Pippen wrote:You assumed a "largest number" and you could indeed prove that this very "largest number" is not the largest, because of the successor of it. So your "largest number" was not the largest number right from the beginning, as the proof shows.
Let's break down assuming a largest number. We assume that a largest natural number
exists. If it does exist, there is
only one of them, and we can give it a name, so that we can designate it. We do not have to actually find the largest number. We can talk about it, anyway. Just like they talk about the UNSUB on "Criminal Minds" without knowing who it is.
Next, with what we know about natural numbers, we can show that
if the largest natural number exists, there is a larger natural number than
it. ("It" is a pronoun that lets us talk about the supposed largest natural number.) That fact is a contradiction, which shows that one of our assumptions is false.
Well, one of our assumptions is that there is a largest natural number. Another is that we can talk about it, even if it does not exist. Another is that every natural number has a successor. Another is that the successor of a natural number is greater than that number. The last two we are sure of, because of what we know about natural numbers. The first assumption, that the largest natural number exists, is the one we want to give up. However, it may well be that the second assumption is false, that we cannot really talk about something that does not exist, or about something that may not exist. But if the largest natural number existed, there would be no trouble with talking about it, so the second assumption is a problem only when the first assumption is false. So we still end up with the first assumption being false.
I know I kind of went around the bush there, but I wanted to be clear that we did not
identify the largest natural number, we simply talked about it as though it existed.
Therefore we are now open to look for another assumed largest number.
No, because we never looked for one in the first place. It is not like we found one and then said, this is not it. We simply talked about one as though it existed. Once we show that one does not exist, what is there to look for?
How does a variable work?
I have seen some old problems that do not use variables. They are often not easy to understand. Even the easy ones are like, "Heap, its third, . . ."
You can think of variables as linguistic devices that refer to things and help us keep track of what we are saying. Like pronouns.
How can an "x" stand for one object out of a set, but at the same time cover all object of a set, so that if you prove something for x it is proven for all objects of the set?
Suppose that something is true for each member in a set. We can then use "x" to denote some member of the set (without necessarily knowing which one) and prove something else about it, using
only what is true for each member of the set. Then we know that what we have proved about it is true for each member of the set. It is not that "x" covers all members of the set, it is that we restricted our proof to what is true about each member of the set.