infinity is not a number
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Indeed. Does infinity as a real share all properties of other reals and obey the same rules?
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Expected this. Rudin of course has a brief section on extended reals but more by way of defining rules wrt infinity in relation to other reals. After all the extended reals do not constitute a field.
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Conway's surreal numbers form a humongous field that contains the real numbers and many more infinities. If you want to extend the real numbers, the rule is that the real numbers contained in your extension must behave exactly like the ones we know.
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