
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 2, 2017 9:22 PM


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. >

