## FANDOM

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The position theory serves as a radical extension to the Veblen function. The Veblen function allows transfinitely many variables arranged on a row, each having an ordinal as its position, e.g. $$\phi(1[\omega])$$

The position theory adds more positions to Veblen function. The traditional positions are now $$0$$-positions. A $$1$$-position is a function from $$0$$-positions to ordinals, s.t. only finite positions are mapped to non-zero ordinals. They are ordered like decimal numbers. If positions $$p,q$$ both map all positions $$[\alpha']_0 > [\alpha]_0$$ to equal ordinals and $$p([\alpha]_0) > q([\alpha]_0)$$ then $$p$$ is stronger than $$q$$. The first $$1$$-position that does not degenerate to a $$0$$-position is $$[[1]_0 \mapsto 1]_1$$. The Veblen function $$\alpha \mapsto \phi(1[[1]_0 \mapsto 1]_1,\alpha[0]_0)$$ enumerates the fixed points of $$\gamma \mapsto \phi(1[\gamma]_0)$$. In particular, $$\phi(1[[1]_0 \mapsto 1]_1)$$ is the Large Veblen Ordinal.

Likewise, $$2$$-positions are functions from $$1$$-positions to ordinals, and $$3$$-positions are functions from $$2$$-positions to ordinals, etc. An $$\omega$$-position is a function from $$0$$-positions to $$\alpha$$-positions where $$\alpha$$ is any natural number. The first $$\omega$$-position that does not degenerate is $$[[1]_0 \mapsto [1]_0]_\omega$$. The function $$\phi(1[[1]_0 \mapsto [1]_0]_\omega,\alpha[0]_0)$$ enumerates the fixed points of $$\gamma \mapsto \phi(1[\gamma]_0),\phi(1[[\gamma]_0 \mapsto 1]_1),\phi(1[[[\gamma]_0 \mapsto 1]_1 \mapsto 1]_2),\cdots$$. It's not hard to see that $$\phi(1[[1]_0 \mapsto [1]_0]_\omega)$$ is $$\vartheta(\varepsilon_{\Omega + 1})$$.

The $$\omega + 1$$-positions are similar to $$1$$-positions, replacing the $$0$$-positions with $$\omega$$-positions in the definition.

This system is much like the BEAF array system. When restricted to finite $$0$$-positions, $$1$$-positions give rise to dimensional arrays. Transfinite $$0$$-positions impose structures on the dimensions, thus rows, planes, realms, etc. of dimensions, and tetrational arrays. Likewise, $$1$$-positions correspond to pentational arrays, etc. The first non-degenerate $$\omega$$-position is in the next legion. By mapping $$[1]_0$$ to different positions, all legion arrays can be built. Legion legion legion legion etc. arrays use up to $$[\omega]_0$$. Lugion arrays use up to $$[\omega^\omega]_0$$. The meameamealokkapoowa oompa array probably uses only $$[\varepsilon_0]_0$$. Nothing has been said of $$\omega + 1$$-positions, not to say $$\omega^2$$-positions.