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What would be the consequence of restricting multiplication by Zero to only Finite Cardinals?

Would this lead to contradictions? How could it be achieved?

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    Restricting is usually done when the definition is not "uniform", in the case of cardinal multiplication it is irrelevant whether or not the cardinals are finite or infinite. The result is that restriction of multiplication by zero to finite cardinals makes little to no sense.2012-01-20
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    Obviously, you couldn't get by without zero in ordinary arithmetic. You absolutely need Zero. But, because the arithmetic of infinite cardinals is as it is, it seems to me that you don't need it. It's something extra. You can't get to zero by subtracting cardinals.2012-01-20
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    This is because often the subtraction is undefined. Multiplication, however, is defined *always* and in particular if one set is empty and the second is infinite.2012-01-20
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    So, even though it is a completely arbitrary and silly thing to do, what trouble would you get into if you just decided to not do it at all?2012-01-20
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    Leaving multiplication by zero undefined when the other factor is infinite would unnecessarily complicate what is now a very simple definition.2012-01-20
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    If you restrict multiplication and adjust the statements of theorems appropriately (so that the statements are only taken "if multiplication is defined"), then you cannot possibly introduce new contradictions into your theory.2012-01-20
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    I agree that it would do that. It creates an unnecessary split between the Finite and Infinite case, but I would like a concrete example of a calculation that starts with Infinite Cardinals and leads up to the use of multiplication by Zero.2012-01-20
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    I think that wasn't clear. I'll restate it. Could you give me a calculation starting with infinite cardinals that requires multiplication by zero.2012-01-20

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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".

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    I'm having trouble following this. Why isn't it behaving nicely there?2012-01-20
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    @mathNotebook: Because on one side you have $|\varnothing|$ which is fine, and on the other you have $\aleph_0\cdot 0$ which is not defined anymore; if something undefined is equal to something else which is defined then there is a problem with your definition.2012-01-20
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    I see. Thank you very much.2012-01-20