Infinity is not a number
This morning I spent the better part of an hour writing a review on Walt Hickey's recent article about controversial math facts and then promptly deleted my work by trusting the browser with too much text and not saving partial work. I'm going to rewrite the part of the article I actually care about and leave the rest of the review in the bitbucket.
I'm only going to look at two of the answers, #3 and #9. (slide 14 and slide 49 for those who don't want to read the whole presentation).
Let's take #9 first. This is a proof of why the harmonic summation (let's call that H) diverges to positive infinity. Hickey takes a similar summation (let's call it A) and expresses it in two forms. One form shows that H is obviously greater than A. The other form shows that A is obviously infinite. The proof concludes that if H is greater than something infinite, it must also be infinite. Great explanation, solid proof; I have no complaints here.
But then there is #3. In this section, Hickey claims that there are the same number of positive even numbers as counting numbers. He sets up a 1-to-1 correspondence between the to sets and concludes that they are both countably infinite and therefore it anyone who says that one set has more elements than the other is wrong.
So close, but fail. The fundamental misunderstanding here comes from the reflexive property of numbers which states Every number is equal to itself
This isn't exactly the most controversial property ever. The problem is when you try to apply that rule to infinity, it doesn't work because infinity is not a number.
In problem 9, the key step of the proof was that H > A. Since H was infinity and A was infinity, that means that in this case infinity > infinity, not infinity = infinity. This is fine, since infinity is not a number and therefore doesn't have to follow the reflexive property.
Applying the same logic to problem 3, we see that it makes no sense to say that there is the same number of elements in the sets because both cardinalities (set sizes) are infinite. The moral of the story is that infinity doesn't play nice with concepts like equals, addition, multiplication and the like, so be careful about treating it like a number, or before you know it you'll be claiming that infinity minus infinity is zero or infinity divided by infinity is one, which turn out to be variants of the same mistake Hickey made.
I'm only going to look at two of the answers, #3 and #9. (slide 14 and slide 49 for those who don't want to read the whole presentation).
Let's take #9 first. This is a proof of why the harmonic summation (let's call that H) diverges to positive infinity. Hickey takes a similar summation (let's call it A) and expresses it in two forms. One form shows that H is obviously greater than A. The other form shows that A is obviously infinite. The proof concludes that if H is greater than something infinite, it must also be infinite. Great explanation, solid proof; I have no complaints here.
But then there is #3. In this section, Hickey claims that there are the same number of positive even numbers as counting numbers. He sets up a 1-to-1 correspondence between the to sets and concludes that they are both countably infinite and therefore it anyone who says that one set has more elements than the other is wrong.
So close, but fail. The fundamental misunderstanding here comes from the reflexive property of numbers which states Every number is equal to itself
This isn't exactly the most controversial property ever. The problem is when you try to apply that rule to infinity, it doesn't work because infinity is not a number.
In problem 9, the key step of the proof was that H > A. Since H was infinity and A was infinity, that means that in this case infinity > infinity, not infinity = infinity. This is fine, since infinity is not a number and therefore doesn't have to follow the reflexive property.
Applying the same logic to problem 3, we see that it makes no sense to say that there is the same number of elements in the sets because both cardinalities (set sizes) are infinite. The moral of the story is that infinity doesn't play nice with concepts like equals, addition, multiplication and the like, so be careful about treating it like a number, or before you know it you'll be claiming that infinity minus infinity is zero or infinity divided by infinity is one, which turn out to be variants of the same mistake Hickey made.