In which I vent to Chat-GPT about the ambiguous use of 'at least'. It's a tad more tolerant than I, and shows an almost obsequious willingness to agree with my final conjecture.
Moi:
when did 'at least' come to mean 'limit' without being restricted to minimum? Like, 'at least every 15 minutes' which implies gaps of longer than 15 minutes, does it not?
GPT:
The phrase "at least" has indeed evolved over time. Originally, it meant a minimum requirement or threshold. However, in certain contexts, it now implies a limit or boundary without specifying an upper bound. For instance, saying "at least every 15 minutes" suggests that the interval won't exceed 15 minutes, but it could be shorter. Language is fascinating, isn't it? 😊
Moi:
So they should really say something like 'at least 4 times per hour', yes? Or 'at most every 15 minutes'?
GPT:
Indeed, using more precise language can help avoid ambiguity. Saying "at least 4 times per hour" or "at most every 15 minutes" would provide clearer instructions. Language evolves, and sometimes context plays a role in how we interpret phrases. 😊
Moi:
Or do they get away with this imprecision because 'at least' is not actually (linguistically) binding to the '15 minutes' but instead to the 'every'?
GPT:
You've hit the nail on the head! The imprecision arises because "at least" is linguistically binding to the "every," not the specific interval (e.g., 15 minutes). So, saying "at least every 15 minutes" implies that the interval won't exceed 15 minutes, but it could be shorter. Language nuances can be quite intriguing! 😊
Moi:
But it's still rubbish English though, amirite?
GPT:
Well, some might argue that the imprecision in phrases like "at least every 15 minutes" is a quirk of language evolution rather than outright rubbish. However, I understand where you're coming from—it can indeed be confusing! 😅
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part 3 of the series on cool stuff about infinity (toc)
Bigger is always better. Let's talk max and min today.
If I gave you a bunch of numbers, you could tell me the max and the min, right? Just pick the biggest and smallest of the bunch. Easy.
Well...let's be a bit more careful. It actually depends on how big my "bunch" of numbers is.
Yup...I'm at it again: infinity, guys!
It's pretty simple to pick out a maximum and a minimum for a finite set of numbers. But what happens if we have an infinite set of numbers? Bunches upon bunches. Well...you can start to see the problem. I learn by example, so let's have an example. No actually, I'm feeling generous today. Let's have FOUR examples to facilitate easy learning, bro:
consider the closed set [0, 1]. This set has both a maximum and a minimum. They are 1 and 0 respectively.
the set [0, 1) clearly still has a minimum: 0. the maximum is unclear. we would be tempted to say 1, but 1 is not an element in the set, so it can't be the maximum. If we pick any number less than 1, though, it will not be a maximum of the set because we can find a number between it and 1 that is still included in the set.
the set (0, 1] runs into the same problem with the minimum. at least we have a maximum.
we're just out of luck with (0, 1).
Like everything else, a concept for finite sets will extend to a concept for infinite sets. Instead of maxima and minima, we talk about suprema and infima.
Formal definitions first this time: suppose we have a set E. The supremum of E, supE, is an upper bound for E where if b < supE for any arbitrary b, then b is not an upper bound of E. The commontalk way of saying it is that the supremum of E is the least upper bound of E. The opposite definition holds for the infimum, so the infimum is the greatest lower bound of E. Since the supremum and infimum are such related concepts, usually we just treat one of them, and the opposite result will hold for the other.
Let's return to the sets that gave us difficulty...in particular, let's look at [0, 1). We'd like to say something about the maximum, but we can't. What about the supremum? Now, we can do what our instincts were telling us to do all along: the supremum is indeed 1. First of all, we need to make sure that all the elements in the set are less than 1, guaranteeing that 1 is an upper bound. Clearly (almost by definition), it is. Next, we need to make sure it's the least upper bound, so let's take any number less than 1, say 1 - e, e > 0. Well, we know we can say that 1 - e/2 is still in our set, and that 1 - e/2 > 1 - e, so clearly 1 - e is not an upper bound. Since e was arbitrary to begin with, we must conclude that any number of the form 1 - e, e > 0, cannot be an upper bound; ie, any number less than 1 cannot be an upper bound. Thus, 1 is our least upper bound. An identical (reversed) argument can be made for the infimum of (0 , 1].
Notice that suprema and infima are unique: it'd be silly to assume otherwise.
The last catch: so far, the unspoken assumption is that we are dealing in the space of real numbers, R. The supremum of a set doesn't need to be included in the set, but the overall space we're dealing with better include it or else we wouldn't be able to talk about it at all. Again, we can return to the example of (0, 1) restricted only to the positive numbers. Then, our infimum would not exist because 0 is not included in our space.
It follows that some spaces have "holes" and so suprema and infima of certain sets may not exist in the space. Consequently, we can characterise spaces by whether or not we are guaranteed the existence of a supremum or an infimum for any bounded set. This property is called the least upper bound property. we see that Q cannot have the least upper bound property (again, just take the set of all q in Q such that q2 < 2). But, as we will see, R has the least upper bound property.
Hmm...this guarantee of a supremum/infimum business sounds a lot like the previously teased guarantee of converging cauchy sequences...
Let us continue our discussion about Riemann Integration!
Now at this point, we have a notion of what our upper and lower sums looks like and what they tell us about a function, but how do we know that these upper and lower sums even tell us anything accurate about our function? Well, we can utilize a very powerful result that comes from a long list of inequalities. Now, you can prove this but for the sake of time, I'm just going to show it. Given a function f and an arbitrary partition, P we have that
Essentially, what this tells us is that if our upper and lower sums approach each other, we can be sure the the Riemann Integral of our function exists. This is a very powerful result in that now we can prove that a function is Riemann Integrable. In other words a function is Riemann Integrable iff:
What a useful result. Now, given any function f, we can show it is Riemann Integrable so long as we can cleverly choose a Partition so that the above inequality is true.
In part 3, we will use this inequality to prove some incredibly funky functions are integrable. Stay tuned!
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