Comment author:Janos2
11 January 2008 08:10:02PM
0 points
[-]

Ben:

Well, that depends on your number system. For some purposes +infinity is a very useful value to have. For instance if you consider the extended nonnegative reals (i.e. including +infinity) then every measurable nonnegative extended-real-valued function on a measure space actually has a well-defined extended-nonnegative-real-values integral. There are all kinds of mathematical structures where an infinity element (or many) is indispensable. It's a matter of context. The question of what is a "number" is I think very vague given how many interesting number-like notions mathematicians have come up with. But unquestionably "infinity" is not a natural number, or a real number, or a complex number.

Probability theory, on the other hand, would have to change shape if we comfortably wanted to exclude 0 probabilities. What we now call measures would be wrong for the job. I don't know how it would look, but I find the standard description intuitively appealing enough that I don't think it should be changed. It's probably true that for a Bayesian inference engine of some sort, whose purpose is to find likelihoods of propositions given evidence, the "probabilities" it keeps track of shouldn't become 0 or 1. If there's a rich theory there focussing on how to practically do this stuff (and I bet there is, although I know nothing of it beyond Bayes' Theorem, which is a simple result) then ignoring the possibility of 0s and 1s makes sense there: for example you can use the log odds. But in general probability theory? No.

## Comments (128)

OldBen:

Well, that depends on your number system. For some purposes +infinity is a very useful value to have. For instance if you consider the extended nonnegative reals (i.e. including +infinity) then every measurable nonnegative extended-real-valued function on a measure space actually has a well-defined extended-nonnegative-real-values integral. There are all kinds of mathematical structures where an infinity element (or many) is indispensable. It's a matter of context. The question of what is a "number" is I think very vague given how many interesting number-like notions mathematicians have come up with. But unquestionably "infinity" is not a natural number, or a real number, or a complex number.

Probability theory, on the other hand, would have to change shape if we comfortably wanted to exclude 0 probabilities. What we now call measures would be wrong for the job. I don't know how it would look, but I find the standard description intuitively appealing enough that I don't think it should be changed. It's probably true that for a Bayesian inference engine of some sort, whose purpose is to find likelihoods of propositions given evidence, the "probabilities" it keeps track of shouldn't become 0 or 1. If there's a rich theory there focussing on how to practically do this stuff (and I bet there is, although I know nothing of it beyond Bayes' Theorem, which is a simple result) then ignoring the possibility of 0s and 1s makes sense there: for example you can use the log odds. But in general probability theory? No.