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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

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 Jim Burns Posts: 831 Registered: 9/26/15
Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Posted: Oct 3, 2017 10:20 AM

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
>>>>>>> ...?

>>>>>>
>>>>>> 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.

> Sorry, no. The meaning of "..." is absolutely clear in this
> context and

Is it clear to you? Really?

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.

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