Language as Set Theory

The revolution in our understanding of the logic of names began with a basic question: Where do the meanings of words live? There are two likely habitats. One is the world, where we find the things that a word refers to. The other is in the head, where we find people’s understanding of how a word may be used. For anyone interested in language as a window into the mind, the external world might seem to be an unpromising habitat. The word cat, for example, refers to the set of all the cats t...

 Infinity Times Zero is Not Zero

The problem is that the laws of addition and multiplication you are using hold for natural numbers, but infinity is not a natural number, so these laws do not apply. If they did, you could use a similar argument that multiplying anything by infinity, no matter how small, gives infinity, thus ∞×0=∞. More sophisticated arguments can also be made, like ∞×0=limx→∞(x×1/x)=1. Clearly all these different values for ∞×0 mean that ∞ cannot be treated like other numbers. In order to ...