
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 3, 2017 10:44 AM


Jim Burns used his keyboard to write : > On 10/3/2017 3:21 AM, netzweltler wrote: >> 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. > >> Sorry, no. The meaning of "..." is absolutely clear in this >> context and > > Is it clear to you? Really? > > I ask because the basis for your whole complaint, in many > threads, is that '...' means "infinitely many commands" in > some way but then you're all "Whoa! that makes no sense, guys". > It does not look to me as though _what you think_ '...' > means in this context is at all clear _to you_ . > > ( _What you think_ it means is not what it means. This is > _my_ point.) > >> we both know that there is a decimal place for each n ? N >> in 0.999... > > And what does that mean? Have you traded one thing that needs > explaining for another thing that needs explaining? It's not > very useful to do that. > > You refer to N here. What is N? Do you need to use the > successor operation infinitely many times to say what N is? > Is it clear to you what that means? > >  > N has a finite description. It is the minimal inductive set > with 0 and successor x > x+1. > > When we use N to describe something, for example, the set of all > finite initial expansions of 0.999..., > { 0.9, 0.99, 0.999, ... } > the use of N will not make that description infinite. > > The value that we assign to 0.999... is the least upper bound > of the set of all finite initial expansions of 0.999... > This is a _definition_ . > > And that assigned value is 1. Not "nearly 1", exactly 1.
Indeed, and of what use is a decimal point in the naturals? I think that he should at least use rationals if he wants to use decimal points. In that case, it seems obvious to me that 0.999 repeating, if different from 1.000 repeating, would have another rational between them since rationals are dense. If that is so, then 0.999 repeating is not as close as he thinks it is to 1.000 repeating.
Of course they are representing the same number after all, and even in the reals one cannot be close enough to the other because they are dense too  always a number between two 'different' numbers and never a number between two representations of the same number.

