Talk:Richard Dedekind
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| Mathematics grading: | B Class | High Importance | Field: Mathematicians | ||
This sentence is incomprehensible. Can someone fix it?
"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. "
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 present in I.
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 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.
- That sentence is indeed incomprehensible and I have simply eliminated it from the entry. Wikipedia's mathematical entries are in good hands, and that holds for the entries ideal and ideal number. Hence I feel no compelling need to explain these concepts carefully here, but others are free to dissent.
- I have rewritten this entry, adding links and references, simply because it, like all too many Wikipedia 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 used to attribute Dedekind's notion of ideal to his Theory of Algebraic Integers, when MacTutor attributes ideals to the 3rd ed. of the Dirichlet lectures. I have gone with MacTutor; correct me if I'm wrong.
- I have a hunch that anyone wishing to edit this entry further should first read Stillwell's Intro to his 1996 translation to the Theory of Algebraic Integers.
[edit] He lived most of his life?
He was born, lived most of his life, and died in Braunschweig (often called "Brunswick" in English).
- What part of his life didn't he live? --Ihope127 23:01, 8 September 2006 (UTC)
[edit] Dedekind's definition of an infinite set
This artical claims that Dedekind thought a set was infinite if
- 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,"
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 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." Boyton 06:39, 5 November 2006 (UTC)
