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| David R Tribble |
 Posted: Wed Aug 30, 2006 5:14 pm |
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[quote:6ad3a4cdef]But since no natural requires an infinite bit string, that is irrelevant.
[/quote:6ad3a4cdef]
Tony Orlow schrieb:
[quote:6ad3a4cdef]If no natural requires an infinite bit string, even the very largest,
and all bit positions are included in it, then no infinite set of bit
positions is required.
[/quote:6ad3a4cdef]
mueckenh wrote:
[quote:6ad3a4cdef]Tony, please drop the "very largest". The rest is ok. If no natural
requires an infinite set then all naturals together do not require an
infinite set. If the contrary is asserted (and if infinity is a number
larger than any finite number), then we may ask which natural is the
first such that all of its predecessors require an infinite number of
bit positions.
[/quote:6ad3a4cdef]
There is no natural with an infinite number of bits. Every natural
is a finite binary string. There is no natural with an infinite
number of predecessors. Every natural has an infinite number
of successors, however. (Which agrees with what you just said
to Tony about there being no "very largest" natural.)
It is truly breathtaking that there are people who cannot grasp
the elementary concept that if there is no largest finite natural
then there must be an infinite number of them. Even children
understand that there is no biggest number (because you can
always add one to any number you think is the largest), and
therefore "numbers never end" (to quote my six-year old
daughter). |
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| David R Tribble |
| Posted: Wed Aug 30, 2006 5:29 pm |
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Dik T. Winter schrieb:
[quote:987b96cf5b]But indeed. If the ordinality of a set is infinite (w or larger), then
so is its cardinality (aleph-0 or larger). On the other hand, if the
cardinality of a set is infinite (aleph-0 or larger), and if it can be
well-ordered, so is its ordinality after well-ordering (w or larger).
[/quote:987b96cf5b]
mueckenh wrote:
[quote:987b96cf5b]As long as the set of natural numbers includes only finite numbers, its
cardinal number is also finite.
[/quote:987b96cf5b]
What is that cardinal number? Do you have a name for it?
Tony calls it "alpha", the cardinality of the finite but "unbounded"
set of all finite naturals, and also the largest natural. It appears
to have the curious property of being a finite natural but having
no finite successor. He's never provided an estimate of how
large it is, though. Perhaps you have a better idea of this? |
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| Virgil |
| Posted: Wed Aug 30, 2006 6:18 pm |
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In article <1156363640.845840.187460@75g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
[quote:211274d495]The cardinal number aleph_0 is infinite while the ordinal number
remains finite in order to have infinitely many finite numbers.
[/quote:211274d495]
Nonsense.
[quote:211274d495]For my part, I agree that the set of finite
naturals is finite, though unbounded,
In that case you are not using standard mathematical terminology. I
have no idea what a finite but unbounded set is.
That's why you cannot understand mathematics. You fall back behind
Cantor. He knew it.
[/quote:211274d495]
Actually, Cantor knew better than that. |
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| Virgil |
| Posted: Wed Aug 30, 2006 6:20 pm |
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In article <1156363950.351759.83230@m79g2000cwm.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
[quote:110438eabb]I do not see how I could avoid my conclusion. But if you are so sure
then give us at least one example how you completely index a number
without covering all the preceding numbers.
[/quote:110438eabb]
Given the number 5, I index it with 1. In this process, I do not need
to "cover" 2 or 3 or 4. |
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| Virgil |
| Posted: Wed Aug 30, 2006 6:22 pm |
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In article <1156364007.547996.270100@i3g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
[quote:eff4cacba2]According to Cantor: The infinite set of finite numbers. aleph_0 is
actually infinite, no natural number is actually infinite.
[/quote:eff4cacba2]
Pretty good! |
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| Virgil |
| Posted: Wed Aug 30, 2006 6:23 pm |
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In article <1156363768.777975.223810@m73g2000cwd.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
[quote:5f2288166a]Dik T. Winter schrieb:
It is the same with the staircase: If the total height is H, then there
must be at least one stair of height H.
Wrong. Think asymptote.
Now I understand your idea and your error. You intermingle the height
of the staircase and its least upper bound, which is not the same. As
long as you cannot find a stair that has height 1 you cannot assert
that the staircase had the height 1.
[/quote:5f2288166a]
Then an open interval from 0 to 1 cannot have length 1. |
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| Dik T. Winter |
| Posted: Wed Aug 30, 2006 8:11 pm |
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In article <virgil-026B14.15185630082006@news.usenetmonster.com> Virgil <virgil@comcast.net> writes:
[quote:a188d7cf3b]In article <1156363640.845840.187460@75g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
The cardinal number aleph_0 is infinite while the ordinal number
remains finite in order to have infinitely many finite numbers.
Nonsense.
For my part, I agree that the set of finite
naturals is finite, though unbounded,
In that case you are not using standard mathematical terminology. I
have no idea what a finite but unbounded set is.
That's why you cannot understand mathematics. You fall back behind
Cantor. He knew it.
Actually, Cantor knew better than that.
[/quote:a188d7cf3b]
Actually Cantor made an error. It was in one of his first papers where
he wrote that you also could sort of count with transfinite numbers, and
you the numbers you could count where those of the same class. If I
remember correctly (I do not have the book here at home) he made the
error that you needed a number of a higher class when counting all
numbers of a particular class. I will try to find it tomorrow (it
is in the first article where he uses omega).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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| Dik T. Winter |
| Posted: Thu Aug 31, 2006 8:59 am |
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In article <J4u33q.LCA@cwi.nl> "Dik T. Winter" <Dik.Winter@cwi.nl> writes:
[quote:a877213806]In article <virgil-026B14.15185630082006@news.usenetmonster.com> Virgil <virgil@comcast.net> writes:
In article <1156363640.845840.187460@75g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
The cardinal number aleph_0 is infinite while the ordinal number
remains finite in order to have infinitely many finite numbers.
Nonsense.
For my part, I agree that the set of finite
naturals is finite, though unbounded,
In that case you are not using standard mathematical terminology. I
have no idea what a finite but unbounded set is.
That's why you cannot understand mathematics. You fall back behind
Cantor. He knew it.
Actually, Cantor knew better than that.
Actually Cantor made an error. It was in one of his first papers where
he wrote that you also could sort of count with transfinite numbers, and
you the numbers you could count where those of the same class.
[/quote:a877213806]
Found it. First some definitions used: the first cardinality is aleph-0,
numbers of the first class are the finite numbers, numbers of the second
class are the ordinal numbers belonging to aleph-0. Now the quote (in
translation) on page 169 of the "Gesammelte Abhandlungen":
... each set of the first cardinality is countable with numbers of the
second class and only through them ...
The crucial word here is "countable". What does it mean? If you give
it a strict interpretation the sentence is false: the naturals (of the
first cardinality) are (in their standard order) "countable" by the
numbers of the first class... I think this is the source of the
confusion.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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| Guest |
| Posted: Sat Sep 02, 2006 7:42 am |
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Dik T. Winter schrieb:
[quote:8d7cf005fe]Quantifier dyslexia is the assumption that there is a number that
counts all natural numbers, i.e. which is larger than all natural
numbers.
That is not quantifier dyslexia. That is an error by Cantor I already
did comment on. That is, when he writes that to count all natural
numbers you need numbers of the second cardinality (or was it the first?
[/quote:8d7cf005fe]
Die erste Zahlenklasse (I) ist die Menge der endlichen ganzen Zahlen 1,
2, 3, ..., auf sie folgt die zweite Zahlenklasse (II), bestehend aus
gewissen in bestimmter Sukzession einander folgenden unendlichen ganzen
Zahlen; erst nachdem die zweite Zahlenklasse definiert ist, kommt man
zur dritten, dann zur vierten usw.
[quote:8d7cf005fe]I disremember what he called aleph-0), he was wrong. To count all natural
numbers you need only natural numbers.
[/quote:8d7cf005fe]
Great! So we have complete consensus now. Excuse me that I did overlook
your previous statement concerning this.
[quote:8d7cf005fe]On the other hand, the "size"
of the set of natural numbers is not a natural number.
[/quote:8d7cf005fe]
It is not a number at all.
[quote:8d7cf005fe]
Note, you may critique Cantor's set theory, but that was set theory in
its infancy. It still did contain inconsistencies and errors. Since
that time quite a bit has been developed and corrected.
[/quote:8d7cf005fe]
In particular by intermingling the different meanings of infinity.
[quote:8d7cf005fe]
Right. Because 0.111... has only finite digit positions we have:
(1) *each* digit position can be indexed
(2) *each* digit position can be covered
but not
(3) the number can be covered.
[/quote:8d7cf005fe]
Your "each" means in symbols of logic: "A = (for) all".
The number is nothing than all of its digit positions.
Therefore your statement is a self contradiction.
[quote:8d7cf005fe]Because (3) would mean that there is a number that covers *all* digits,
and as that number also indexes a digit position, there is a last finite
digit position. But that is not the case.
[/quote:8d7cf005fe]
Correct. Therefore 0.111... would imply a self-contradiction if it
existed. Conclusion: 0,111... does not exist.
Regards, WM |
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| Guest |
| Posted: Sat Sep 02, 2006 5:09 pm |
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Dik T. Winter schrieb:
[quote:96c20dc840]In article <1156842898.805375.169610@h48g2000cwc.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:
Dik T. Winter schrieb:
...
You gave an example how a number of the form 0,111...1 with n digits
indexes the n-th digit but does not cover all digits with m =<n ???
No, because that does not exist.
How then can you believe and assert that the number 0.111... which has
*only finitely indexable positions* could be completely indexed but not
covered.
Because there are infinitely many finitely indexable positions. Do you
see the way in which the first number you give "0.111...1" differs from
the second number you give "0.111..."?
[/quote:96c20dc840]
For indexing we have exactly the same question.
Regards, WM |
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| Guest |
| Posted: Sat Sep 02, 2006 5:12 pm |
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David R Tribble schrieb:
[quote:35e92c99f2]Dik T. Winter schrieb:
But indeed. If the ordinality of a set is infinite (w or larger), then
so is its cardinality (aleph-0 or larger). On the other hand, if the
cardinality of a set is infinite (aleph-0 or larger), and if it can be
well-ordered, so is its ordinality after well-ordering (w or larger).
mueckenh wrote:
As long as the set of natural numbers includes only finite numbers, its
cardinal number is also finite.
What is that cardinal number? Do you have a name for it?
[/quote:35e92c99f2]
The set is potentially infinite. It has no cardinal number in the sense
of set theory. All we can attach to it is the number of elements known
or existing. Disregarding physical constraints we can assume that the
elements of the set of natural numbers are counted by the largest
natural number temporarily known. With respect to physical constraints
the number of elements is less than 10^100.
[quote:35e92c99f2]
Tony calls it "alpha", the cardinality of the finite but "unbounded"
set of all finite naturals, and also the largest natural. It appears
to have the curious property of being a finite natural but having
no finite successor. He's never provided an estimate of how
large it is, though. Perhaps you have a better idea of this?
[/quote:35e92c99f2]
Sorry, there is no largest natural and no natural is infinite.
Regards, WM |
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| Virgil |
| Posted: Sat Sep 02, 2006 5:40 pm |
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In article <1157227963.905905.309230@e3g2000cwe.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:
[quote:5e43b1af26]David R Tribble schrieb:
Dik T. Winter schrieb:
But indeed. If the ordinality of a set is infinite (w or larger), then
so is its cardinality (aleph-0 or larger). On the other hand, if the
cardinality of a set is infinite (aleph-0 or larger), and if it can be
well-ordered, so is its ordinality after well-ordering (w or larger).
mueckenh wrote:
As long as the set of natural numbers includes only finite numbers, its
cardinal number is also finite.
What is that cardinal number? Do you have a name for it?
The set is potentially infinite. It has no cardinal number in the sense
of set theory.
[/quote:5e43b1af26]
Does "Mueckenh" claim that the set of all naturals is incapable of being
bijected with any other set? That is what "Mueckenh" is implying when he
claims that the set of all naturals does not have any cardinality in the
sense of set theory. |
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| Dik T. Winter |
| Posted: Sat Sep 02, 2006 10:55 pm |
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In article <1157193775.450075.276010@b28g2000cwb.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:
[quote:28df7822a0]Dik T. Winter schrieb:
Quantifier dyslexia is the assumption that there is a number that
counts all natural numbers, i.e. which is larger than all natural
numbers.
That is not quantifier dyslexia. That is an error by Cantor I already
did comment on. That is, when he writes that to count all natural
numbers you need numbers of the second cardinality (or was it the first?
Die erste Zahlenklasse (I) ist die Menge der endlichen ganzen Zahlen 1,
2, 3, ..., auf sie folgt die zweite Zahlenklasse (II), bestehend aus
gewissen in bestimmter Sukzession einander folgenden unendlichen ganzen
Zahlen; erst nachdem die zweite Zahlenklasse definiert ist, kommt man
zur dritten, dann zur vierten usw.
I disremember what he called aleph-0), he was wrong. To count all natural
numbers you need only natural numbers.
Great! So we have complete consensus now. Excuse me that I did overlook
your previous statement concerning this.
[/quote:28df7822a0]
I suspect not, because I think you are misreading about the error I wrote
about.
[quote:28df7822a0]On the other hand, the "size"
of the set of natural numbers is not a natural number.
It is not a number at all.
[/quote:28df7822a0]
Well, we mathematicians are willing to call things numbers whenever we feel
like it. The term is not sacred. Cayley numbers, cardinal numbers,
p-adic numbers, cardinal numbers, and you can go on. Whenever you want
to talk particularly about some kind of numbers you should state what
kind you are talking about. But I have stated this before.
But if you wish, provide a definition of "number". As far as I know,
there is not one in mathematics.
[quote:28df7822a0]Note, you may critique Cantor's set theory, but that was set theory in
its infancy. It still did contain inconsistencies and errors. Since
that time quite a bit has been developed and corrected.
In particular by intermingling the different meanings of infinity.
[/quote:28df7822a0]
In particular by just separating the meanings of potential and actual
infinity.
[quote:28df7822a0]Right. Because 0.111... has only finite digit positions we have:
(1) *each* digit position can be indexed
(2) *each* digit position can be covered
but not
(3) the number can be covered.
Your "each" means in symbols of logic: "A = (for) all".
The number is nothing than all of its digit positions.
Therefore your statement is a self contradiction.
[/quote:28df7822a0]
Where? In logical terms (A meaning "for all" and E meaning "there is"):
(1) A{p = digit position} E{q = list item} {such that q indexes p}
(2) A{p = digit position} E{q = list item} {such that q covers p}
but not
(3) E{q = list item} A{p = digit position} {such that q covers p}
this is simply false, just as:
(4) E{q = list item} A{p = digit position} {such that q indexes p}
is false.
Again, your interpretation of "for all" is false. It does *not*
mean "for all at once", it means "for each one individually". If
you disagree, and think they mean the same, in your opinion also
the statement:
(4) there is a number that indexes all digit positions.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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| Dik T. Winter |
| Posted: Sat Sep 02, 2006 10:55 pm |
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Guest
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In article <1157227790.457614.271810@74g2000cwt.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:
[quote:df7c2a95e5]Dik T. Winter schrieb:
In article <1156842898.805375.169610@h48g2000cwc.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:
Dik T. Winter schrieb:
...
You gave an example how a number of the form 0,111...1 with
n digits indexes the n-th digit but does not cover all digits
with m =<n ???
No, because that does not exist.
How then can you believe and assert that the number 0.111... which has
*only finitely indexable positions* could be completely indexed but not
covered.
Because there are infinitely many finitely indexable positions. Do you
see the way in which the first number you give "0.111...1" differs from
the second number you give "0.111..."?
For indexing we have exactly the same question.
[/quote:df7c2a95e5]
Does not make sense. Again, do you see the way in which those two numbers
differ? Why do you never give a straight answer to such plain questions?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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| Dik T. Winter |
| Posted: Sat Sep 02, 2006 10:55 pm |
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In article <1157227721.868719.283000@p79g2000cwp.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:
[quote:c1afc07a35]Dik T. Winter schrieb:
....
You are trying to make it difficult, using UTF-8. But this one takes the
cake.
What is UTF-8?
[/quote:c1afc07a35]
From your article:
Content-Type: text/plain; charset="utf-8"
[quote:c1afc07a35]Again translated (why do you post so much German in an English speaking
newsgroup while you should know that most readers are not able to read
German?):
Because I have the German text available and because I do not want to
be blamed of mistranslating.
[/quote:c1afc07a35]
So you can blame other persons of mistranslating?
[quote:c1afc07a35]Cantor: So while a changing quantity x that successively takes the
various values of finite numbers 1, 2, 3, ..., v, ... , is a
potential infinite, on the other hand, a through the axioms completely
determined set (N) of all integral finite number is an example of an
actually finite quantity.
Not through the axioms, but through "a law" (ein Gesetz).
[/quote:c1afc07a35]
What is the difference? He uses his (I think, but that is from memory)
second completion law. In current terminology, that is an axiom.
[quote:c1afc07a35]Nice that you found the quote I have alluded to, and that you did deny
of existing, but that I could not find back.
What Cantor is stating here (and I did already indicate that in an
earlier response), is, translated to current set theory:
The set N is potentially infinite,
No. The changing quantity is here a variable.
the size of N is actually infinite.
(In current terminology a set is not a quantity.)
Cantor's changing quantity is a variable. A set is never potentially
infinite according to Cantor.
[/quote:c1afc07a35]
You missed my "in current set theory"?
To be honest. If you did not go any further than Cantor, you may find
quite a few contradictions. This does not say anything about set
theory as developed beyond Cantor. Stating that set theory is
inconsistent because Cantor was inconsistent is simply false.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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