Hey everyone, ​I want to discuss the philosophical motivations behind each of the ZFC axioms. ​Axioms are mathematically true by definition, but what philosophical worldview actually justifies them? For example, does the Axiom of Infinity require strict Platonism, or is it just about our cognitive ability to imagine such concepts? What about the philosophical reasoning behind the Axiom of Choice or Regularity? ​I'd love to hear your thoughts on the reasoning that grounds these axioms, or get recommendations for philosophers who have deeply explored the "why" behind ZFC.

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

A Candidate Boundary-Recursive Interpretation of Cantor's Theorem

I've been exploring an alternative interpretation of Cantor's theorem that keeps the diagonal proof intact but proposes a different interpretation of what it demonstrates. I'd appreciate feedback on where this framework succeeds, where it fails, and whether anything similar already exists in the literature.

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Step 1 — Cantor's Definition of Size

Cantor defines two sets to have the same size if there exists a bijection between them.

For finite sets this agrees with counting.

For infinite sets it replaces counting entirely.

For example,

ℕ ↔ Even Numbers

via

f(n)=2n

shows that the natural numbers and the even numbers have the same cardinality.

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Step 2 — Cantor's Theorem

Cantor then proves there is no bijection

A ↔ ℘(A)

using diagonalization.

The standard conclusion is

|℘(A)| > |A|

which produces the hierarchy

ℵ₀ → 𝔠 → 2^𝔠 → …

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Sigma Observation

The diagonal proof unquestionably constructs an object outside every proposed complete correspondence.

My question is whether the proof necessarily establishes larger infinities, or whether it establishes something weaker and more general:

«Every completed representation of an unbounded generative system admits another valid representational transform.»

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Sigma Boundary Theory

Suppose mathematics is studying an unbounded generative system.

The recursive process becomes

Reachable System → Draw Boundary → Treat Boundary as Object → Apply Valid Transform → New Boundary → Repeat

The recursion occurs in the representations—not necessarily in infinity itself.

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Boundary Interpretation

Under this interpretation:

• A power set is not viewed primarily as a "larger infinity."
• It is viewed as a boundary-lifting transform.
• Diagonalization demonstrates that no completed representation is terminal.

Instead of reading Cantor's theorem as

«"There exists a larger infinity,"»

the same proof may be read as

«"Every completed representation of an unbounded generative system admits another representational closure."»

The mathematics of diagonalization is unchanged.

Only the interpretation changes.

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Candidate Replacement Primitive

Rather than ordering mathematical objects by cardinality,

|A| < |B|

Sigma proposes ordering representations by recursive closure:

Closure₀ → Closure₁ → Closure₂ → …

The hierarchy becomes a hierarchy of boundary closures rather than a hierarchy of infinities.

Infinity itself is treated as a single unbounded phenomenon.

What grows is the sequence of completed representations constructed around it.

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Candidate Boundary Escape Theorem

Every reflective completed representation of an unbounded generative system admits another valid representational transform.

Equivalently,

Reachable System → Draw Boundary → Treat Boundary as Object → Apply Valid Transform → New Boundary → Repeat

No completed representation is terminal.

Two systems are Sigma-equivalent if

1. They generate the same reachable universe.
2. Every valid transform of one corresponds to a valid transform of the other.
3. Neither admits a boundary escape that the other does not.

---

The Question

I'm not claiming this disproves Cantor's theorem.

I'm asking whether this provides a viable alternative interpretation of the theorem.

Specifically:

• Does diagonalization require the ontology of multiple infinities?
• Or is it sufficient to interpret it as demonstrating the nonexistence of a terminal representation of an unbounded generative system?

I'd appreciate rigorous criticism. If this framework fails, I'd like to know exactly where. If it resembles existing work in category theory, type theory, domain theory, or another area, I'd also appreciate references.

Hey everyone! I'm u/Arlo_Tinkerman, the founding moderator of r/InfinityMath.

This is a place to explore infinity from every mathematical perspective.

Cantorian, constructivist, finitist, ultrafinitist, undecided, or developing something new, all are welcome here. The point is not to gather only people who already agree. The point is to put different ideas into contact with each other and see which arguments hold up.

Infinity is one of those subjects that can get people surprisingly worked up. It sits underneath set theory, number systems, limits, geometry, calculus, logic, and the foundations of mathematics. It also raises basic questions that are easy to state but difficult to settle.

• What does it mean for something to be infinite?
• Can an infinite collection be complete?
• Is infinity actual, potential, or something else?
• What does it mean to account for all members of an unending sequence?
• How should whole numbers, fractions, decimals, limits, and sets be understood in relation to infinity?

Those questions are open for discussion here.

What to Post

You are welcome to post:

• Proofs and proposed proofs
• Objections to established or alternative views
• Questions about infinity
• Arguments involving actual or potential infinity
• Discussions of Cantorian set theory
• Constructivist, finitist, or ultrafinitist approaches
• New mathematical frameworks or interpretations
• Explanations of difficult ideas
• Historical material about infinity
• Diagrams, models, or thought experiments
• Requests for criticism or review

I will also be sharing some of my own work here. My current project examines infinity through generative systems, whole-number counting, fractional structure, positional notation, and what I call the prime extent. I do not expect anyone to accept those ideas merely because I post them. I want people to inspect the definitions, premises, operations, and inferential steps.

The same standard should apply to every position, including established ones.

Community Vibe

Disagreement is expected. Hostility is not required.

Criticize the argument rather than the person making it. Try to explain where an argument fails instead of merely labeling it wrong. When possible, identify the specific premise, definition, equation, or inferential step you reject.

Likewise, anyone presenting a new idea should expect serious objections. A strong challenge is useful when it exposes an error, forces a definition to become clearer, or reveals that two people are using the same word differently.

The goal is not enforced agreement. The goal is stronger ideas through serious discussion.

How to Get Started

1. Introduce yourself in the comments and mention what interests you about infinity.
2. Share a question, proof, objection, diagram, or argument.
3. Invite someone who would contribute thoughtfully, including someone who may disagree with you.
4. Let me know if you are interested in helping moderate as the community grows.

Thanks for joining at the beginning. I hope this becomes a place where people can question familiar assumptions, defend established mathematics, propose alternatives, and sharpen one another's thinking.

A Proposal for a Context-Specific Interpretation of "Infinity Divided by Infinity"

Author: Ritesh Wankhede

Abstract

This paper proposes a specific way to interpret the expression "infinity divided by infinity." It does not claim standard mathematics is wrong. It presents one interpretation, valid only under clearly stated conditions.

The core idea: when infinitely many objects are matched to infinitely many recipients so that every recipient gets exactly one object and every object goes to exactly one recipient, the distribution can be understood as one object per recipient — within that specific matching.

1. The Question

If infinitely many objects are distributed among infinitely many recipients, with each recipient getting exactly one object and no object left over, what does that distribution represent?

2. What "Infinite" Means Here

For this proposal, infinite means a collection that never ends — like the counting numbers 1, 2, 3, 4, 5, ... — with no final number.

3. The Setup

Assume:

There are infinitely many people.

There are infinitely many chocolates.

Every person receives exactly one chocolate.

Every chocolate goes to exactly one person.

4. The Example

Person 1 gets Chocolate 1.

Person 2 gets Chocolate 2.

Person 3 gets Chocolate 3.

This continues forever: Person n always gets Chocolate n.

No chocolate is shared between two people. No person goes without.

5. The Reasoning

With finite numbers, if 100 chocolates are split evenly among 100 people, each person gets exactly one. This proposal simply extends that same idea to the infinite case above: since every person is matched to exactly one chocolate, the natural reading of the distribution is one chocolate per person.

6. The Proposed Interpretation

This does not claim "infinity divided by infinity always equals one" as a general rule.

It claims something narrower: when infinite objects are matched to infinite recipients so that each side is used exactly once, that specific matching can be read as one object per recipient.

7. Where This Does NOT Apply

This idea is limited to the exact matching described above. It is not claimed to apply to:

Calculus or limits involving infinity

Every possible way of pairing two infinite collections

Every mathematical system

Other ways of matching the same two infinite groups might not give this same "one-to-one" feel — that's a real limitation, and I don't yet have a way to resolve it. It's part of what I want feedback on.

8. Open Questions

Is there a rigorous way to express this idea?

Does the interpretation change if the matching rule between people and chocolates changes?

Can this idea be pushed further without leading to contradictions?

9. Conclusion

When infinitely many objects are matched one-to-one with infinitely many recipients, each recipient receives exactly one object. Within this specific matching, the distribution corresponds to one object per recipient. This is offered as a starting idea for discussion, not a finished theorem.

Author's Note

I'm a Class 12 student from Mumbai with a strong interest in mathematics. I know this connects to ideas already studied by mathematicians, and I'd like to learn where my reasoning holds up and where it needs refining.

"Is 0.999.. with a infinite number of 9's equal to 1" has sparked a number of online debates. I believe the crux of the debate is the question is not worded precisely enough. There are different notions of infinite which lead to different answers. Adding to the confusion, "0.999..." is math notation with a specific meaning, but some people aren't thinking of that when they use it in a sentence.

Below are my two proofs for "Does 0.999 (repeating) equal one?" using less ambiguous notions of infinity. The answer is that it does and it doesn't but you need to be specific about what you mean in a way that we aren't used to.

"0. followed by a specific but infinitely large integer number (H) of 9's"**

I will denote this quantity as 0.{H 9's}. This is LESS than 1.*

Proof:

1. Assume two numbers** are equal if and only if their difference is zero
2. Therefore 0.{H 9's} is 1 if and only if abs(1 - 0.{H 9's}) = 0
3. Consider a specific but infinitely large integer** value of H
4. There exists H+1
5. And abs(1 - 0.{H 9's}) > abs(1 - 0.{H+1 9's})
6. This implies abs(1 - 0.{H 9's}) > 0
7. (1.) and (6.) imply 0.{H 9's} is not 1

Unsatisfyingly, this doesn't prove things like "The concept of H is valid", or "H < a different notion of infinity", "H+1 > H", or "H+1 exists". If you want to read up on this, look up hyperreal numbers, hyperintegers, and nonstandard analysis.

"0. followed by a 9 for every standard natural number"

I will denote this quantity as 0.999... . This is EQUAL to one. I think this is what most people think of when they hear "0.999 repeating infinitely".*

Proof:

1. Assume two numbers** are equal if and only if their difference is zero
2. Therefore 0.999... is 1 if and only if abs(1 - 0.999...) = 0
3. Assume that abs(1 - 0.999...) is a positive number**
4. Consider a specific nonzero positive number** H
5. There exists a number of nines in 0.999... such that abs(1 - 0.999...) < H
6. (1.) and (5.) imply abs(1 - 0.999...) is not H
7. (4.) and (6.) imply abs(1 - 0.999...) is not a nonzero positive number**
8. The only positive number that is not a nonzero positive number** is zero
9. This implies abs(1 - 0.999...) is zero
10. (1.) and (9.) imply 0.999... = 1

Unsatisfyingly, this doesn't prove "0.999... always has enough nines to make abs(1 - 0.999...) < H". If you want to read up on this, look up hyperreal numbers, hyperintegers, and nonstandard analysis.

...Its been a long time since I wrote a proof ...Im sure some of my wording isn't great ...I wanted to make this semi-readable for "common folk"...

*There is a better way to write 0.{H 9's} and 0.999... but I can't write it here because it requires LaTeX formatting.

**"number" should be replaced with "hyperreal number" and "integer" should be replaced with hyperinteger" but I wanted to make the proof readable to people who don't know what those are.

Although I know the science that the universe is infinite if it's flat or saddle shaped and may not be if it's closed curved, I can't understand infinity when I consider a simple thought experiment. I would really appreciate help from anybody who can align my brain with reality.

In my thought experiment, before the Big Bang, everything is just a singularity sitting in one point. Ignoring heat, lack of visibility, etc I can imagine the universe exploding and after 1 second it is some size larger than the singularity.. for the sake of argument let's say it has expanded 100 miles in each direction..

So now we have a sphere with a radius of 100 miles.

Obviously the earth doesn't exist yet, but let's assume that the cosmic ingredients for our solar system are halfway between the center starting point and the hundred miles to the edge of the sphere that we're inside of. Let's let's put a point on the inside edge of the sphere, the edge of the universe and nothing, and call it x.

X is currently 50 miles away in one direction and the center of the universe is 50 mi in the opposite direction.

Time goes on, the bubble keeps expanding and at some point our solar system has formed and we're here and we can look out.

We may not be able to see x because it could have moved away from us faster than the speed of light.

However from my previous thinking, it seems logical that there is a distance to x, which in this case might be half of the radius of the universe at this point in time.

Or maybe the expansion is faster farther away at point x so maybe the scale is wrong.. maybe the stuff earth was made from is closer to the center than it is to x at this point.

But if we consider the uneven rates, we should still be able to say x is this distance and the center of the universe is y distance.

Of course I'm not considering any other variables like uneven density or anything else.

I also know of the analogy that we are on the skin of a balloon that is being blown up, but once again the distance of that skin from the originating point seems like it would also give us a distance.

There's another analogy that seems appropriate.. one of the ways that we can prove tectonic movement is to measure the increase of distance of Europe from America every year... Measure how far they have moved away from the Mid-Atlantic fault and do the math to understand how long ago the two separated from the falt.

So why can't we run the math backwards just like we did with plate tectonics to figure out how far everything has moved?

I know I'm missing something big, what is it?

In mathematics, infinity is not treated as a vague or mystical idea—it is a fully structured concept.

We don’t just say “infinity exists” as a single thing. Mathematics actually refines it into different forms. In set theory, for example, there are different sizes of infinity. The set of natural numbers is infinite, but the set of real numbers is a strictly larger type of infinity. This leads to a hierarchy of infinities, where some infinities are provably bigger than others.

So in math, infinity is not a failure or an edge case—it is something that can be rigorously defined, compared, and explored inside a consistent logical system.

Physics, however, seems to treat infinity very differently.

When infinities appear in physical theories, they are almost never accepted as “real answers.” Instead, they are usually treated as warning signs that something has gone wrong in the model.

For example:

• In general relativity, black hole singularities produce infinite density.
• In quantum field theory, certain calculations initially produce divergent (infinite) results.
• In many cases, physicists introduce techniques like renormalization to remove or reframe these infinities so that meaningful finite predictions emerge.

So in physics, infinity is often not the endpoint of understanding—it is usually a signal that the theory is being pushed beyond its valid range.

This creates a really interesting contrast.

In mathematics:

• Infinity is structured.
• Infinity is classified.
• There are multiple infinities with precise relationships between them.

In physics:

• Infinity is usually problematic.
• It often indicates a breakdown of the theory.
• It is something to be removed, regulated, or replaced.

Yet despite this difference, mathematics and physics are otherwise extremely tightly connected. In almost every other domain—calculus, differential equations, geometry, linear algebra, symmetry theory—mathematics provides the language of physics, and physics often guides new mathematical development.

Which makes infinity stand out as one of the rare points where the connection feels less direct or at least less “clean.”

This leads to a deeper question:

Is this because mathematics is dealing with a purely abstract concept of infinity that is not meant to correspond directly to physical reality?

Or is it because the infinities we see in physics are not “real infinities,” but instead signs that our current theories are incomplete, and that a deeper framework might eventually remove them entirely?

Or, more fundamentally, are mathematical infinity and physical infinity actually different kinds of concepts that only share the same name, but not the same meaning?

Because in a strange way, it almost feels like:

• Mathematics is comfortable exploring infinity as a fully defined structure.
• Physics is only comfortable with infinity when it can be eliminated or interpreted away.

So why is infinity one of the very few concepts where mathematics and physics seem to diverge so sharply in interpretation, despite their otherwise extremely deep and successful relationship?