
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 3, 2017 3:55 PM


netzweltler laid this down on his screen : > 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...
Okay, so the natural number is only an index for positions in the endless (ad infinitum) string of nines in the decimal expansion.
We're still in the reals, good. :) Sorry for my confusion.

