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(κ,μ). Thus, if we're given λ and κ we may always solve the equation in variable μ
κμ=λ

if and only if κλ. Indeed, if κλ, then
κλ=max(κ,λ)=λ.
On the other hand, if κ>λ
κμ=max(κ,μ)κ>λ.


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