What would be the consequence of restricting multiplication by Zero to only Finite Cardinals?
Would this lead to contradictions? How could it be achieved?
closed as not a real question by Asaf Karagila, tomasz, rschwieb, Emily, BenjaLim Sep 23 '12 at 23:37
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Fact: If $|A|=0$ then $A=\varnothing$.
Cardinal arithmetics is just a definition allowing us to observe what is the cardinality of sets created by unions, or by products of sets. If we disallow $\kappa\cdot 0$ for infinite $\kappa$, consider this:
$$\mathbb N\times\varnothing = \varnothing\Rightarrow |\mathbb N\times\varnothing|=0\Rightarrow |\mathbb N|\times0=0$$
We have that cardinality no longer behave nicely. This means that what was simple to define and very natural to begin with will now require elaborate tricks to overcome.
Cardinality, in such case, cannot be defined using bijections, since from one end of the spectrum there exists a bijection from $\mathbb N\times\varnothing$ to $\varnothing$; however the cardinality of the former is "undefined".