Infinities, Existence, and Uniqueness

Getting a teacher off track is always fun in class. There is usually a student who is good at convincing the teacher to tell a personal story, or go on a tangent about the subject. Today in differential equations we went on an infinities tangent. We were discussing the “largest” interval a function could be valid on, and we came to the conclusion that the interval was (-2,∞).

Someone from the front of the class asked, “well, wouldn’t (-1,∞) be just as big?” And just like that, the class was wildly derailed. The teacher apologized and said he was using a more intuitive sense of “largest,” but the damage was done. My brain had already started to hurt, because everyone was asking about infinities at the same time. The professor was frantically fielding abstract and esoteric questions that had nothing to do with first order differential equations, and a brave student stepped forward to calm the storm by suggesting that people watch Vsauce video on YouTube, “How to Count Past Infinity.” I watched it right before writing this post.

Aleph Null is the cardinal number that represents all finite numbers. I have never been one to enjoy thinking about infinity, or the universe, or things so big it’s impossible to keep track of them. So I was happy to hear there was a simple solution. This is what Aleph Null looks like א‎0. But unfortunately, there is more after Aleph Null, much more, so much more that mathematicians justify the arbitrary infinities they are making up with axioms, and just say they’re true. I’m not going to try and stop them, but it seems too meta to give themselves that much power. I ultimately found no closure, except in the point that there are “uncountable sets,” such as sets that cannot be reached from the bottom. It kind of limits human’s power, which to me gives the numbers more credibility because mathematicians are relinquishing some control and admitting they cannot count up the number. 

After the barrage of complications with infinite intervals was cleared up, my professor wrote “Existence and Uniqueness” on the paper (he projects his writing on printer paper onto the screen), and said “I know, kind of esoteric.” I thought he was joking, but we learned a fundamental principle about the existence and uniqueness of solutions to differential equations. My math class today turned out to be awesome. Besides the infinite discussion, two other friends and I walked to the top of Waldo hall in the 10 minutes between the recitation and lecture. Waldo hall looks like a castle from the outside, and it was just as fun to walk around inside it. We jiggled a precarious bag of pretzels out of the bottom a vending machine that someone earlier that day had probably been very mad about after spending $1.75 just to get them stuck beneath the M&M’s. And we also solved a fun slope fields packet. I am really starting to enjoy the people and format of the class. My homework is done for the week in math so now I’m just touching up my integrating skills and hoping we don’t have any pop-quizes.

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