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Infinity is not a number. It is another way of saying that something gets larger and larger without an end. So, you cannot add 1 to some thing which is not a number. However, in mathematics, there are infinities that could be added, multiplied, or subtracted. There are cardinal, ordinal, surreal and hyperreal numbers to all of which 1 can be meaningfully added. For infinite cardinal numbers, adding one does not change the number. For ordinal numbers, the number can be changed depending on whether you add 1 to the number, or the number to 1 - there is no commutativity. With hyperreal numbers, one can do pretty much anything one would expect of a real number. As to the surreal ones, the arithmetic operations produce magic results increasing the number of various infinities to fantastic proportions. In general, to add 1 to a number means to take the successor of that number. The successor of a set A is the set which consists of all the elements of A along with A itself. In other words the successor of A is A U {A}, and it has "one" more element than A. So, infinity + 1 = infinity U {infinity}.
