I thought about this some more and want to elaborate on what we’re talking about for those who haven’t encountered the question of being “well-defined” in math and might not know what exactly it is we mean.
Example: A definition that implicitly assumes existence of something.
If we have a collection of (real) numbers X, we might want to know what the largest number in that collection is. So let’s define max(X) to mean the largest number in X. Is this well-defined? Sure, I just defined it! But then what is max(X) when X is, say, all positive integers? No positive integer is larger than all the others, so there isn’t a largest number in X as every number in X is smaller than some other one.
Example: A definition that involves a implicit choice.
If we have a (real) number n and write the set of integers as Z, then we might write n+Z to mean all the numbers that may be written as a sum n+k for some integer k. We call n+Z a coset of Z. Note that we are definitely allowing n to be a non-integer value, such as n=1/2. Nothing is wrong with this definition.
But we can add integers, so can we add cosets? Well, let’s try defining what it would mean to add two cosets n+Z and m+Z together. Define n+Z + m+Z = (n+m)+Z, which seems to be the most natural thing to try. But are we done—is this actually well-defined?
Not really, since although we wrote n+Z down that way, we can see that there are other ways to write it, too. We get exactly the same set of numbers if we had instead used (n+1)+Z, so n+Z = (n+1)+Z. So our definition of how to add implicitly assumes that we have a ‘chosen’ way to write down the coset. But luckily, the way we defined it doesn’t actually depend on how we wrote down the coset! For example, (n+1)+Z + m+Z = (n+m+1)+Z = (n+m)+Z. In essence, even though there are multiple ways to write down the cosets n+Z and m+Z, there are also multiple ways to write down (n+m)+Z and the different ways for the first two just give different ways to write the second one. So this can be shown to be well-defined even though it involved an implicit choice.
Question: Does anyone have a good example of this off-hand of these kinds of things in real life? The only non-contrived example I have is an ontological ‘proof’ of God.
I thought about this some more and want to elaborate on what we’re talking about for those who haven’t encountered the question of being “well-defined” in math and might not know what exactly it is we mean.
Example: A definition that implicitly assumes existence of something.
If we have a collection of (real) numbers X, we might want to know what the largest number in that collection is. So let’s define max(X) to mean the largest number in X. Is this well-defined? Sure, I just defined it! But then what is max(X) when X is, say, all positive integers? No positive integer is larger than all the others, so there isn’t a largest number in X as every number in X is smaller than some other one.
Example: A definition that involves a implicit choice.
If we have a (real) number n and write the set of integers as Z, then we might write n+Z to mean all the numbers that may be written as a sum n+k for some integer k. We call n+Z a coset of Z. Note that we are definitely allowing n to be a non-integer value, such as n=1/2. Nothing is wrong with this definition.
But we can add integers, so can we add cosets? Well, let’s try defining what it would mean to add two cosets n+Z and m+Z together. Define n+Z + m+Z = (n+m)+Z, which seems to be the most natural thing to try. But are we done—is this actually well-defined?
Not really, since although we wrote n+Z down that way, we can see that there are other ways to write it, too. We get exactly the same set of numbers if we had instead used (n+1)+Z, so n+Z = (n+1)+Z. So our definition of how to add implicitly assumes that we have a ‘chosen’ way to write down the coset. But luckily, the way we defined it doesn’t actually depend on how we wrote down the coset! For example, (n+1)+Z + m+Z = (n+m+1)+Z = (n+m)+Z. In essence, even though there are multiple ways to write down the cosets n+Z and m+Z, there are also multiple ways to write down (n+m)+Z and the different ways for the first two just give different ways to write the second one. So this can be shown to be well-defined even though it involved an implicit choice.
Question: Does anyone have a good example of this off-hand of these kinds of things in real life? The only non-contrived example I have is an ontological ‘proof’ of God.