3 ms·

It stems from what you mean by "infinity". Most people use the term as if infinity is a number, and that gets you into trouble. You need to say what you mean.

It stems from what you mean by "infinity". Most people use the term as if infinity is a number, and that gets you into trouble. You need to say what you mean. He talked about \infty - 1, and really that means - "take an infinite-sized set and remove an element from it." Now you have the question: Which infinite-sized set? In particular, there are different sizes of infinite sized sets. The set of positive whole numbers, the set of primes, and the set of rationals are all "the same size" (under a reasonable definition of the term). However, the set of real numbers, or the set of subsets of primes, are both "bigger" (in some very real sense). So I was responding to the comment about "\infty - 1 being \infty" causing trouble. When you treat these things properly (whatever that means) then it doesn't cause trouble. At least, not any more, and not for people who understand the care required. tl;dr : If you mention "infinity" you are probably talking about the size of a set.

lmkg 16y ago
This gets into something that you and I discussed earlier, that "infinity" can refer to two different concepts. While your interpretation is well-defined, and reasonable, it's not what people usually mean when they think of that expression. Even knowing the math and notation behind both of them interpretations and the difference between them, I still tend to see \inf-1 as an ill-defined arithmetic (algebraic) expression rather than a well-defined set operation. Note also, that you can treat \inf as an ordinal number, rather than a cardinal or a limit point, and subtracting one as the inverse of the ordinal successor operation. In this case, \inf-1 has a rigorous definition, which exists almost all of the time, and is somewhat closer to the naive, ill-defined algebraic interpretation. Under this definition, \inf-1 does not equal \inf.

lmkg 16y ago

This gets into something that you and I discussed earlier, that "infinity" can refer to two different concepts. While your interpretation is well-defined, and reasonable, it's not what people usually mean when they think of that expression. Even knowing the math and notation behind both of them interpretations and the difference between them, I still tend to see \inf-1 as an ill-defined arithmetic (algebraic) expression rather than a well-defined set operation. Note also, that you can treat \inf as an ordinal number, rather than a cardinal or a limit point, and subtracting one as the inverse of the ordinal successor operation. In this case, \inf-1 has a rigorous definition, which exists almost all of the time, and is somewhat closer to the naive, ill-defined algebraic interpretation. Under this definition, \inf-1 does not equal \inf.