User blog:Qlf2007/The Position Theory

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.