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. In any case, (1/0)*0 can't be defined to make sense.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”. Sometimes people also extend the real numbers to include -\infty, sometimes people don't: it depends on what you want to do with things. So we add the limit in and get a new number larger than all real numbers. Where do these choices come from? They do indeed come from taking limits of things like 1/x as x tends to zero: this is can be made arbitrarily large by taking x arbitrarily close to zero. The thing we commonly add is a symbol \infty and it is true that 1/0=\infty and 1/\infty=0 are definitions we make, but we do not define 0*(1/0) even in this extended set of numbers (just as we do not define 1/0 in R). We have to 'add' something to R to start to make sense of it. So, this is not actually a question about R. Let's switch to calling the number line the Real numbers which we will denote by R. So 1/0 or 1/infinity are not questions about the number line. The first thing to remember is that the number line does not have infinity on it. Just pick x to be bigger than 2/k, and you'll see that the limit is less than their guess, so they are wrong. For instance, suppose they think it's k, where k is really really small. What exactly do your friends think the value of 1/infinity is anyway? By using limits (so it's rigorous) you can show that if they think it's any number other than zero, they are wrong. It is true that in the limit as x->0, 1/x goes to infinity, but I'd be extremely weary of putting equals signs there without mentioning limits or you'll end up proving 1=2.
Clearly this is wrong and is a sign you've assumed you can do a certain operation you shouldn't. I could just double each side to get 2 = 0*infinity, so that means 1=2. Now as x-> infinity, the 4/x and 2/x parts go to zero and you end up with y=3.īecause of this requirement to 'approach infinity' rather than just put in infinity, you can't say things like 1/0 = infinity means 1 = 0*infinity. Rather than just put in x=infinity, you consider what happens when x gets really big (or sometimes when x gets really small).įor instance, what is y = \frac When you're doing this kind of thing you need to instead use 'limits'. For a start, infinity isn't a member of the Real numbers, it's a member of the extended Reals. You cannot just plug in infinity and zero into certain equations like y = 1/x and then expect it to obey the 'normal rules'.