Every time Sheafification of G does the "Law of the Excluded Middle" bit

Added: 2026-05-10

[00:00:00]Now, if you're in the comment section typing away that vector spaces are always freely generated, I want you to stop right there and realize that you just assume the aim of choice.

[00:00:09]>> Hell no to the no.

[00:00:15]>> Rather than proving that there must exist a point C where f of c is zero, I actually just proved that it's impossible for there not to be such a point. If you're thinking, what's the difference? Isn't that the same thing? I want you to know that you just assumed the law of the excluded middle.

[00:00:31]>> Hell no. To the no.

[00:00:36]>> Imagine you are in a completely new scenario with completely different colleagues Alice, Bob, Charlie, and Dan.

[00:00:43]One of them is a mathematician, one's a physicist, one's an engineer, and one's a philosopher. And you don't know who is which. You're a computer scientist, so everyone thought it would be fun and quirky to respond to your yes no questions in foos, bars, and bases. If you're wondering why there are three responses, you assume the law of the excluded middle.

[00:01:11]Working through it carefully, you'll realize that the join and the meet operations will define if you were about to say a complete boolean algebra. Then you just assume the law of the excluded middle.

[00:01:23]>> Hell no to the no.

[00:01:27]>> With this set omega, we can actually see the duality between elements and subsets. Unless you wasted your second year summer doing too much set theory like I did, this might take some time to digest and appreciate. The astute among you might have noticed, however, that I've been assuming the law of the excluded middle.

[00:01:53]While many of us are probably aware that such a statement is not true, the sheath also tells us that this statement is not false. If you think this is nonsensical, then you just assumed the law of the excluded middle.

[00:02:06]>> Hell no. to the no to the >> in. Note that there are essentially only two distinct minus one categories. The one with only one thing in it and the one with nothing in it. Or that would be the case if you assume the law of the excluded middle.

[00:02:25]>> Hell no to the no.

[00:02:28]>> We manage this control flow logic with the conditional branch instruction. The first label tells us where to branch to if the hasint boolean is true. And the second is where to branch to if has int is false. Unfortunately, like most languages, this language also assumes the law of the excluded middle.

[00:02:46]>> Hell no to the no.

[00:02:49]>> If you're wondering why there are ellipses here, you must be new to the channel. Welcome to my little corner of the internet. Please enjoy your stay here. Per the guidelines, this community is very inclusive. Accordingly, to ensure everyone has a good time, I do ask you to do some introspection because you just assumed the law of the excluded middle.

[00:03:18]Gluing the middle is fine.