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Topic: Matheology § 300
Replies: 27   Last Post: Jul 12, 2013 6:58 PM

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Ralf Bader

Posts: 488
Registered: 7/4/05
Re: Matheology § 300
Posted: Jul 11, 2013 3:37 PM
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Julio Di Egidio wrote:

> "fom" <> wrote in message

>> On 7/8/2013 4:45 PM, wrote:
>>> On Monday, 8 July 2013 23:24:10 UTC+2, FredJeffries wrote:
>>>> On Monday, July 8, 2013 10:39:01 AM UTC-7,
>>>> wrote: > > Hahaha. And if enumerated, you think you can be sure to get
>>>> even the last one out???

>>>> Explain it slowly:
>>> If "all" could leave the urn, one must have left without another
>>> remaining there. Or go many together?
>>> Explain it slowly: In scientific applications of "all" there is always
>>> and "end signal", a last one.

>> Except when WM is asked for that number which is the last number.

> WM would not agree there is a last one, mind the confusion. Indeed, I
> think he is quite correct here: he is saying "to have *all*" we need an
> "end-signal", which looks perfectly sensible, IMO.

/We/ certainly do not need an "end-signal". When I was a little child for a
while I had problems to imagine how the sea looks from the shore. I had
been told that there is an endless surface of water but in my imagination
always an "end-signal" in the form of a shore on the opposite side popped
up. The issue could be resolved instantaneously by a picture or actually
seeing the sea. If you need an "end-signal" then this your personal problem
of a psychological nature but it is totally irrelevant for mathematics.

Neither are those stories about urns of any mathematical relevance. They are
pictures which may elucidate some issues but they obviously can also
obscure matters. In set theory /we/ talk about sets, elements of sets,
mappings between sets and so on, but not about Mückenheimian crap. And the
urn story needs to be reshaped in that language of set theory. Then, for
example, there is a mapping f:[0,1) -> P(IN) from the time interval [0,1)
to the power set of the natural numbers which associates to any point of
time t the set of numbers f(t) sitting in the urn at that moment. Set
theoretically there is only that mapping f, and it is not defined at t=1;
the urn story just tells which balls are in the urn at times before 1 but
says nothing about the moment t=1. However that mapping f does not need
an "end-signal" to be able to be undefined at an endpoint of its interval
of definition and to be defined at all points in [0,1).

Then there is a set-theoretical gadget in the form of lim_(t->1) f(t) which
exists under favourable circumstances and then it is a subset of IN and it
may be taken as f(1). And if that urn experiment could be carried out in
reality (but there is not the slightest hint how anything of that kind
should be accomplished and therefore it is also completely open whether any
kind of "end-signal" would be technically involved in such experimental
setup) and the final result could be inspected one might find that it
coincides with lim_(t->1) f(t), or one might find that it differs from this
limit. This then are empirical observations which may consolidate into laws
of physics but they do not say anything about set theory which is used as a
language to describe these empirical or physical issues.

In the Cantor diagonal argument there is e.g. a mapping f:INxIN -> D =
{0,...,9} such that f(i,j) is the jth decimal of the ith real number in
the "list". Real numbers, say from the interval [0,1) are seen here as
mappings from IN to D giving their decimal expansion (neglecting for the
moment a slight incorrectness due to the non-uniqeness of decimal
expansions). And given f as above we can create g:IN -> D by g(i) = f(i,i)
and this g is a real number and no limit needs to be taken to obtain it.
Nothing like the extension of the domain of definition as in the urn
situation is involved here, contrary to Mückenheim's crap. And I would say
that this is such easy stuff that one must be a real idiot if one neither
comprehends it within ten years nor gets the feeling during that time that
one should give up on it and spend one's time in some more fruitful way.
And I don't consider it of any relevance if an idiot disagrees.

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