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Topic: Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Replies: 42   Last Post: Oct 9, 2017 11:53 AM

 Messages: [ Previous | Next ]
 FromTheRafters Posts: 248 Registered: 12/20/15
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted: Oct 3, 2017 8:20 AM

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
>>>>>>> ...?

>>>>>>
>>>>>> 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 finitely-many, finite-length
>> 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

>>>
>
> 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.

Date Subject Author
10/2/17 Guest
10/2/17 netzweltler
10/2/17 Jim Burns
10/3/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 bursejan@gmail.com
10/4/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 Jim Burns
10/4/17 netzweltler
10/4/17 Jim Burns
10/4/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 FromTheRafters
10/6/17 netzweltler
10/6/17 Jim Burns
10/7/17 FromTheRafters
10/8/17 FromTheRafters
10/8/17 netzweltler
10/8/17 Jim Burns
10/8/17 netzweltler
10/8/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/7/17 Jim Burns