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spiral figure representing both finite and transfinite ordinal numbers
Credit: Pop-up casket & Fool

This spiral diagram represents all ordinal numbers less than ωω. The first (outermost) turn of the spiral represents the finite ordinal numbers, which are the regular counting numbers starting with zero. As the spiral completes its first turn (at the top of the diagram), the ordinal numbers approach infinity, or more precisely ω, the first transfinite ordinal number (identified with the set of all counting numbers, a "countably infinite" set, the cardinality of which corresponds to the first transfinite cardinal number, called 0). The ordinal numbers continue from this point in the second turn of the spiral with ω + 1, ω + 2, and so forth. (A special ordinal arithmetic is defined to give meaning to these expressions, since the + symbol here does not represent the addition of two real numbers.) Halfway through the second turn of the spiral (at the bottom) the numbers approach ω + ω, or ω · 2. The ordinal numbers continue with ω · 2 + 1 through ω · 2 + ω = ω · 3 (three-quarters of the way through the second turn, or at the "9 o'clock" position), then through ω · 4, and so forth, up to ω · ω = ω2 at the top. (As with addition, the multiplication and exponentiation operations have definitions that work with transfinite numbers.) As one would expect, the ordinals continue in the third turn of the spiral with ω2 + 1 through ω2 + ω, then through ω2 + ω2 = ω2 · 2, up to ω2 · ω = ω3 at the top of the third turn. Continuing in this way, the ordinals increase by one power of ω for each turn of the spiral, approaching ωω in the middle of the diagram, as the spiral makes a countably infinite number of turns. This process can actually continue (not shown in this diagram) through \omega^{\omega^\omega} and \omega^{\omega^{\omega^\omega}}, and so on, approaching the first uncountable ordinal number, ε0, which (according to the continuum hypothesis) corresponds to only the second transfinite cardinal number, 1. Georg Cantor proved in 1874 that the cardinality of the continuum (i.e., of the real numbers) is larger than that of the natural numbers (\mathfrak c > \aleph_0), but the identification of this larger cardinality with the second transfinite cardinal (\mathfrak c = \aleph_1) can neither be proved or disproved within the standard version of axiomatic set theory called Zermelo–Fraenkel set theory, whether or not one also assumes the axiom of choice.

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