#Infinity times 0 plus

This same logic lets us quickly prove that “infinity type 1 plus anything is infinity type 1”, where by “anything” I mean “any finite number of elements”. Thus, F is bijective, and therefore A and B have “the same number” of elements-namely, they both have “infinity type 1” elements! Clearly, no two elements in A get mapped to the same element in B (so F is injective), and every element in B is eventually “hit” by something in A (so F is surjective). In other words, if F is our function from A to B, then, and so on, and, and so on. Now, let’s define our function to map, in order, the odd elements in A to the positive elements in B, and the even elements in A to the negative elements in B. Namely, take the subset, and also do the same for the even ones. Defining this function and seeing that it is indeed bijective is actually quite simple.įirst, let us take the subset of A that contains only its odd positive whole numbers. This is, after all, the very definition of “having the same cardinality”. I’ll prove that A and B have the same cardinality by doing the only thing that I can do: define a bijective function from A to B. Thankfully we don’t need to rely on my intuition for this problem. Half of me wants to say that they have the same cardinality because, whereas the other half of me wants to say no, they don’t have the same cardinality because B clearly has “more” elements than A. I claim that they do in fact have the same cardinality, but before proving this let’s reflect on why this is a non-trivial statement. Namely, let us ask whether or not and have the same cardinality. So let us not worry about our intuitive notion of what “twice as big means”, and rather just inquire about our rigorous definition of cardinality. Namely, if I have a set SET1 which can be put into bijection with, and I have another set SET2 which can be put into bijection with, then SET1 is “twice as big” as SET2 only if. Recall, however, that our idea of what “twice as big” means now has a very formal meaning. The set B is in some clear way “twice as big” as A, since we define B by essentially adding another copy of A to it (and just adding a trivial “minus” sign in front of some of the numbers). We can make this rigorous by adding into our set every single negative number. Namely, let’s define a set by taking every element in and “creating another copy of it”. Let us define a set that is in a very obvious way “infinity type 1 times 2”. Actually, what we’ll show, is that “infinity type 1 times 2 is infinity type 1”. What we’ll focus on in this lesson is giving precise meaning to the phrase “infinity times 2 is infinity”. So enough with the fluffy philosophy-let’s do some math. Indeed, the rigor and beauty of these ideas are such that we’ll be forced to somehow believe that there “really are” many kinds of infinities (scare quotes because I won’t discuss here what it means to “really be”). While all of this might sound a bit strange or a bit “fake”, we’ll soon see that these ideas are in fact completely rigorous. Or maybe it’s the case that there are some things that we could add to “infinity type 1” to give us “infinity type 1” again, and other things which would give us “infinity type 2”. Then we need to know whether “infinity type 1 plus anything is infinity type 1” or if “infinity type 1 plus anything is infinity type 2”.

Suppose for a second that there is an “infinity type 2” (there is, but since we haven’t proved it yet let’s just suppose it’s there). In particular, we need to know which kind of infinity plus anything is which kind of infinity. However, if I claim that there are different kinds of infinity (which is what I’m claiming), then we need to be more careful when we make statements like “infinity plus anything is infinity”.

We all already know that “infinity plus anything is infinity” and “infinity times anything (other than 0) is infinity”, and other sort of “obvious” statements like these. This is indeed the case (and much more!), but before we can truly appreciate this fact we need to first get used to what this definition of “infinity type 1” really means. It is implicit in my calling this new number “infinity type 1” that there is likely an “infinity type 2”. In particular, analogously to how we define a set A to have N elements if we can define a bijective function from the set to A, we also define a set A to have “infinity type 1” elements if we can define a bijective function from the set to A. Last lesson we used our abstract way of counting the number of elements in a set to extend the “normal” number system to include a number which we called “infinity type 1”.