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Wasn't that Cantor? According to
Stephen Hawking's book "God Created the Integers" he claims "Dedekind took the natural numbers as the paradigm example of an infinite set and defined a set as infinite if the natural numbers could be put into a one-to-one correspondence with that set, or a subset of
553:
I have rewritten this entry, adding links and references, simply because it, like all too many
Knowledge biographical entries for mathematicians and philosophers, was a shabby affair. I think what I have done here is largely a matter of polish, detail, and organization, with one exception. The entry
513:
This means that if you take any ring and then take any set of elements in R that is a group by itself and obey all of the laws of normal addition (e.g. a+b=b+a, a+0=a) then that set is an Ideal if any element of R can be multiplied on the left and on the right of any elemnt of I and the result is in
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An ideal doen't aply to specific numbers. It is a set of numbers. Specifically, if R is a ring and I a nonempty subset of R than I is an ideal if: I is a group under the additive operation of R, and given any element of R, say r, and any element of I, say a, then ra is in present in I and ar is
644:
As for
Hawking, unfortunately he is wrong, Dedekind was the first to employ that definition -- he even mentions in a footnote that he presented it to Cantor in sept. 1882. See e.g. J. Ferreiros, "Labyrinth of Thought" (Birkhäuser, 1999) on this and related issues.
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In the article you write "This is known as
Dedekind's theorem.". It should say: "This is known as the concept of Dedekind-infinite set; it appears originally in section 5 of Was sind und was sollen die Zahlen?"
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Bell is notoriously unreliable and should not under any circumstances be used as a source of biographical information. The whole section should be checked in detail and referenced to reliable sources.
483:"We can easily see that for arbitrary whole numbers m and n and if for their such 'classes' 'class' (m) is part of 'class' of (n) (we write then as (m)/(n)) only and only then if m divide n. "
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That sentence is indeed incomprehensible and I have simply eliminated it from the entry. Knowledge's mathematical entries are in good hands, and that holds for the entries
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Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the
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He invoked similarity to give the first precise definition of an infinite set: a set is infinite when it is "similar to a proper part of itself,"
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it. ... Cantor ... defined a set as being infinite if it could be put into a one-to-one correspondence with a proper subset of itself."
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I have a hunch that anyone wishing to edit this entry further should first read
Stillwell's Intro to his 1996 translation to the
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I i.e. a*r and r*a both equal an element in I, but not always the same element. In fact usually they are not equal.
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This{{Clarify|date=October 2011}} is known as
Dedekind's theorem.{{Citation needed|date=October 2011}}
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He was born, lived most of his life, and died in
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This artical claims that
Dedekind thought a set was infinite if
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This sentence is incomprehensible. Can someone fix it?
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used to attribute Dedekind's notion of ideal to his
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