Gödel’s Incompleteness Theorems

Godel’s Incompleteness Theorems are some of the most important results in logic, that I’ve used in many forms day to day. “any formal system (such as a system of the natural numbers), there are certain true statements about the system which cannot be proven by the system itself.” https://listverse.com/2013/05/05/10-coolest-mathematics-results/ This, although seemingly technical, means to solve a problem you need to look outside the problem itself. It isn’t necessarily the theorem itself, but the state of mind. We often think that we can find the problems and solutions of a problem on whether something is true or not, without looking past the system itself. It is like determining whether the legal system is just based only on the legal system.  Which leads us to the other Incompleteness Theorem, which basically says you can’t determine if something is consistent based on itself. This comes in handy whenever you are trying to figure out how to handle an issue.

The first step when presented with an issue is to find a good goal, or achievement you want to achieve, even if it’s broad, like, betterment for society, as long as there is an outside goal, you can work backwards to determine if you are considering everything.

The other great use, is that you can never fully figure out the impact of a system by looking at the system itself, consider iphones, while we may enjoy them, is our enjoyment worth the bad conditions of the people working to make them? And what can we do about it? Math is at its heart all about problem solving, and I feel like sometimes it gets undervalued to memorization of what dead people figured out and numbers, but really it is about solving problems given axioms. And this result tells us, we never have it all figured out. (and even if we thought we were at least on the right track, we wouldn’t know it)

One Comment

  1. Austin, this is a very interesting take on something that too many people don’t know about. Only by looking at many different fields and systems did our classic philosophers, mathematicians, and scientists developed a deeper understanding of the world around them. Now, even our society’s “smart people” are tied to one or maybe two issue areas. While I know this isn’t always the case, it certainly seems as though society’s intellectuals are more specialized, which I would argue places them further away from their goals than trying to be the “jack of all trades.” Regardless though, the first step in figuring anything out is gathering information, which will always be fundamentally incomplete. Your post shines a light on that, and in turn results in better decision making if closely adhered to. Great work man, and great topic selection.

    Mitch Dobbs

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