Das Kalenderblatt 090826

25/08/2009 - 21:32 von WM | Report spam
Other Versions of Richard's Paradox
(A) The version given in Principia Mathematica by Whitehead and
Russell is similar to Richard's original version, alas not quite as
exact. Here only the digit 9 is replaced by the digit 0, such that
identities like 1.000... = 0.999... can spoil the result. {{Dies ist
aber nicht der einzige Irrtum in Russell's Lebenswerk.}}
(B) Berry's Paradox, first mentioned in the Principia Mathematica as
fifth of seven paradoxes, is credited to Mr. G. G. Berry of the
Bodleian Library. It uses the least integer not nameable in fewer than
nineteen syllables; in fact, in English it denotes 111,777. But "the
least integer not nameable in fewer than nineteen syllables" is itself
a name consisting of eighteen syllables; hence the least integer not
nameable in fewer than nineteen syllables can be named in eighteen
syllables, which is a contradiction
(C) Berry's Paradox with letters instead of syllables is often related
to the set of all natural numbers which can be defined by less than
100 (or any other large number) letters. As the natural numbers are a
well-ordered set there must be the least number which cannot be
defined by less than 100 letters. But this number was just defined by
65 letters including spatia.
(D) König's Paradox was also published in 1905 by Julius König. All
real numbers which can be defined by a finite number of words form a
subset of the real numbers. If the real numbers can be well-ordered,
then there must be a first real number (according to this order) which
cannot be defined by a finite number of words. But the first real
number which cannot be defined by a finite number of words has just
been defined by a finite number of words.
(E) The smallest natural number without interesting properties
acquires an interesting property by this very lack of any interesting
properties.
(F) A loan of the Paradox of Grelling and Nelson. The number of all
finite definitions is countable. In lexical order we obtain a sequence
of definitions D1, D2, D3, ... Now, it may happen that a definition
defines its own number. This would be the case if D1 read "the
smallest natural number". It may happen, that a definition does not
describe its own number. This would be the case if D2 read "the
smallest natural number". Also the sentence "this definition does not
describe its number" is a finite definition. Let it be Dn. Is n
described by Dn? If yes, then no, and if no, then yes. The dilemma is
irresolvable.
Solution of Richard's Paradox
The solution of Richard's paradox may be explained most easily by
version (C). If all definitions with less than 100 letters already are
given, then also the sequence of letters Z = "the least number which
cannot be defined by less than 100 letters" does define a number. As
an example, let a = 01, b = 02, c = 03 etc. By "example" we have
defined the number
example = 05240113161205
and also the sequence Z would already define a number. Only by the
change of language from numeral to colloquial the apparent paradox
occurs.
{{Im Gegensatz dazu "gibt es" (wer ist "es"?) nach Ansicht der
Mengenlehrer (zu deren Nutz und Frommen diese Erbauungslektüre gedacht
ist)) "reelle" Zahlen, die nicht real(-isierbar) sind, weil sie in
keiner Sprache definiert werden können denn das kartesische
Produkt aller Symbole, aller daraus gebildeten endlichen Wörter und
aller ihrer Interpretationen/Bedeutungen ist abzàhlbar.
Weil aber Zahlen nirgendwoanders existieren als in Speichern, können
Zahlen, die auch dort nicht existieren können, gar nicht existieren}}

Gruß, WM

http://en.wikipedia.org/wiki/Jules_Richard
(Aus der Version vom 19. 4. 2007 entnommen.)
 

Lesen sie die antworten

#1 Albrecht
26/08/2009 - 12:09 | Warnen spam
On 25 Aug., 21:32, WM wrote:
Other Versions of Richard's Paradox
(A) The version given in Principia Mathematica by Whitehead and
Russell is similar to Richard's original version, alas not quite as
exact. Here only the digit 9 is replaced by the digit 0, such that
identities like 1.000... = 0.999... can spoil the result. {{Dies ist
aber nicht der einzige Irrtum in Russell's Lebenswerk.}}
(B) Berry's Paradox, first mentioned in the Principia Mathematica as
fifth of seven paradoxes, is credited to Mr. G. G. Berry of the
Bodleian Library. It uses the least integer not nameable in fewer than
nineteen syllables; in fact, in English it denotes 111,777. But "the
least integer not nameable in fewer than nineteen syllables" is itself
a name consisting of eighteen syllables; hence the least integer not
nameable in fewer than nineteen syllables can be named in eighteen
syllables, which is a contradiction
(C) Berry's Paradox with letters instead of syllables is often related
to the set of all natural numbers which can be defined by less than
100 (or any other large number) letters. As the natural numbers are a
well-ordered set there must be the least number which cannot be
defined by less than 100 letters. But this number was just defined by
65 letters including spatia.
(D) König's Paradox was also published in 1905 by Julius König. All
real numbers which can be defined by a finite number of words form a
subset of the real numbers. If the real numbers can be well-ordered,
then there must be a first real number (according to this order) which
cannot be defined by a finite number of words. But the first real
number which cannot be defined by a finite number of words has just
been defined by a finite number of words.
(E) The smallest natural number without interesting properties
acquires an interesting property by this very lack of any interesting
properties.
(F) A loan of the Paradox of Grelling and Nelson. The number of all
finite definitions is countable. In lexical order we obtain a sequence
of definitions D1, D2, D3, ... Now, it may happen that a definition
defines its own number. This would be the case if D1 read "the
smallest natural number". It may happen, that a definition does not
describe its own number. This would be the case if D2 read "the
smallest natural number". Also the sentence "this definition does not
describe its number" is a finite definition. Let it be Dn. Is n
described by Dn? If yes, then no, and if no, then yes. The dilemma is
irresolvable.
Solution of Richard's Paradox
The solution of Richard's paradox may be explained most easily by
version (C). If all definitions with less than 100 letters already are
given, then also the sequence of letters Z = "the least number which
cannot be defined by less than 100 letters" does define a number. As
an example, let a = 01, b = 02, c = 03 etc. By "example" we have
defined the number
example = 05240113161205
and also the sequence Z would already define a number. Only by the
change of language from numeral to colloquial the apparent paradox
occurs.
{{Im Gegensatz dazu "gibt es" (wer ist "es"?) nach Ansicht der
Mengenlehrer (zu deren Nutz und Frommen diese Erbauungslektüre gedacht
ist)) "reelle" Zahlen, die nicht real(-isierbar) sind, weil sie in
keiner Sprache definiert werden können denn das kartesische
Produkt aller  Symbole, aller daraus gebildeten endlichen Wörter und
aller ihrer Interpretationen/Bedeutungen ist abzàhlbar.
Weil aber Zahlen nirgendwoanders existieren als in Speichern, können
Zahlen, die auch dort nicht existieren können, gar nicht existieren}}

Gruß, WM

http://en.wikipedia.org/wiki/Jules_Richard
(Aus der Version vom 19. 4. 2007 entnommen.)



Über die Frage, ob Zahlen existieren und wenn ja, wie, kann man sich
offensichtlich trefflich streiten. Mein Standpunkt ist folgender:

Wir erfahren die Aspekte unserer Realitàt als i.d.R. aus Teilen
zusammengesetzten Konglomerate. Ein Stein, ein Gedanke, ein Molekül,
ein Text, ein Baum, eine Menschenmasse, ein Haus, ein Staat, ...
bestehen aus Untereinheiten, die wiederum aus Untereinheiten
zusammengesetzt sind.

Ist ein solches Konglomerat eindeutig definiert, und ist eindeutig
definiert was dessen Untereinheiten ausmacht, so kommt diesem
Konglomerat in Bezug auf diese Untereinheiten eine Zahl, und zwar eine
natürlicher, zu. Damit ist eine Zahl eine Eigenschaft eines
Konglomerates und insofern existent wie etwa die Eigenschaft Farbe,
Oberflàchenbeschaffenheit, Masse, Form, etc. existent sind.


Gruß
Albrecht

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