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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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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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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Ostensive definitions for queer experiences

While I’m still on the subject of Wittgenstein’s private language arguments, I’d like to say more about how it relates to queer experiences.

You might notice that I’ve never stated exactly what the private language argument is. It isn’t really a formal argument, in the sense of having premises and a conclusion. Rather, the private language argument refers to a cluster of issues regarding personal experiences. For example, what does “pain” refer to, if anything? When I experience a thing, how do I identify it as pain or not pain? How do I know that it is similar to what other people are feeling when they refer to pain?

You must realize that I am not formally trained in philosophy. I’ve never read Wittgenstein first-hand and don’t know precisely what he says. But it seems to me that the private language argument is wasted on philosophers, when it’s so directly relevant to queer experiences. How does one know that one is experiencing sexual or romantic attraction? How about gender dysphoria? This isn’t philosophical abstraction to us, it’s something we live through and discuss amongst ourselves extensively. I would bet that it is also relevant to other minority experiences, such as chronic pain, depression, or aphantasia.

Usually, when we define a word, we explain it in terms of other words. But clearly we can’t do this for every word, because the definitions would eventually become circular. If you think about it, there is a way around this.  You can define a word by pointing to examples of it. For example, I can define an ant by pointing at one, or I can define an octahedron by pointing at one. This is called an ostensive definition.
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My beetle is an elephant

This is a repost of an article from less than a year ago, which went on The Asexual Agenda.  I was recently reminded of this article, and I intend to say more on the subject.

Sciatrix once created an influential metaphor for attraction: it’s like everyone has an invisible elephant that only they can see.  These invisible elephants are apparently very important in society, but hardly anyone can be bothered to describe them because it’s assumed that everyone has their own elephant and can see for themselves.

Ludwig Wittgenstein, one of the most important philosophers of the 20th century, once described a thought experiment: Suppose that everyone has a box with a “beetle” inside it, but each person can only see their own “beetle”.  Wittgenstein argues that when we talk about “beetles”, we are only referring to that which is in the box.  It doesn’t matter if the boxes actually contain different things, or if the things change over time, or if the boxes are actually empty.  (watch this video)

That feeling when philosophical thought experiments become directly applicable to your daily life. [Read more…]

On judging people of the past

A bold statement: People of the past should almost always be judged by today’s standards. This results in thinking of a lot of historical figures as horrible people. So yeah, I’ll say it: most historical figures were horrible people. Some of them were horrible because their surrounding culture was horrible, and others were just plain horrible.

My basic reasoning: Moral judgment isn’t for people of the past. The people of the past are dead, and their actions are already foregone conclusions. Moral judgement is for people of the present. I do not wish for people of the present to valorize or emulate people of the past just because they were great by the standards of their own time. I strive for the perpetual improvement of humankind, not the stagnation of virtue. [Read more…]