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This paper explores properties and applications of an ordered subset of thequadratic integer ring $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$. Thenumbers are shown to exhibit a parity triplet, as opposed to the familiareven/odd doublet of the regular integers. Operations on these numbers aredefined and used to generate a succinct recurrence relation for thewell-studied Fibonacci diatomic sequence, providing the means for generatinganalogues to the famed Calkin-Wilf and Stern-Brocot trees. Two related fractalgeometries are presented and explored, one of which exhibits several identitiesbetween the Fibonacci numbers and golden ratio, providing a unique geometricexpression of the Fibonacci words and serving as a powerful tool forquantifying the cardinality of ordinal sets. The properties of the presentedset of numbers illuminate the symmetries behind ordinals in general, as well asprovide perspective on the natural numbers and raise questions about thedynamics of transfinite values. In particular, the first transfinite ordinal$\omega$ is shown to be logically consistent with a value whose cardinality isdual: both zero and one. Considerations of these points and opportunities forfurther study are discussed.

On ordinal dynamics and the multiplicity of transfinite cardinality