By Georg Cantor
Covers addition, multiplication and exponentiation of cardinal numbers, smallest transfinite cardinal numbers, ordinal forms of uncomplicated ordered aggregates and operations on ordinal forms. Develops thought of well-ordered aggregates; investigates ordinal numbers of well-ordered aggregates and extra.
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Additional info for Contributions to the Founding of the Theory of Transfinite Numbers. Georg Cantor
PRELIMINARIES A’ ” ,A a2 +A”-O Then the induced sequences Ker(d’) +Ker(d) +Ker(d”) and Coker(d’) +Coker(d) +Coker(d”) areexact. Define afunction 6 : Ker(d”) +Coker(d’) asfollows. For x E Ker(d“) let q ( y ) = x. Then P2d(y) = 0, so d(y) = p1(z) for some z E B’. Define 6 ( x ) = [ z ] E Coker(d’). Then 6 is well defined morphism of groups, and the sequence 8 Ker(d) --f Ker(d”) +Coker(d’) + Coker(d) is exact. 21. Consider the exact sequence in an exact category. Then 19 is a retraction if and only if u is a coretraction.
Then the diagram can be extended to a pullback fi and only fi A' + A is the kernel of the composition A -+B -+ B". Proof. Suppose that A' -+A is the kernel A +B -+ B". Then A' + A +B +B" is 0, and so since B' -+B is the kernel of B +B" we get a unique morphism A'+B' making ( 1 ) commutative. Suppose that X + A - + B = X + B ' + B . Then X +A +B -+ B" is zero, hence there is a unique morphism X -+A' such that X+A'-+A = X - t A . Then also X+A'-+B'+B = X+A'-+A+B = X+B'+B, and so since B' --+ B is a monomorphism it follows that X +A' -+B' = X +B'.
Then X+A'+A+B = X+A'+B'-+B = X+B'+B = X+A+B. 16. THE 21 9 LEMMA Since A + B is a monomorphism this means that X + A ' + A = X + A . Consequently we have shown that A' + A is the kernel o f A -+ B",or, in other words, that O+A'-+A-+B" is exact. By duality it follows that A->B"+C"+O is exact. Now since A" +B" is the kernel of B" -+C" we see that the factorization of A --+B" through its image is just A +A" +B". Exactness of 0 +A' +A -+A" +O now follows. 2 (First Noether Isomorphism Theorem). Let B c A2 c A , in an exact category.