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Originally Posted by Merkin Muffley
This is common usage in, for example, measure theory. It is not universal.
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Actually, cardinal and ordinal numbers have little to do with the extended real numbers which are commonly used in measure theory. The use of the extended reals in measure theory is nothing more and nothing less than a notational convenience.
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Originally Posted by Merken Muffley
Of course it does, capitalization notwithstanding. If you define infinity one way and apply one set of rules to it, this constrains by logical consistency your ability to apply other sets of rules to it.
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Sorry, but that is simply incorrect. Cardinal and ordinal arithmetic is quite logical, and it permits the operations of addition and multiplication. It has absolutely nothing to do with any sort of definition of 1/0.
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Originally Posted by Merken Muffley
It is defined in the Riemann sphere, where zero times infinity is undefined.
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Yes, you can make that convention, and it works topologically. But the "point at infinity" in the Riemann sphere is not an infinite cardinal number or anything other than the point added to form the one-point-compactification of the plane. It has nothing to do with the usual notion of infinity as applied to the definition of infinite sets, for instance. The Riemann sphere is a useful manifold, but it is not a number system. This is a rather bogus example, since it is based on notation and nothing more. As I said earlier, you can DEFINE 1/0 to be infinity. In the case of the Riemann sphere that is exactly what is done, but in doing that you lose other things and it does not carry over to other situations. The Riemann sphere is not a field, it is not a ring, it is just a comples manifold.
This is a bogus example, and brings no light to the topic being discussed.