User blog:Edwin Shade/On The Formalization, Oxymoronical Over-complicated Elucidification and Parameterization Of A Transfinite Generalization Of Taranovsky’s Notation, (O.T.F.O.O.E.A.P.O.F.T.G.O.T. for short)

In this blog post I will posit, elucidate, and analyze an intuitively graspable generalization of Taranovsky’s ordinal notation which provably supersedes Taranovsky’s original notation by transfinite diagonalization over the ordering of entries in expressions of the form \(C(\alpha_1,\alpha_2)\), whilst preserving the notions of ‘’standard form’’ and multiple cardinalities.

To begin with it will be of great assistance if in our mental cogitation we are able to formulate a visualization of the notations about which we speak, so that by juxtaposition of the two formulations, one visual, the other formal, we thereby come to a fuller understanding of the complete work.

Note that in Taranovsky’s notation, (in which familiarity is hereby assumed), any conceivable expression will always be of the form \(C(\alpha_1,\alpha_2)\), in which \(\alpha_1\) and \(\alpha_2\) are either base constituents of Taranovsky’s notation escaping capsulation through lesser expressions, (i.e. \(C(0,0)\), \(\Omega_n\) where \(n\in\mathbb{N^+}\) ), or are merely other expressions of the form \(C(\alpha_1,\alpha_2)\) themselves. Thus by considering the expression \(C(\alpha_1,\alpha_2)\) as isomorphic to a vertically bipartioned equiangular quadrilateral containing two smaller visual constructions isomorphic to \(\alpha_1\) and \(\alpha_2\), from left to right respectively, we may progressively build a visual representation of any conceivable expression within Taranovsky’s notation. \(C(0,0)\) can be modeled with an empty vertically bipartioned equiangular quadrilateral, and \(\Omega_n\) can be modeled with multiple colors for each cardinality, black for \(\Omega_1\), crimson for \(\Omega_2\), cobalt for \(\Omega_3\), yellow for \(\Omega_4\), green for \(\Omega_5\), and so on as you desire. The end pictorial product pleasantly produces a Mondrian-esque pallette.

[It's a work in progress, which means I can't finish this in one day, but will nonetheless be contributing to this blog post frequently. I haven't yet reached the transfinite generalization yet, but when I do it will be agreed upon it is very powerful.]