“Mächtigkeit†versus “Kardinalität�
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In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
set-theory ho.history-overview
asked Aug 16 at 2:59
Drew Armstrong
1,390829
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In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
set-theory ho.history-overview
asked Aug 16 at 2:59
Drew Armstrong
1,390829
2
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
1
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
2
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
set-theory ho.history-overview
asked Aug 16 at 2:59
Drew Armstrong
1,390829
In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
set-theory ho.history-overview
asked Aug 16 at 2:59
Drew Armstrong
1,390829
asked Aug 16 at 2:59
Drew Armstrong
1,390829
asked Aug 16 at 2:59
Drew Armstrong
1,390829
asked Aug 16 at 2:59
Drew Armstrong
1,390829
1,390829
2
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
1
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
2
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14
add a comment |Â
2
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
1
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
2
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14
2
2
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
1
1
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
2
2
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14
add a comment |Â
2 Answers
2
active
oldest
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up vote
13
down vote
Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):
The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
add a comment |Â
up vote
7
down vote
The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
13
down vote
Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):
The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
add a comment |Â
up vote
13
down vote
Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):
The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
add a comment |Â
up vote
13
down vote
up vote
13
down vote
Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):
The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
Here is how Cantor introduced "Mächtigkeiten" in Ueber eine elementare Frage der Mannigfaltigketislehre (1890):
The "Mächtigkeiten" represent the unique and necessary generalisation of the finite "Cardinal numbers", they are nothing other than infinitely large Cardinal numbers, and they share the same reality and definiteness.
So it seems that, at least initially, Cantor did not speak of "Kardinalität", which was linked to the existing term of "cardinal numbers", a term from the 16th century meaning "principal numbers". He introduced a new term "Mächtigkeiten" for infinitely large cardinal numbers. The dictionary I consulted lists 1935 as the first use of the term "cardinality". Because "Mächtigkeiten" is not easily transferred to the English language, the shift to "Kardinalität" and "cardinality" seems a natural one.
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
edited Aug 16 at 6:50
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
answered Aug 16 at 6:40
Carlo Beenakker
68.5k8154258
68.5k8154258
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
add a comment |Â
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
2
2
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
I recall having seen the word "power" used for the size of infinite sets in an older text, but I can't recall where.
– David Roberts
Aug 16 at 6:54
1
1
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
The natural cognate of "Mächtigkeit" in English is "mightiness", though it seems to have never been used as a translation, "power" being used instead, as pointed out by David Roberts. An example of an old book where it is used is: books.google.dk/… The phrase "power of the continuum" also seems to have lasted a bit longer than "power" as a general term for cardinality.
– Robert Furber
Aug 16 at 12:21
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
There is also "puissance" in French, which can often be seen in old articles by Kuratowski or Sierpinski.
– Robert Furber
Aug 16 at 12:25
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
Yes, "power" is sen in old writings in English. The power of the real line is $2^aleph_0$.
– Gerald Edgar
Aug 16 at 12:42
2
2
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
The one place (as far as I know) where "power" is still so common in English that "cardinality" would sound strange is the model-theoretic notion of "categoricity in power".
– Andreas Blass
Aug 16 at 12:48
add a comment |Â
up vote
7
down vote
The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
add a comment |Â
up vote
7
down vote
The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
add a comment |Â
up vote
7
down vote
up vote
7
down vote
The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
The two terms "Mächtigkeit" and "Kardinalität" do indeed mean the same in Cantor's set theory; also today, for instance in lectures in German and Swiss universities, the terms are used as synonyms for the same concept.
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
answered Aug 16 at 5:33
Dominic van der Zypen
12.6k43167
12.6k43167
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
add a comment |Â
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
3
3
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
I would feel more confident in your answer if there was a direct quote from Cantor supporting it. The oldest reference on that Wikipedia page is a 1928 set theory text, by which time axiomatic set theory had undergone all kinds of conceptual developments (Zermelo, Skolem, Fraekel, von Neumann etc).
– David Roberts
Aug 16 at 5:53
add a comment |Â
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2
(warning: bad German ahead) Lawvere certainly seems to think that the difference between Menge and Kardinale is one that has been lost ncatlab.org/nlab/show/… . I'm not completely familiar with Mächtigkeit and Kardinalität, but I guess the former would be translated 'potency', so that sets (Mengen?) are equipollent when there is an isomorphism between them. The Kardinale associated to a Menge is a different sort of object, and so I guess that's why one could talk of the Kardinalität as something distinct
– David Roberts
Aug 16 at 5:02
1
Do you mean "to Cantor", when you say "in Cantor's set theory"? Or in the broader sense of set theory as it generally stood at its beginning?
– David Roberts
Aug 16 at 5:55
2
Looking at your recent questions, it might be worth mentioning that there is also a separate site for History of Science and Mathematics. At least some of the top ho.history-overview answerers on MO have an account also on that site; for example, Carlo Beenakker, Francois Ziegler or Alexandre Eremenko.
– Martin Sleziak
Aug 16 at 7:17
I'm voting to close this question just because it would be suited for HSM (not because I think it is a bad question per se)
– Yemon Choi
Aug 16 at 23:14