Monday, 27 February 2017

elementary set theory - Reference Request: Cardinal Division



In this link, Division of cardinals, someone asks a question about cardinal division and references a Wikipedia page about it. The Wikipedia page does not give a reference to their statement, but I'd really like to know one. Does anyone know specifically (preferably book and page number) where I can find this and a proof?


Answer




This follows from the fact that cardinal multiplication is not very interesting for infinite cardinals. Namely, if κ and μ are infinite, κμ=max. Thus, if we're given \lambda and \kappa we may always solve the equation in variable \mu
\kappa \cdot \mu = \lambda if and only if \kappa \leq \lambda. Indeed, if \kappa \leq \lambda, then
\kappa \cdot \lambda = \max(\kappa,\lambda) = \lambda. On the other hand, if \kappa > \lambda
\kappa \cdot \mu = \max(\kappa,\mu) \geq \kappa > \lambda.


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