Gödel’s Second Incompleteness Theorem Explained

This is a followup to an earlier post where I talked about Gödel’s First Incompleteness Theorem. Here, I discuss the Second Incompleteness Theorem, and further implications.

Could you remind me what the theorem was?

The theorem states that a consistent formal system cannot prove its own consistency.

As previously discussed, there are a couple qualifiers. The formal system must include some amount of arithmetic, and must have a computable set of axioms.

What does consistency mean?

A system is consistent if it cannot prove any contradictions. A system is inconsistent if it can prove a contradiction.

Contradictions sound bad. Are they bad?

Yes. The Explosion Principle states that if you can prove a direct contradiction, then you can prove absolutely any statement.

Here’s how the Explosion Principle works. Suppose A and not-A are both provable. Now consider statement B. “(A implies B) or (not-A implies B)” is a tautology. Since both A and not-A, that means we can prove B. Following the same procedure we can also prove not-B.

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Gay loneliness: critiques and counter-critiques

Recently I read the article “Together Alone: The Epidemic of Gay Loneliness” by Michael Hobbes. It’s about the physical and mental health issues gay men face, even in absence of overt bigotry. Based on personal stories and talking to researchers, Hobbes identifies two causes. The first one is “minority stress”–we’re made aware of our marginalized status constantly.  And even if we’re in a friendly environment right now, the minority stress was already pounded into us as kids.

The second cause, says Hobbes, is gay culture itself. Well, you get a bunch of people together, all of whom have dealt with minority stress, and it turns out they don’t form a big happy family. Hobbes talks about meanness, often in the form of racism, body policing, and masculinity policing. He laments that for many gay men, hookup apps are the primary way they really interact with other gay people.

I am mostly sympathetic to this article. I’ve long thought the health disparities suffered by gay men (and by other minority groups as well) are an elephant in the room. Instead we talk so much about same-sex marriage, bathroom bills, job and housing discrimination, and bullying. And while these are all important issues, it seems like they were chosen not on the basis of being important, but on the basis of being amenable to public policy changes. Health and economic disparities are tougher to address, because we often don’t know what causes them, much less how to solve them.

But here I will raise a few criticisms of Hobbes’ article, and also discuss other people’s critiques.
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Linkspam: March 10th, 2017

It’s my monthly linkspam!

4chan: The Skeleton Key to the Rise of Trump – To be honest, the first thing that struck me about this long article, was the name of the author, Dale Beran.  Isn’t he the webcomics legend behind A Lesson Is Learned But The Damage Is Irreversible? Yeah, so I’m a webcomics geek. I also remember Dale Beran for this comic–I happen to share his negative opinion of cars.

I’m not sure how far I would vouch for this article. There’s some nice insider perspective on the nihilistic culture of 4chan, and a better explanation of Anonymous than I’ve ever seen from mainstream news outlets. (I remember news outlets saying “anonymous” just refers to people who obscure their identity. But but that’s not what Anonymous is, can you even internet?) It veers a bit much into depicting 4channers as failures who live with their parents. I have lots of friends and relatives who live with their parents–it’s a cheaper way to live in times of economic hardship and I consider this stereotyping to be classist.

I like the bit about models of what men are supposed to strive for in life–either they get a wife and kids, or else they’re supposed to be “players”. Being an ace activist I emphatically reject these models and question whether they’re any good even for non-ace people. Dale Beran suggests that many young men are trapped in these models, and when I question them it’s like I’m saying their problems are in their heads. Something to mull over.

Laurie Penny shared her experience touring with Milo Yiannopolous. This covers some of the same territory as the previous article, but is more compact and focused.

Drug Watch: New Addyi Marketing Campaign, “Find My Spark” – Addyi is the drug recently approved (on weak evidence) to treat Hypoactive Sexual Desire Disorder in women.  After disappointing sales, they are trying a marketing campaign.  The campaign encourages women who want sex less often than their partners to see this as a medical issue.
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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…]

Plantinga’s private language

One of the great things about arguments for gods and the supernatural, is that you can always look back at them and find new problems. Alvin Plantinga’s arguments are especially lovely in this regard. Having been recently been thinking of Wittgenstein’s private language arguments, it occurred to me that somewhere in there is a rebuttal to Plantinga’s evolutionary argument against naturalism.

The evolutionary argument against naturalism argues that if both evolution and naturalism are true, then we cannot trust our own rational faculties, and therefore cannot trust our belief that evolution and naturalism are true.  The reasoning goes that naturalistic evolution does not specifically produce true beliefs, but rather produces adaptive beliefs. An adaptive belief does not need to be true, it just needs to produce adaptive behavior. For example, rather than believing that you should run away from a tiger because it will eat you, you might believe that you should run away from a tiger because that’s the best way to pet the tiger (Plantinga’s example). The number of false beliefs that produce adaptive behavior is much larger than the number of true beliefs that produce adaptive behavior. Therefore, most beliefs are probably false.

There are numerous issues with this argument, a few of which you might be shouting at the screen. From a scientist’s perspective, Plantinga appears to be ignorant of how evolution actually works. Evolution does not necessarily produce the most adaptive traits, certainly not immediately. If you have false but adaptive beliefs at one point in time, it is questionable whether those beliefs would continue to be adaptive when your descendants find themselves in slightly different environments. Also, Plantinga ignores that brain efficiency is an adaptive trait. I would imagine that a brain which produces true beliefs via reasoning is far more efficient than a brain that produces false but adaptive beliefs via some mysterious yet reliable process.
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