What would be the consequence of restricting multiplication by Zero to only Finite Cardinals?
Would this lead to contradictions? How could it be achieved?
|show 3 more comments|
It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
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".