Exploring the Meaning and Implications of Real Numbers Raised to Cardinal Powers

When mathematicians, especially those working within the realms of set theory and algebra, deal with the idea of raising a real number to the power of a cardinal number, they quickly encounter a multitude of challenges and complexities. As Martín García points out, the question of what it means to raise a real number to the power of a cardinal number is itself meaningless without a clear and well-defined context.

Cardinality and Real Powers

Cardinal numbers, which represent the size of sets, have well-established arithmetic, and there is a well-established definition for raising a positive real number to a real power. However, the concept of raising a real number to a cardinal power is fraught with ambiguity.

For the purposes of this discussion, let us assume that we have defined the power in a reasonably consistent way. Specifically, we will consider (2^{aleph_0}) to be the cardinality of the set of all subsets of the natural numbers. If we interpret the number 2 as a cardinal number, then (2^{aleph_0}) still represents the same cardinality as the power set of the natural numbers. This leads to an interesting result:

(1.1^{aleph_0} leq 2^{aleph_0})

On the other hand, if we use the notation (9aleph_0 aleph_0), then:

(1.1^{aleph_0} 1.1^{9aleph_0} 1.1^{aleph_0^9} geq 2^{aleph_0})

From these observations, it seems that:

(1.1^{aleph_0} 2^{aleph_0})

However, as Martín García notes, the exact value of this expression is not necessarily (aleph_1). It could be strictly greater, and in the absence of the Axiom of Choice, it could even be incomparable.

Revisiting the Expression (x^{aleph_0})

The question may still seem valid: what does an expression of the form (x^{aleph_0}) actually mean in the first place? The notation (2^X) for the power set of (X) provides a useful parallel. We can denote the power set of a set (X) by (2^X), and this corresponds to the set of all functions from (X) to ({0, 1}). This reveals a deeper structure:

Powers of Cardinals

More generally, for two sets (X) and (Y), we denote the set of functions from (X) to (Y) by (Y^X). The notation (2^X) aligns perfectly with the power set notation, which we reconcile by noting that (2 {0, 1}). The von Neumann construction further extends this to ordinals, defining (0 varnothing) and (n {0, 1, dots, n-1}). From this, we can verify that (2^3 8), as expected.

The first infinite ordinal (omega {0, 1, 2, dots}) is the union of all the natural numbers.

The Cardinality of Sets

Cardinals are defined as the cardinalities of sets. The cardinality of a set (X) is the least ordinal (alpha) such that (X) is in bijection with (alpha). For example, the cardinality of (mathbb{N}) is (omega), denoted as (aleph_0) to signify that we are talking about cardinals rather than ordinals.

Exponentiation of cardinal numbers is then straightforward: if (X kappa) and (Y mu) are cardinals, then (kappa^{mu} X^Y).

The Implications of Real Numbers and Cardinals

Given that (1.1) cannot be a cardinal number since sets do not have (1.1) elements, the expression (1.1^{aleph_0}) is inherently meaningless. If we were to assume otherwise, we would still face another hurdle: the Continuum Hypothesis. G?del and Cohen demonstrated that the statement (2^{aleph_0} aleph_1) is independent of ZFC, meaning that it cannot be proved true or false using just the ZFC axioms.