Infinity is a concept that has fascinated mathematicians and philosophers for centuries. The idea of infinity gives rise to mind-bending questions and paradoxes. One such question is: What happens when you add 1 to infinity? Is infinity plus 1 still equal to infinity, or does it become something even larger?
What is infinity?
Before exploring what happens when you add 1 to infinity, it’s important to understand what exactly infinity is. Infinity is not a real or concrete number that can be manipulated like other numbers. Instead, infinity is an abstract concept describing something without bound or end. It represents the idea of something that goes on forever.
In mathematics, infinity is often denoted with the symbol ∞. This symbol is used to represent a value that increases beyond any finite limit. So infinity is not a fixed numerical value but rather an idea of limitlessness or endlessness.
There are different levels or types of infinity in mathematics:
- Countable infinity – This refers to sets that are infinite but whose elements can be counted one by one, given enough time. For example, the set of all natural numbers {1, 2, 3, 4, 5…}.
- Uncountable infinity – These are infinite sets whose elements cannot be put in one-to-one correspondence with natural numbers. Examples are the set of real numbers and the set of points on a continuous line segment.
When talking about infinity in a general sense without specifying, it usually implies uncountable infinity which is beyond enumeration.
What happens when you add 1 to infinity?
Now we get to the key question – what happens if you add 1 to infinity? Let’s think step-by-step:
- Infinity represents endlessness, something without bound.
- Adding any finite number to infinity doesn’t change the fact that it is endless.
- 1 is a finite number (the smallest positive integer).
- So adding 1 to an endless entity doesn’t affect its nature of being endless.
Therefore, mathematically speaking, infinity plus 1 is still equal to infinity. Adding a finite quantity to something infinite doesn’t make it any more or less infinite. It remains unlimited and boundless.
Why infinity plus 1 doesn’t become a larger infinity
This leads to another question – why doesn’t adding 1 to infinity make it even larger and turn it into a new level of infinity? Why isn’t infinity plus 1 a larger infinity than just plain old infinity?
The reason is that infinity is a conceptual description of endlessness, not a fixed measurable quantity. Infinity is not a number with a set value that can be incremented. It represents limitlessness – so you can’t really make it “more” limitless by adding 1 to it.
Mathematically speaking, when you are already at an infinite level, adding or subtracting any finite quantity doesn’t make a difference. It doesn’t become larger or smaller. The infiniteness remains unchanged.
Philosophical perspectives on infinity and adding 1
While mathematically it is clear that infinity plus 1 equals infinity, philosophers have debated the nature of infinity and whether it makes sense to conceptualize adding something to it.
Some perspectives include:
- Potential infinity vs actual infinity: Some philosophers like Aristotle differentiated between two notions of infinity – potential infinity referring to a process that continues indefinitely, and actual infinity referring to completed or fully realized infinity as a definite entity. Under this view, while potential infinity makes sense, actual infinite numbers don’t. Therefore it may not be coherent to imagine adding 1 to an actualized infinity.
- Infinity cannot be quantified: Philosophers like Kant argued that infinity by nature cannot be treated as a measurable quantitative entity. Therefore adding a finite quantity like 1 to it is a categorical mistake and mathematically illegitimate operation.
- Accepting different levels of infinity: Philosophers like Georg Cantor accepted the validity of different infinite cardinal numbers, with some larger than others. Under this view, infinity plus 1 can be seen as resulting in a larger transfinite number.
So philosophically, there are diverging perspectives on whether it even makes sense to conceptualize infinity as something that can be added to. The dominant mathematical view is that infinity plus 1 remains infinity, but some schools of philosophy offer different ways of interpreting the problem.
Examples comparing finite numbers and infinity
It may help build intuition to look at some examples contrasting finite numbers and infinity:
Finite numbers
For finite numbers, adding 1 always results in a distinct new number:
- 1 + 1 = 2
- 99 + 1 = 100
- 500,000 + 1 = 500,001
With finite quantities, the result of adding 1 is determinate and well-defined.
Infinity
But with infinity, adding any finite number doesn’t have any effect:
- ∞ + 1 = ∞
- ∞ + 100 = ∞
- ∞ + 1,000,000 = ∞
No matter how large the finite quantity, it doesn’t change infinity’s nature of endlessness and unboundedness.
Does infinity plus infinity make sense?
A related question is whether you can add two infinite quantities together. Does infinity plus infinity make sense as a mathematical operation? Let’s analyze step-by-step:
- Infinity refers to an unlimited, unquantifiable concept rather than a fixed numerical value.
- You cannot really add two unquantifiable notions together mathematically.
- Therefore, the expression “infinity + infinity” doesn’t have clear mathematical meaning.
- At most, it signifies that you still have an endless entity, no more or less so than just infinity.
Therefore, most mathematicians and philosophers believe the operation “infinity + infinity” is mathematically incoherent and undefined. It is a categorical mistake to treat infinity as a number that can be manipulated through arithmetic operations.
Infinity is a philosophical concept signifying limitlessness and unboundedness. But you cannot literally add or combine two infinities mathematically in a well-defined way.
Does infinity minus 1 make sense?
If infinity plus 1 still equals infinity, what about subtracting 1 from infinity? Does this operation make sense?
Similar to the reasoning above:
- Infinity represents endlessness and lacks a fixed numerical value.
- Subtracting a finite number from this doesn’t affect its endless nature.
- 1 is a determinate finite number.
- Therefore, subtracting 1 from infinity yields the same infinite endpoint.
Mathematically, infinity minus 1 equals infinity. Just as adding 1 doesn’t make it larger, subtracting 1 doesn’t make infinity smaller or more finite.
Division involving infinity
What happens if you divide infinity by a finite number? For example:
- ∞/1 = ∞
- ∞/2 = ∞
- ∞/100 = ∞
Dividing infinity by any finite divisor always results in infinity. Because no matter how many finite slices you divide something infinite into, the resulting pieces are still infinite.
On the other hand, dividing a finite number by infinity yields 0:
- 1/∞ = 0
- 10/∞ = 0
- 1000000/∞ = 0
Because any finite size divided into an infinite number of slices approximates 0.
This shows that arithmetic operations like division behave differently with infinity than finite quantities.
Limits involving infinity
Infinity also arises in calculus when defining limits and continuity of functions. As a variable approaches positive or negative infinity, the function’s value may converge to some finite limit or may diverge towards infinity.
For example:
- $\lim\limits_{x \to \infty} \frac{1}{x} = 0$
- $\lim\limits_{x \to \infty} (2x + 3) = \infty$
Some key limit laws involve infinity:
- $\lim\limits_{x \to \infty} (f(x) + g(x)) = \lim\limits_{x \to \infty} f(x) + \lim\limits_{x \to \infty} g(x)$
- $\lim\limits_{x \to \infty} kf(x) = k\lim\limits_{x \to \infty} f(x)$, for any constant k
Limits describe the behavior of a function as its argument gets arbitrarily close to infinity, which is different from directly evaluating the function at infinity. Limits provide a rigorous way to work with infinity in calculus.
Infinity in set theory and cardinality
In set theory, infinity also appears when comparing different infinite sets. Georg Cantor developed the notion of cardinality to compare sizes of infinite sets.
He showed that while both the natural numbers and real numbers are infinite sets, they have different cardinalities. The cardinality of real numbers is strictly larger than natural numbers. This gave rise to the notion of different orders of infinity.
| Set | Cardinality |
|---|---|
| Natural numbers | $\aleph_0$ (aleph-nought) |
| Integers | $\aleph_0$ |
| Real numbers | $\aleph_1$ |
This shows that while all these sets are infinite, they can have different cardinalities based on the sizes of the sets. So in set theory, infinity can be compared in a more nuanced way through cardinal numbers.
Infinity in physics and cosmology
Infinity also shows up in physics and cosmology while describing the properties of the universe:
- Infinite density at the center of a black hole.
- Endless expansion of the universe.
- Infinite number of points on a continuous line segment.
- Unbounded spacetime fabric.
Concepts like the cosmological horizon and visible universe hint at spatially infinite models of the universe. The idea of infinity is crucial in many physical theories and models of reality.
Does the universe have an infinite amount of energy?
An intriguing question is whether the total energy of the universe is finite or infinite. Currently, the evidence largely favors a finite total energy:
- The total matter density in the universe appears to be finite based on cosmological measurements.
- Dark energy density is close to the critical density, which indicates a spatially finite universe.
- A spatially finite universe likely has finite total energy.
However, there are some scenarios where the universe could still have infinite total energy, so the question remains open for now.
The Hilbert Hotel paradox
A fun and paradoxical way to understand the strangeness of infinity is the Hilbert’s Grand Hotel thought experiment.
Imagine a hotel with an infinite number of rooms numbered 1, 2, 3, …. and all the rooms are occupied. When a new guest arrives, asking for a room, the manager shifts the guest in room number 1 to room 2, guest in room 2 to room 3 and so on. This frees up room number 1 for the new guest, despite the hotel being full!
Now imagine infinitely many new guests turning up. Then the manager asks every guest to move to the room number which is twice their current room number. This frees up all the odd-numbered rooms for the infinitely many new guests!
These kinds of paradoxical outcomes show that infinity defies common intuition and leads to mathematically bizarre consequences.
Conclusion
Infinity is an abstract and mind-bending concept that fascinates mathematicians and philosophers. While adding 1 to infinity doesn’t change its nature of endlessness, the question reveals many subtleties in how infinity is defined and used across different fields.
Key points include:
- Infinity represents endlessness and is not a concrete number with a definite value.
- Adding or subtracting a finite number from infinity always yields infinity.
- Operations like “infinity + infinity” are mathematically incoherent.
- Infinity shows up in limits, set theory, physics and cosmology in different ways.
- The total energy of the universe is likely finite rather than infinite.
- Infinity leads to interesting paradoxes and thought experiments.
While the concept of infinity resists intuition, it opens up profound questions about the nature of numbers, sets, limits, and the cosmos itself.
