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Topic: § 417 An implication of actual infinity
Replies: 3   Last Post: Jan 11, 2014 6:52 PM

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 Ben Bacarisse Posts: 751 Registered: 7/4/07
Re: § 417 An implication of actual infinity
Posted: Jan 10, 2014 3:19 PM

WM <wolfgang.mueckenheim@hs-augsburg.de> writes:

> Am Freitag, 10. Januar 2014 01:52:42 UTC+1 schrieb Ben Bacarisse:
<snip>
>> The system
>> you are talking about in which numbers are "created by a union" and
>> unions can "fail to produce" numbers is your own invention and its
>> limitation are of your own making. I hope you don't tell you students
>> any of this nonsense.

>
> I tell them every year (some courses even every semester) that a
> strictly increasing sequence does not contain its limit.

Your ability to avoid the issue is phenomenal. Do you tell them that
limit does not exist? That was the point. It does in mathematics, it
does not (apparently) in WMaths. I am glad you tell them some true
things about sequences and limits, but my question was whether you tell

>> digit sequences. I repeat it here in case you missed it. The context
>> is the list of rationals 0.0, 0.1, 0.11, 0.111, and so on:
>>
>> (a) every digit of the anti-diagonal is 1
>> (b) 1/9 has the decimal expansion that puts a 1 at every index n
>> (c) 0.111... expresses a number where every index n has digit 1
>> (d) 0.111... = 1/9

>
>
> By the finite definition of the list its antidiagonal has a 1 at every
> place. Therefore the antidiagonal is 1/9 = "0.111...". These are
> finite definitions that put a 1 at every place indexed by n. But
> "every" means only every finite place such that infinitely many places
> remain - in eternity.

So which list element has these properties? That 'the antidiagonal is
1/9 = "0.111..."' is enough to show that it differs from every list
element.

>> (e) the anti-diagonal is not 1/9
>
> 1/9 cannot be obtained from the anti-diagonal when it is listed digit
> by digit.

Of course! The anti-diagonal is 1/9 and it is not 1/9. Why did I not
see that simple explanation? It has the advantage that whatever anyone
says about the anti-diagonal you can say they are wrong.

<snip>
--
Ben.

Date Subject Author
1/10/14 Ben Bacarisse
1/11/14 wolfgang.mueckenheim@hs-augsburg.de
1/11/14 Virgil