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The short answer is:

We don't know. The question "Is there an infinity between Aleph-0 and Aleph-1?" is mathematically undecidable from the standard axioms (ZFC) of set theory.

To understand why, let's break it down.

1. What Do Aleph-0 and Aleph-1 Mean?

Aleph-0 (ℵ₀): This is the smallest infinity. It represents the cardinality (size) of the set of all natural numbers {1, 2, 3, ...}. Any set that can be put into a one-to-one correspondence with the natural numbers is called "countable" and has size ℵ₀.
Aleph-1 (ℵ₁): This is defined as the next smallest cardinal number after Aleph-0. It is the cardinality of the set of all countable ordinal numbers.

The crucial link is to the real numbers. The cardinality of the real numbers (the "continuum") is denoted by c or 2^(ℵ₀).

So, the real question becomes: Is 2^(ℵ₀), the size of the continuum, equal to ℵ₁?

2. The Continuum Hypothesis (CH)

This exact question is the subject of the Continuum Hypothesis (CH), proposed by Georg Cantor in 1878.

The Continuum Hypothesis states: There is no set whose cardinality is strictly between that of the integers (ℵ₀) and the real numbers (2^(ℵ₀)). In other words, 2^(ℵ₀) = ℵ₁.

If CH is true, then the answer to your question is NO, there is no infinity between Aleph-0 and Aleph-1.

If CH is false, then the answer is YES, there is at least one infinity between them, meaning 2^(ℵ₀) is larger than ℵ₁.

3. The Groundbreaking Discovery: Independence

The situation was resolved in a shocking way in the 20th century:

Kurt Gödel (1940): Showed that the Continuum Hypothesis cannot be disproven from the standard ZFC axioms of set theory. That is, if ZFC is consistent, then ZFC + CH is also consistent. CH is not false.
Paul Cohen (1963): Showed that the Continuum Hypothesis cannot be proven from the ZFC axioms. That is, if ZFC is consistent, then ZFC + ¬CH (the negation of CH) is also consistent. CH is not true.

Together, this means the Continuum Hypothesis is independent of ZFC. You can assume it's true, or you can assume it's false, and either way, you won't create a contradiction within standard set theory.

Analogy: The Parallel Postulate

This is similar to the geometry of the ancient world. For centuries, mathematicians tried to prove Euclid's Fifth Postulate (the parallel postulate) from his other axioms.

They eventually discovered that it was independent.
You can assume it's true, and you get "flat" Euclidean geometry.
You can assume it's false, and you get perfectly consistent "curved" non-Euclidean geometries (like spherical or hyperbolic geometry).

Similarly, in set theory:

You can assume CH is true, and you get one kind of mathematical universe.
You can assume CH is false, and you get a different, but equally valid, mathematical universe.

Most modern set theorists who lean towards a "false" CH believe that 2^(ℵ₀) is actually much larger than ℵ₁, meaning there is not just one, but a vast, infinite number of infinities between the integers and the real numbers.

Summary

Assumption	Is there an infinity between ℵ₀ and ℵ₁?	What is the size of the continuum `2^(ℵ₀)`?
Continuum Hypothesis is TRUE	No	`2^(ℵ₀) = ℵ₁`
Continuum Hypothesis is FALSE	Yes	`2^(ℵ₀) > ℵ₁` (often much greater)

Conclusion: Your question is the Continuum Hypothesis itself, and it is one of the most famous undecidable problems in all of mathematics. There is no universally accepted answer.

This is a fantastic and crucial question. The relationship between the unending hierarchy of mathematical infinities and the physics of our universe is deep, subtle, and one of the most important open areas where pure mathematics confronts empirical reality.

The short answer is: It is primarily a mathematical construct, but it has profound, and sometimes shocking, implications for physics. The "unending universe" of infinities is a feature of the logical landscape of mathematics, but whether our physical universe makes use of all, or even any, of these vast infinities is a wide-open question.

Here’s a breakdown of the connection:

1. Where Physics is Perfectly Happy with the Smallest Infinity (ℵ₀)

For most of classical and modern physics, countable infinity (ℵ₀) is sufficient and necessary. We use it to describe things that are discrete but unbounded:

Quantum Mechanics: The number of possible energy states in a system can be countably infinite. The dimension of the Hilbert space (the abstract space of quantum states) for many systems is ℵ₀.
Spacetime Points: In both classical and quantum field theory, we often treat space and time as a continuum, which is a much larger infinity (2^(ℵ₀), the size of the real numbers). We need this to do calculus.

So, even basic physics requires at least one level of infinity beyond the merely countable.

2. Where the "Unending Hierarchy" Becomes Physically Relevant: The Nature of Spacetime

This is where things get controversial and deeply connected to your question. The mathematical possibility of ever-larger infinities forces physicists to ask: What is the true nature of physical reality at its most fundamental level?

The Continuum Hypothesis and Spacetime: Is spacetime truly a continuous manifold (with the cardinality of the real numbers), or is it discrete at the Planck scale? This is one of the biggest questions in quantum gravity.
- If spacetime is a true continuum, then physics must contend with the infinity of the real numbers (2^(ℵ₀)).
- If spacetime is discrete, then perhaps only countable infinities (ℵ₀) are physically realized. Many approaches to quantum gravity (like Loop Quantum Gravity) suggest this.
The Multiverse and "All Possible Worlds": This is where the higher infinities might truly come into play.
- In the Many-Worlds Interpretation of quantum mechanics, the "wavefunction of the universe" is constantly branching. The number of branches is staggering, but is it "merely" countably infinite (ℵ₀), or does it require the infinity of the continuum (2^(ℵ₀))? Most models suggest the latter.
- In cosmological inflation, the theory of the "eternal inflating multiverse" suggests an endlessly reproducing cosmos containing an infinite number of "pocket universes." The size of this collection could be countably infinite or even larger. The mathematical structure describing the entire multiverse could be vast enough to require these more abstract concepts of infinity.

3. The Ultimate Constraint: Computability and the End of Physics

There is a powerful, almost philosophical, argument that connects the hierarchy of infinities directly to the limits of physics. It's grounded in the concept of computation.

The Physical Church-Turing Thesis: This is the hypothesis that a universal Turing machine can simulate any physical process. In other words, the universe is, at its core, computable.
A Computable Universe is a Countable Universe: Anything that can be computed or simulated by a finite set of rules and a finite (but unbounded) amount of time is, by definition, dealing with a countable set of states and operations. A Turing machine can only handle discrete symbols.

The stunning implication: If the Physical Church-Turing Thesis is true, then the universe, at its most fundamental level, cannot make use of the vast majority of the mathematical infinities. It would be confined to the realm of the countable (ℵ₀) and the computable. The unending universe of larger infinities (ℵ₁, ℵ₂, ...) would remain a magnificent, unreachable "Platonic heaven" that our physical universe can describe in its laws but can never physically instantiate.

Conclusion: A Spectrum of Possibilities

So, to answer your question directly:

Strictly Mathematical? The hierarchy itself is a proven feature of mathematical logic (ZFC set theory). It is a necessary consequence of our standard definitions.
Relevant to Physics? Absolutely. It forces us to ask fundamental questions:
- Is the universe fundamentally continuous or discrete?
- How big is the multiverse, if it exists?
- Are the laws of physics ultimately computable?

The relationship can be summarized as follows: Mathematics provides a vast, unending "menu" of possible logical structures. Physics is the process of discovering which, if any, of these structures our universe has chosen to use. The higher infinities may be physically unrealized, but their mere existence as mathematical possibilities shapes our deepest questions about the nature of reality.