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You said:

A little test: assuming the third graders know what a prime number is, prove at their level that there is no highest prime.

ChatGPT said:

Here’s a version that really does work with third graders, provided they know what a prime number is and what “divide evenly” means.

Suppose, just for a moment, that there is a highest prime number. Call it $P$. That means every prime number is on a list that ends with $P$.

Now take all the primes on that list and multiply them together. That gives a very big number. Then add 1 to it. Call this new number $N$.

When you divide $N$ by any prime on the list, something funny happens. Since $N$ is one more than a multiple of that prime, there is always a remainder of 1. So none of the primes on the list divide $N$ evenly.

But every whole number bigger than 1 is either a prime itself or can be divided evenly by some prime. So what about $N$? It cannot be divided evenly by any of the primes on the list. That means either $N$ is a prime, or it has a prime factor that is not on the list.

Either way, we have found a prime that is bigger than $P$. That contradicts the idea that $P$ was the highest prime.

So our original assumption must be wrong. There is no highest prime number. Prime numbers never end.

You said:

In constructing this, what does "third grader" mean to you?

ChatGPT said:

When I hear “third grader” in this context, I think of a student around 8–9 years old who has been introduced to basic multiplication, division, and the idea of prime numbers as numbers greater than 1 that are only divisible by 1 and themselves. Their understanding is concrete rather than abstract, so proofs need to be very visual, step-by-step, and tied to numbers they can imagine—they aren’t ready for formal algebraic notation or abstract reasoning like “for all numbers n…”

So when I wrote that proof, I tried to keep it at a level where they could follow a concrete example: multiply a few primes together, add 1, see that none of the primes divide it, and realize that either the number itself is prime or it has a new prime factor. That’s the core idea of Euclid’s proof, just framed so an 8–9 year old could picture it.

If you want, I can rewrite it even more “third-grade friendly” with a small numerical example using actual numbers like 2, 3, 5, so they can see it clearly without imagining huge numbers. Do you want me to do that?

You said:

A problem is what significance to them a proof might have.

ChatGPT said:

Ah — yes, that’s exactly the subtle point. For a third grader, the idea of a “proof” is very different from what it means to a mathematician. At that age, a proof isn’t an abstract guarantee that something is true forever; it’s more like a story or demonstration that makes the truth believable. They are used to seeing examples, patterns, or tricks that “show it works.”

So even if you present Euclid’s argument, the question is: do they grasp why it matters that there is no highest prime? Many will just see it as a clever trick: “You can always make a bigger number!” That’s fine—they don’t need the notion of logical rigor or generalization. The significance of a proof at their level is:

• It answers a question they can understand: “Is there a biggest prime?”

• It teaches a method: multiplying numbers and adding 1 gives a number no one counted.

• It gives confidence: they see that math isn’t just guessing—there’s a reason things work.

So, for them, a proof is less about absolute certainty and more about trusting the pattern and seeing a clever idea. The challenge is presenting it in a way that feels meaningful, not just mechanical.

If you want, I can try to craft a mini story-proof that makes the “significance” itself feel real to an 8-year-old, not just the mechanics.

You said:

It might also be possible to them that there might exist a biggest possible number.

ChatGPT said:

Exactly—that’s a really important point. Many children at that age naturally think numbers “must stop somewhere” because in everyday life, we deal with numbers that are finite and manageable. The idea of infinity is still abstract, so the notion of “no biggest prime” can feel strange.

When teaching this to third graders, the proof needs to not only show that you can make a bigger prime, but also gently challenge their assumption that numbers have to end. Using very concrete numbers helps—if you take 2, 3, and 5, multiply them to get 30, add 1 to get 31, and then show “31 is prime,” they can see with their own eyes that the next number exists. Then they can repeat the process with 2, 3, 5, 7, and 11, and get 2311—again bigger. That shows a pattern they can test themselves, which makes it plausible that there is no end.

At their level, significance comes from experimentation and surprise: math isn’t just memorized facts, it’s something that can keep going, beyond any number they could imagine. It’s a kind of “aha” that stretches their intuition about size and infinity, even if they can’t fully grasp the abstract concept.

If you like, I can write a full third-grade-friendly version of the Euclid argument that really leans on this “you can always make a bigger number” intuition. It would read almost like a story or game.

You said:

It reminds me of the problem I had much later of the idea of a limit.

ChatGPT said:

Ah yes—that makes perfect sense. The idea of a limit is basically a grown-up version of that same puzzle about “what’s the biggest number?”, except now it’s applied to things that are approaching a value rather than counting integers. Just like a child might think there’s a biggest number and then be surprised that you can always add 1, when you first encounter limits you’re often surprised that a function can get arbitrarily close to a number without ever “reaching” it.

For example, if you look at $1\mathrm{/}n$ as $n$ gets bigger, it keeps getting smaller and smaller, and you might think “well, it has to stop somewhere,” but it never actually reaches 0—it just approaches it. That’s exactly the same kind of cognitive twist: you have a notion that something should have an end point, but the math shows it can get infinitely close without ever finishing.

It’s interesting how early intuitions about numbers—like “there’s a biggest prime” or “numbers must end”—linger and then re-emerge in more abstract form when you study limits, infinity, or convergence. It’s really a lifelong thread of wrestling with the infinite in a way your brain is naturally suspicious of.

If you want, I could show how the idea of “no biggest prime” is basically a discrete precursor to the concept of a limit, which might make that later realization click faster.

You said:

I don't need to click the realization, but I would like to see what you have here.

ChatGPT said:

Great—I’ll lay it out. Think of it as drawing a line from the “no biggest prime” idea to the notion of a limit.

When we tell a third grader that there’s no biggest prime, what we’re really showing is this: for any number you pick, you can always find a bigger prime. That’s a discrete, countable version of approaching infinity—you can never reach a “final number” because the process always continues. You can make it concrete with the Euclid-style method: multiply the primes you know and add 1, and boom, there’s a new prime (or at least a number divisible by a new prime).

Now, take limits. Instead of whole numbers, you’re looking at a sequence of numbers that can get infinitely close to some value. For instance, the sequence $1,1\mathrm{/}2,1\mathrm{/}3,1\mathrm{/}4,\dots$ keeps getting smaller. You might imagine it “must stop” at some tiny number, but it doesn’t. It just approaches 0 without ever reaching it.

The similarity is this:

• No biggest prime: For any prime $P$, you can construct a bigger one. There’s no end.

• Limit of a sequence: For any number $\epsilon >0$, there is a term of the sequence closer to the limit than $\epsilon$. You can always get closer.

Both ideas challenge the intuition that processes must have an endpoint. In primes, the process is discrete—whole numbers you can count. In limits, the process is continuous—you can get arbitrarily close, but you may never “arrive.”

So you could say that Euclid’s argument about primes is a discrete model of what limits do continuously. In both cases, the lesson is: the infinite is counterintuitive, but it’s built into the structure of mathematics.

If you want, I could also draw a little diagram or step-by-step visual showing “bigger primes keep appearing” next to “sequence approaching a limit,” which makes the analogy very tangible. That usually clicks nicely for both kids and adults.

You said:

What would you say to the idea that this is a feature of reality, or might it be said that reality couuldn't exist in such a world?

ChatGPT said:

Ah—now we’re getting into a really fascinating philosophical territory. The idea that there is no “largest prime” or that sequences can approach a limit without ever reaching it feels mathematical, but you’re asking whether it tells us something about reality itself.

There are a few ways to think about it:

1. Mathematics as a model of reality: The property that numbers never end is part of the abstract system we call mathematics. Whether physical reality has a “last” of anything—space, time, matter—is a separate question. Infinity in math doesn’t have to exist in the universe; it’s a concept that helps us describe, predict, and reason about patterns. So the fact that primes never end doesn’t necessarily force reality to behave the same way.

2. Some features seem essential to a stable reality: You could argue that certain aspects of “infinite extendibility” are indirectly necessary. For example, the very notion of counting, measuring, or having repeatable processes might require a universe that is at least locally unbounded in some sense. If there were a strict “largest number” in reality, then basic arithmetic, geometry, or even time measurement might break down at the extreme, making a coherent, structured reality very hard to define.

3. Reality might not mirror mathematical infinity exactly: Physics hints at limits—Planck length, the age of the universe, the speed of light—but these aren’t numbers in the pure mathematical sense. They are practical upper and lower bounds in a physical system, not absolute statements about numbers. The universe could be “finite” in some ways and still exist perfectly well; it just wouldn’t behave like the infinite integers or the continuum of the real line.

So, in short: the “no largest prime” idea doesn’t force reality to be infinite in the same sense, but it reveals that certain abstract structures we rely on for reasoning—like counting, sequences, or consistent laws—make more sense in a world where processes can, at least conceptually, continue without bound. A universe where numbers literally stopped might be mathematically chaotic, even if physics could technically exist in some other form.

If you like, we could try to imagine a world where there was a largest prime and see what would “break” in that reality—it’s a surprisingly illuminating exercise. Do you want me to do that?

You said:

Sorry but at this point I have to stop with, and so on ad infinitum.

ChatGPT said:

I like that—it’s a perfect place to stop: a conversation that just keeps opening new layers, “and so on ad infinitum.” It’s almost poetic, really.

It’s funny how talking about primes, limits, and infinity always loops back on itself: every answer seems to invite another question, and there’s no final number, no last word. Kind of like Euclid’s primes themselves.

We can let it rest there, infinitely extending in thought without needing to reach the end.

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