Tuesday, 15 December 2009

set theory - Is it possible to decrease the rank of known structures?

Recall that for a set x its rank alpha is the least ordinal such that xinValpha+1. Or in other words: x is built up out of alpha levels of braces and the empty set.



I think with the usual constructions of numbers (cartesian products, sets of equivalence classes, Dedekind cuts, etc.), we have



rank(mathbbN)=omega,rank(mathbbZ)=omega+4,rank(mathbbQ)=omega+8,rank(mathbbR)=omega+10



Now it is possible to find a bijection mathbbZcongmathbbN, so that there is a copy of mathbbZ of smaller rank, namely omega. But this is, of course, nonsense. We should also consider the ring structure on mathbbZ, which is given by two maps mathbbZtimesmathbbZtomathbbZ. Therefore we might ask the following:



Is there a ring (R,+,), which is isomorphic to (mathbbZ,+,), but rank(R,+,)<rank(mathbbZ,+,)? What about the other rings above?



Of course, this question is just out of curiosity. I doubt that anybody cares about these bounds of ranks (if not, please let me know).

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