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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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 netzweltler Posts: 473 From: Germany Registered: 8/6/10
Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Posted: Oct 3, 2017 3:25 PM
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Am Dienstag, 3. Oktober 2017 16:20:25 UTC+2 schrieb Jim Burns:
> 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 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?
>
> 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.)

Sorry, I don't get what you are trying to teach me.

> > 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

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