Division by zero is not allowed because it yields something undefined. It’s not a number. It’s not precise. It was excluded, and math became cleaner.
Now here’s the speculation: maybe infinity is the same kind of mistake. A non-number that gets treated like a number, smuggled in under symbols like “…” or phrases like “for all.”
What if most mathematical paradoxes come not from infinity, but from this style of reasoning? The part where we go from:
“for any natural number k” to “for all natural numbers”
without acknowledging that something changed.
The idea
Let’s add precision.
We don’t reject any number. Any size. Any density. But we define the data more precisely before we start reasoning. Once you choose a bound k, you can’t introduce anything beyond it inside the same proof. That’s it.
Maybe call this the Precision Discipline.
It’s not minimalism. It’s not rejection. It’s just: pick your set of assumptions and stick to them without using undefined non numbers which later may be reversed to something else.
Where this shows up: Cantor’s diagonal
The classic diagonal argument says: list all binary sequences. Flip the nth bit in the nth row. You get a new sequence not in the list. Infinite sets are uncountable. Done.
But here’s the problem:
You have a list of sequences of length k.
Your new diagonal sequence tries to reach digit k+1.
With precision discipline you can’t do that. You’re only allowed to talk about sequences up to length k — not bring in data that wasn’t part of the assumptions.
So the paradox doesn’t happen. You never construct the anti-diagonal sequence. The argument halts, not because math broke, but because it respected the bound.
Russell’s paradox
It disappears if we apply precisions discipline. When you work with non-numbers (like infinity), contradiction naturally arises. We can’t define “all” with precision. What does “all” even mean? It’s infinite — it’s not a number, it’s not precise, it’s vague.
Maybe such paradoxes could be avoided — and everything else kept — if we tried to be more precise and avoided using non-numbers.
A tiny example: escape the paradox, keep the result
Suppose we want to prove: “There is a prime number greater than any given number k.”
Normally, we say: “for all k, there exists a prime p > k.” And we might feel the need to appeal to some set of all primes.
But under precision discipline, we rephrase:
For any given k, construct p = product of all primes up to k, plus 1.
Then p is either prime, or divisible by a prime not in the list. Either way, you get a new prime > k.
No infinity. No paradox. Just for any. Not for all — because all is infinite, it’s not a number.
And we can still use one rule, one proof that works for any — just don’t mix in proof data from one assumption with data from another.
What about geometry?
What’s a line?
Is it a “set of all points”? “All” is not a number unless we have resolution, density—that’s where by skipping this information we introduce infinity. How can adding any numbers of objects with 0 dimension create 1 dimension. Infinity introduces dimension and postulates that infinity times zero has value.
Line is a concept not a set of points, a category. We can generate set of points from a line, we can define a line using set of points, but not the other way around. Just like we can sample a function at some inputs, but that doesn’t define the function. Different rules may generate the same samples, but samples can’t define what rule was used.
Line isn’t a set of points. It’s something different. It may be redefined as a set of smaller lines with some precision — it can be defined using set of points, but it’s not, it’s something more. It’s a function, a category of numbers. Is function a set of points? No, it’s a rule, something distinct from its output.
Missing precision is: a function of a line may produce output, a set of points, but only given specific density. In other cases we introduce unknown—not quantifiable infinite precision.
We can create multiple lines that looks the same using different rules, different precisions. A whole category of points is treated as a single object. Rules that create the same output are freely juggled and we have whole category of Banach-Tarski paradoxes.
Final thought
Maybe mathematics is just a little bit imprecise — and through that, undisciplined.
Shouldn’t we treat infinity the same way we treat division by zero?
It’s not a number. Don’t use it unless you mean something finite, declared, precise.
You still get geometry. You still get limits. You still get primes. You just stop falling into paradoxes built from undefined assumptions.
“For any” is different than “for all” — isn’t that true?
Infinity Is Not a Number
Division by zero is not allowed because it yields something undefined. It’s not a number. It’s not precise. It was excluded, and math became cleaner.
Now here’s the speculation: maybe infinity is the same kind of mistake. A non-number that gets treated like a number, smuggled in under symbols like “…” or phrases like “for all.”
What if most mathematical paradoxes come not from infinity, but from this style of reasoning? The part where we go from:
without acknowledging that something changed.
The idea
Let’s add precision.
We don’t reject any number. Any size. Any density. But we define the data more precisely before we start reasoning. Once you choose a bound k, you can’t introduce anything beyond it inside the same proof. That’s it.
Maybe call this the Precision Discipline.
It’s not minimalism. It’s not rejection. It’s just: pick your set of assumptions and stick to them without using undefined non numbers which later may be reversed to something else.
Where this shows up: Cantor’s diagonal
The classic diagonal argument says: list all binary sequences. Flip the nth bit in the nth row. You get a new sequence not in the list. Infinite sets are uncountable. Done.
But here’s the problem:
You have a list of sequences of length k.
Your new diagonal sequence tries to reach digit k+1.
With precision discipline you can’t do that. You’re only allowed to talk about sequences up to length k — not bring in data that wasn’t part of the assumptions.
So the paradox doesn’t happen. You never construct the anti-diagonal sequence. The argument halts, not because math broke, but because it respected the bound.
Russell’s paradox
It disappears if we apply precisions discipline. When you work with non-numbers (like infinity), contradiction naturally arises. We can’t define “all” with precision. What does “all” even mean? It’s infinite — it’s not a number, it’s not precise, it’s vague.
Maybe such paradoxes could be avoided — and everything else kept — if we tried to be more precise and avoided using non-numbers.
A tiny example: escape the paradox, keep the result
Suppose we want to prove: “There is a prime number greater than any given number k.”
Normally, we say: “for all k, there exists a prime p > k.” And we might feel the need to appeal to some set of all primes.
But under precision discipline, we rephrase:
Then p is either prime, or divisible by a prime not in the list. Either way, you get a new prime > k.
No infinity. No paradox. Just for any. Not for all — because all is infinite, it’s not a number.
And we can still use one rule, one proof that works for any — just don’t mix in proof data from one assumption with data from another.
What about geometry?
What’s a line?
Is it a “set of all points”? “All” is not a number unless we have resolution, density—that’s where by skipping this information we introduce infinity. How can adding any numbers of objects with 0 dimension create 1 dimension. Infinity introduces dimension and postulates that infinity times zero has value.
Line is a concept not a set of points, a category. We can generate set of points from a line, we can define a line using set of points, but not the other way around. Just like we can sample a function at some inputs, but that doesn’t define the function. Different rules may generate the same samples, but samples can’t define what rule was used.
Line isn’t a set of points. It’s something different. It may be redefined as a set of smaller lines with some precision — it can be defined using set of points, but it’s not, it’s something more. It’s a function, a category of numbers. Is function a set of points? No, it’s a rule, something distinct from its output.
Missing precision is: a function of a line may produce output, a set of points, but only given specific density. In other cases we introduce unknown—not quantifiable infinite precision.
We can create multiple lines that looks the same using different rules, different precisions. A whole category of points is treated as a single object. Rules that create the same output are freely juggled and we have whole category of Banach-Tarski paradoxes.
Final thought
Maybe mathematics is just a little bit imprecise — and through that, undisciplined.
Shouldn’t we treat infinity the same way we treat division by zero?
You still get geometry. You still get limits. You still get primes. You just stop falling into paradoxes built from undefined assumptions.
“For any” is different than “for all” — isn’t that true?