Thinking beyond the Paradox

Background

Russell’s paradox is a famous problem in set theory that asks: consider the set (S) of all sets that do not contain themselves. Does (S) contain itself? If it does, it shouldn’t (because it only includes sets that don’t contain themselves), but if it doesn’t, it should (because it fits the rule of sets that don’t contain themselves).

It’s a logical loop that breaks naive set theory. A similar paradox exists in the “barber paradox”: imagine a barber in a town who shaves everyone who does not shave themselves. The question is: does the barber shave himself? If he does, he shouldn’t (since he only shaves people who don’t shave themselves), but if he doesn’t, he should (because he fits the rule of people who don’t shave themselves). Both paradoxes show a self-referential loop that leads to a contradiction.

So, I was at work in Paris with a colleague, and we were bored out of our minds at around 5 p.m. He randomly brought up something about Russell’s paradox. something about sets and at first, I was just confused. I mean, I’m an aerospace engineering grad, not a mathematician or a logician. But at the time, I was mostly through The Beginning of Infinity by David Deutsch, so naturally, I started trying to think about it in a quantum framework.

A couple of hours later, I blurted out my first attempt: “Well, in a quantum universe, it’s not a paradox, right? The set either contains itself or it doesn’t, depending on the universe.” It seemed clever for about five seconds, until I realised it was a total cop out. It just dodged the paradox instead of actually dealing with it. So, I left it for a while.

Then, about seven months later, I started reading The Fabric of Reality, and with the blind confidence of someone who has read exactly two David Deutsch books, I self-promoted myself to physicist. Armed with this delusion I decided to take another swing at Russell’s paradox. But this time, I wasn’t just going to wiggle my way out of it. I was going to break it till the explanation is satisfying.

The Quantum Spark

It started with this wild thought: what if Russell’s paradox isn’t a paradox in a quantum physics world? Like, in quantum mechanics, stuff can be in superposition—both true and false until you measure it. So I pictured this set ( S ) (the one with all sets that don’t contain themselves) just chilling in a superposition of “in itself” and “not in itself.” No collapse, no problem, right? In one universe, maybe ( S ) contains itself; in another, it doesn’t. Many worlds style. I was like, “Like if it never picks a side, where’s the paradox?” It’s not forced to contradict itself if it’s just vibing in this blurry quantum state.

But then I hit a wall. Even in quantum land, the paradox isn’t about what happens after. It is more about defining ( S ) in the first place. Superposition’s cool, but the self-reference still loops back and bites you.

For example:
Imagine you have a magic box that can be either open or closed. You define the rule that the box is both open and closed at the same time, like a quantum superposition.

Now, you say the box can only be opened if it’s both open and closed. But here’s the problem: you can’t open it because it’s stuck in its own rule. It’s constantly referencing itself. if it’s open, it can’t be closed, and if it’s closed, it can’t be open.

Similarly, in Russell’s paradox, the set (S) keeps referencing itself in a way that makes it impossible to define without creating a contradiction. The paradox is baked into the definition, no matter how you try to handle it.

Hence I needed more needed to think relatively outside the box;

The “Me” Thought Experiment

So I pivoted. I started thinking about myself, like, “I am me,” but there are all these versions of me across time. Snapshots, you know? There’s me right now (x_now), then me yesterday (x₁), me tomorrow (x₂), all the way to me when I kick the bucket (x_end). What if I made a set S of all “me’s” except the current me? So:

S = {x₁, x₂, x₃, …}

but x_now isn’t in it.

Here’s the thing though: as time moves, x_now shifts. Today’s “now” becomes tomorrow’s “past me,” and S keeps updating. I thought, “This isn’t really a paradox right lol!” It’s dynamic, S never has to ask if it contains itself in that doomed Russell way. It’s just all the “me’s” I’ve been, trailing behind the current me. Quantum fits here too since maybe x_now is in a superposition of being me and not me, shifting across universes or moments.

But then I tripped. My life’s finite. There’s a last me (x_end), and if S is all “me’s” except “now,” what happens at the end? Does the last me include itself? It started feeling shaky, like the paradox sneaking back in. Finite stuff always screws you. But there is an endpoint, and that’s where it gets messy.

Going Infinite

So I ditched the “me” thing and went a bit bigger, like never-ending bigger. What if S isn’t about finite snapshots but infinite, growing stuff? Imagine this: S is the set of all growing infinities that never end. Like:

• X1 = {1, 2, 3, …}

• X2 = {2, 4, 6, …}

• X3 = {3, 6, 9, …}

Each one is an infinite sequence, and S is all of them, forever expanding. It’s not a static “all things”; it’s a process that keeps chugging along, adding more infinities. But here’s the twist. Not all infinities are built the same. Sure, each Xi is countable, like the natural numbers (call it aleph-zero), but S could be stacking up to something wilder; say, the uncountable infinity of all possible sequences (call it two-to-the-aleph-zero), as Cantor showed and Deutsch riffs on in The Beginning of Infinity. It’s not just more of the same; it’s a hierarchy of bigness, growing beyond what you can list.

Now, does S contain itself? In the classic paradox, that’s the trap. But here, S is one of these growing infinities too since it’s infinite and changing, just like the Xi’s. If S contains itself, it’s fine since it’s just another layer of infinity piling on, maybe jumping to a higher tier of endlessness. If it doesn’t, that’s fine too since S keeps growing, and the question never settles. No contradiction, because it’s never locked in place. It’s like, ‘Sure, I might contain myself, but I’m always shifting and growing, so good luck pinning me down’

Deutsch and Wolfram Lightbulbs

This clicked when I started riffing on David Deutsch and Stephen Wolfram. In The Beginning of Infinity, Deutsch talks about reality as this endless, constructive thing; Cantor vibes, infinite hierarchies. My ( S ) is like that; a Cantor-like environment where infinities stack up forever. It’s not a fixed set; it’s a machine that spits out more and more, no endgame.

Then Wolfram’s A New Kind of Science hit me. He’s all about how we’re computationally bounded, our brains and systems can’t handle the full scope of infinite, irreducible stuff. Russell’s paradox feels real because we try to pin ( S ) down, like, “Answer now!” But if ( S ) is unbounded, always growing, the paradox is just us being too human, too limited. We see a contradiction because we can’t compute the never-ending part.

Back to Quantum

Tying it back to my quantum spark. If ( S ) never collapses, it’s the same deal. Superposition or many-worlds, doesn’t matter. ( S ) can be “in itself” and “not in itself” or just stay blurry. The growing infinity thing is like that; no snapshot, no collapse, no paradox. It’s fluid, infinite, and free.

Final Thoughts

Maybe the paradox is a product of the way we try to think about things from a human, bounded perspective. Humans like to create order in truly chaotic universe to appeal to our fallible senses, perhaps thinking outside the box and embracing chaos is the first step to true discovery. I dunno though, just another human just spewing words cause I can. There might be value there.