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On the Wikipedia page on Cardinal Numbers, Cardinal Arithmetic including multiplication is defined. For finite cardinals there is multiplication by zero, but for infinite cardinals only defines multiplication for nonzero cardinals. Is multiplication of an infinite cardinal by zero undefined? If so, why is it?

Also does $\kappa\cdot\mu= \max\{\kappa,\mu\}$ simply means that the multiplication of the two is simply the cardinality of the higher cardinal? Why is this?

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    Of course any infinite cardinal is nonzero... if you want to multiply an infinite cardinal by a finite cardinal, the result is equal to the infinite cardinal if the finite one is nonzero and zero otherwise.2012-01-20
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    But what is Zero times a Cardinal? Is it undefined? Why?2012-01-20
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    Your claims are not true. The [article in question](http://en.wikipedia.org/wiki/Cardinal_number#Cardinal_multiplication) defines multiplication for any pair of cardinals. It even notes explicitly that multiplication by $0$ gives $0$. The statement you've picked out is not a *definition*. It is a *theorem*, and the reason the cardinals are required to be nonzero is that the result wouldn't be true otherwise.2012-01-20
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    That is for FINITE CARDINALS. Look underneath that.2012-01-20
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    Notice where it says: "Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then..." https://en.wikipedia.org/wiki/Cardinal_numbers2012-01-20
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    "It is a theorem, and the reason the cardinals are required to be nonzero is that the result wouldn't be true otherwise" What result wouldn't be true? The multiplication? So it is proven as a theorem because it wouldn't be true otherwise?2012-01-20
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    Chris Eagle’s link in fact tells you that $\kappa\cdot 0=0\cdot\kappa=0$ for *all* cardinals. The statement that you quote in your last comment restricts $\kappa$ and $\lambda$ to non-zero cardinals precisely *because* it would be false if one were $0$: the product would then be $0$, not $\max\{\kappa,\mu\}$.2012-01-20
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    OK. THANK YOU. ...I see what he was saying now. These comments were very helpful.2012-01-20
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    The only way to thank users of the forum who have been of help to you is by upvoting their answers and accepting top answers. I understand that you cannot upvote because you need atleast 15 reputations to do so. But, certainly, you can select an answer by clicking the tick mark on the side of the answer.2012-01-20

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