finite sums, a set of > >>>>>> numbers. We can informally write that set as > >>>>>> { 0.9, 0.99, 0.999, ... } > >>>>>> That is an infinite set, but we can give it a finite description. > >>>>>> > >>>>>> (Our finite description won't use '...'. The meaning of > >>>>>> '...' depends upon it being obvious. If we are discussing > >>>>>> what '...' means, it must not be obvious, so we ought to > >>>>>> avoid using '...') > >>>>>> > >>>>>> There is number which is the unique least upper bound of that set. > >>>>>> The least upper bound is a finite description of that number. > >>>>>> > >>>>>> 0.999... means "the least upper bound of the set > >>>>>> { 0.9, 0.99, 0.999, ... }". > >>>>>> That number can be show to be 1, by reasoning in a finite manner > >>>>>> from these finite descriptions of what we mean. > >>>>>> > >>>>>> If you give 0.999... some meaning other than what we mean, > >>>>>> and then it turns out there are problems of some sort with > >>>>>> your meaning, than that is your problem, not ours. > >>> > > > > Sorry, no. The meaning of "..." is absolutely clear in this context and we > > both know that there is a decimal place for each n ? N in 0.999... > > But 0.999 repeating is a rational number, no need for repeating > decimals at all in the naturals. Repeating zeros is okay I guess, but > why use them in the naturals. In the rationals and reals, repeating > zeros are called 'terminating' decimal expansions and the trailing > zeros are elided.
I'm not sure if you got what I meant. Let me rephrase it:
The meaning of "..." is absolutely clear in this context and we both know that there is a nth decimal place for each n ? N in 0.999...

