Sometimes you say something that sounds right, but when you actually look at it, the whole thing comes undone. I noticed this with the phrase “everything is nuanced.” On the surface, it feels thoughtful. But if everything is nuanced, shouldn’t that claim be nuanced too? If it isn’t, it doesn’t live up to itself. If it is, it can’t be taken as an absolute. You end up going in circles.

Another example is “there are no right answers in philosophy.” If that’s true, then we’ve just found a right answer, which breaks the rule.

Or, “you shouldn’t make generalizations.” That’s a generalization.

Or, “truth is always relative.” If that’s true, even that claim is relative, which means sometimes it isn’t true.

I thought this was just a weird language quirk until I learned that these kinds of paradoxes have actually shown up in philosophy and math in ways that matter.

A classic example is the liar paradox: “this statement is false.” If the statement is true, then what it says must hold, so it’s false. But if it’s false, then what it says isn’t the case, so it must be true. It makes my head spin. ¹

Philosophers and logicians realized this challenged the whole idea of truth. If “truth” lets you make a statement that flips on itself, then truth itself can break down. So, what do you do with a concept that defeats its own definition?

Alfred Tarski tackled this head-on in the 1930s. He noticed that paradoxes like the liar statement only happen if you let a language “talk about itself.” For example, in English, you can say “This sentence is false,” but you can also try to define “truth” for English using only English. Tarski’s solution was to separate the language you use to talk (call it the “object language”) from the language you use to describe truth about sentences in the object language (call that the “meta-language”). ²

How does this work?

Suppose you have a simple language for talking about the weather:

“It is raining” would be object language. But if you want to say, “The sentence ‘It is raining’ is true,” you step outside to a meta-language.

However, Tarski’s hierarchy only works for strict, formal languages. In real life, we mix object language and meta-language all the time, so the fix doesn’t apply to ordinary conversation. Plus, some paradoxes like sentences that say “this sentence is not true in any meta-language” still slip through, which create new problems called “revenge paradoxes.” ^{1, 2}

Another approach to fixing the liar paradox done by some logicians was introducing new options beyond “true” and “false.” For example, Jan Łukasiewicz created three-valued logic: a statement could be true, false, or “undefined.” So the liar sentence would just be labeled “undefined.” ^{3, 4}

These questions in philosophy had many parallels in mathematics. In the early 1900s, Bertrand Russell was working with sets, which are basically groups of things. Most sets are simple, like “the set of all chairs in this room.” But Russell asked about the set of all sets that don’t contain themselves. ⁵

Here’s a way to picture it:

Imagine you have sets like “the set of all apples” (just apples), or “the set of all sets” (every set you can imagine). Most sets don’t include themselves. For example, “the set of all apples” is not itself an apple, so it doesn’t contain itself. But the “set of all sets” does contain itself, because it contains every set, including itself.

Now, try to build “the set of all sets that do not contain themselves,” and ask yourself: “does this set include itself as a member?”

If it does contain itself, then it shouldn’t, because it’s only supposed to include sets that don’t contain themselves. But if it doesn’t contain itself, then it fits its own rule, so it should be included.

This is why Russell realized that the idea of “all sets” is broken if you allow unlimited self-reference. ⁵

When Russell showed this to Frege (who was building set theory to ground all of math), Frege realized his entire foundation was ruined. Russell himself worked for years with Whitehead to design a new “type theory” where you build sets in levels, so no set can ever contain itself or refer to itself directly. ^{5, 6}

Modern set theory (like Zermelo-Fraenkel set theory) just bans sets from being members of themselves, which sidesteps the paradox completely. This is why in modern math, you never talk about “the set of all sets.” ⁷

Even as set theory was being fixed, a new paradox made mathematicians realize there might never be a perfect logical system. In the 1930s, Kurt Gödel showed that any logical system that’s strong enough to handle basic math, like adding and multiplying, will always contain statements that are true, but can’t be proven within the system. ⁸

But first, what’s a “system”? In this context, it’s a set of logical rules for writing proofs, like the formal rules mathematicians use to prove things about numbers.

Gödel realized you could use numbers to “code” any sentence, any proof, and even the rules themselves. This is called Gödel numbering. He constructed a sentence that, using those rules, says, “This statement cannot be proven in this system.” This is very close to the liar paradox, but in the language of math. ⁸

If the system can prove this statement, then it’s inconsistent (it proves something that says it can’t be proven). But if the system can’t prove it, then the statement is true, but it’s true for a reason that the system can never reveal from the inside.

Gödel’s first incompleteness theorem says that every system complex enough to include basic arithmetic has some truths it can never prove on its own terms. His second incompleteness theorem says that the system can’t prove it’s free from contradictions using just its own rules. ⁸

The practical effect was huge: Hilbert and other mathematicians had hoped to find a complete and perfect set of rules to answer every math question. Gödel proved this was impossible. There will always be mathematical truths that no set of rules can ever capture completely, and no system can guarantee it is free of contradictions from within. ⁹

So this information is cool, but what now? A philosopher named Wittgenstein started with the idea that language could be made perfectly precise, and anything outside its limits was simply nonsense. But in his later work, he realized that language isn’t a machine with fixed definitions, but more like a set of games, or “forms of life,” where meaning changes with context and use. I actually wrote another post about definitions that kind of talks about this idea! ¹⁰

For example, in Philosophical Investigations, Wittgenstein imagines someone asking, “What is the definition of ‘game’?” You might try to find a single rule that covers all games, like cards, soccer, chess, tag. But you’ll fail. There’s no one feature all games share. ¹⁰

Wittgenstein’s insight was that a lot of philosophical confusion comes from trying to force language into rigid categories when ordinary life doesn’t work that way. Instead of hunting for strict definitions where none fit, Wittgenstein argued it’s better to notice how words are actually used in real situations. ¹⁰

Paradox and self-reference forced philosophers and mathematicians to rethink their entire approach. The result is a world where logic, math, and language are more robust, but also more humble about their own limits.