Das Kalenderblatt 120127

26/01/2012 - 08:50 von WM | Report spam
Das Kalenderblatt 120127

WM: The problem boils down to the following:
En Am: m =< n <==> Am En m =< n [*]
En Am: m =< n ==> Am En m =< n [**]
You know: Classical logic was obtained from finite sets ... Show me a
finite set that obeys [**] but not [*].
DW: The above is not contested: [**] implies [*].
WM: I said: For complete linear sets [*] is true. You said [*] is
not true, but [**] is true.
Weyl said: Classical logic was obtained from finite sets.
Therefore I asked you: Show me a linear complete finite set, that
makes your claim [**] right and my claim [*] wrong.
DW: What is contested is that:
En Am: m =< n <== Am En m =< n [***]
implies [*]. And *that* is the form you do use.
WM: No. I do not use the implication only, I use the full
equivalence. Of course the equivalence includes the implication [***]
as well as the implication [**]
DW: There is a trivial finite counter-example. Take three dice
where on each of the sides one of the numbers one to nine is printed
(some of them repeated). Say the set is {d1, d2, d3}. Define di < dj
when the probability to throw a higher number with dj than with di is
larger than to 1/2. (I would submit that all this is quite
physical.) There is a set of three dice such that d1 < d2, d2 < d3
and d3 < d1.
And so we have:
Am En d_m < d_n (d1 < d2 < d3 < d1)
but not
En Am d_m < d_n (there is no best die).
WM: Of course there is no best die. Therefore this set is not
linear. Every finite set of natural numbers has a "best" number.
Why do you bother with such nonsense examples? The answer is easy:
Because you have no other examples.
[Dik T. Winter, "Answer to Dik T. Winter", sci. logic, 26. 5. 2009]

{{Zu meiner Sammlung von Beispielen aus Tànzern, Tànzerinnen, roten
Kreisen und grünen Dreiecken, die das Verbot der Quantorenvertauschung
begründen sollen, gesellten sich hier nun die nichtlinearen Würfel,
eng verwandt mit Stàrkevergleichen zwischen Schachspielern oder
Fußballclubs oder mit relativen Beliebtheitswerten von Schauspielern
oder Politikern.
Natürlich sind alle diese Beispiele verfehlt, denn die ganzen
Zahlen bilden eine linear geordnete Menge. Für jede endliche linear
geordnete Menge gilt nun einmal die mit [*] bezeichnete Äquivalenz.
Und da die Logik anhand von endlichen Mengen entwickelt worden ist,
sollte das Verbot der Quantorenvertauschung erst einmal anhand einer
endlichen linear geordneten Menge begründet werden, was absolut
unmöglich ist. Dass die temporàr größte Zahl einer potentiell
unendlichen Menge niemals bekannt sein kann, sollte doch den anhand
von Zermelos Wohlordnungsatz zum Glauben an das Unsichtbare bereiten
Mengenlehrer nicht wirklich von ihrer Nichtexistenz überzeugen.}}

PB: Yet there is increasing scepticism about the objective truth of
the
Continuum Hypothesis and similar statements.
TC: I'm curious about this "increasing scepticism" that you speak
of. Do you have any statistical evidence of increasing scepticism?
WM: Concerning the question of statistical evidence for growing
scepticism I can report my personaI efforts over many years: When I
teach Cantor's diagonal argument, every student understands that the
real numbers are uncountable (because it is
really not hard to understand that argument).
When I represent all the real numbers of the unit interval by the
paths of an infinite binary tree with a countable number of nodes,
every student understands that there cannot be more paths than nodes
(because it is really not hard to understand that argument). No
student of mine has ever argued against that, although that would not
have changed her marks!
In this way I have contributed to increase the scepticism against
transfinite set theory by some hundreds of heads (that are not below
average intelligence).
[Paul Budnik, Tim Chow, "The boundary of objective mathematics", FOM,
18. 3. 2009]

CM: Ultrafinitism is in fact a coherent, if ultimately unfeasible,
philosophical/mathematical theory. While WM's philosophical outlook
might best be described as ultrafinitist, his exposition is confused
and his reasoning is flatly unsound.
WM: Why do you think so? Here is the shortest reasoning: Construct
the binary tree from a countable set of paths. Let someone else find
one of the uncountably many paths suspected to be in the tree. See him
fail. Cp. no. 11 of
http://www.hs-augsburg.de/~mueckenh/GU/Pruefung%20GU0907.pdf
[Chris Menzel, "Would it matter if ZF was inconsistent?", sci.logic,
18. 7. 2009]

PW: I am still waiting for your construction of the tree. This is
done by specifying the set of nodes, and the set of links between
nodes.
WM: A definition can be done by text or by drawing, like here:
http://www.hs-augsburg.de/~mueckenh/GU/GU12c.PPT#361,34,Folie 34
My definition has the property that the number of paths, by logical
reasons, cannot be larger than that of nodes.
PW: If you think you have been successful, as a first step please
determine what N corresponds to the Real 1/3. If there is no such N,
then you have failed.
WM: In my game "conquer the binary tree", node number 1 may
correspond to 1/3. Would that be alright?
[Peter Webb, "Would it matter if ZF was inconsistent?", sci.logic, 21.
7. 2009]

Gruß, WM
 

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#1 Michael Klemm
26/01/2012 - 11:42 | Warnen spam
WM wrote:

WM: The problem boils down to the following:
En Am: m =< n <==> Am En m =< n [*]
En Am: m =< n ==> Am En m =< n [**]



Jedes Element der Menge

{{1}, {1, 2}, {1, 2, 3}, }

besitzt ein maximales Element, die Menge selbst
jedoch nicht.

Gruß
Michael

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