Gödel’s First Incompleteness Theorem explained

Once upon a time, mathematicians thought they would be able to prove everything. The endeavor was known as Hilbert’s Program. They would find a complete and consistent set of axioms, and on this foundation build all of mathematics. (Although to be fair, much of mathematics was already built and was to be placed upon on those foundations retroactively.) And then, if everything went well, they would generate an algorithm that could prove every statement either true or false.

To some extent, Hilbert’s Program was successful. We now have Zermelo-Fraenkel set theory, which is a solid foundation for the vast majority of mathematics. But there are two problems. First, set theory isn’t complete. Second, we can’t prove it’s consistent. And Gödel showed that these problems have no solutions.

Gödel’s First Incompleteness Theorem: No consistent formal system is complete.
Gödel’s Second Incompleteness Theorem: No consistent formal system can prove its own consistency.
(Both of these theorems have additional qualifiers that I’ll get to later.)

Here I will explain the proof for the First Incompleteness Theorem, and a few of its implications. In a later post, I will talk about the Second Incompleteness Theorem.
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Queer men and women: cultural differences

Throughout this post, I refer to queer men and women, but I understand that many of the people in question do not use the word “queer” for themselves. Let’s just acknowledge that and move on.

Here’s a little mystery that my readers can help me out with: What are the cultural differences between queer men’s and queer women’s spaces?

The differences are directly relevant to my life. I am gay, and I have hung out in many spaces for queer men. However, I am also active in online ace communities, which are predominantly made up of women. Occasionally, this causes a disconnect between the cultures I see online, and the cultures I see offline. For example, ace communities experience a lot of gatekeeping, wherein people try to say aces aren’t queer, or else reject the word “queer”. To me this has always felt like absurd internet nonsense, because my impression is queer men don’t engage in the same variety of gatekeeping at all. But the ability to dismiss gatekeeping as absurd is a kind of privilege. I want to understand the differences rather than dismissing them.

Obviously, one of the major differences is the difference between offline and online. But recently, I came to recognize gender as an important factor. I wanted to investigate this further by seeing what other people say, but all I found was a silly Buzzfeed article.  Clearly this warrants more serious discussion.
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Origami: Arrow illusion

So, you know how things in mirrors always have left and right reversed?  This origami model is no exception.

An origami arrow in front of a mirror. The arrow points to the left, while the reflection points to the right.

Arrow Illusion, my design.

The arrow illusion was inspired by a much more impressive optical illusion, the Ambiguous Cylinder Illusion.  Video below the fold. [Read more…]