Contributions to the Founding of the Theory of Transfinite by Georg Cantor

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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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.

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