User blog:Edwin Shade/A Three-Symbol System For Representing Ordinals

-$$|$$ represents 1, or a unit, and $$(x)$$ means to replicate the string inside the parenthesis an $$\omega$$ number of times. The power of this system is that by nesting, (that is, enclosing strings with parenthesis which are inside a second set of parenthesis, (which in turn are part of a set of even larger parenthesis, you are continually diagonalizing, (thus utilizing recursion to create bigger ordinals quickly, (as you'll soon see)))).

One caveat though, is in the parenthetical nesting. Making the parenthesis all equal in size will create ambiguity, so we must make them different sizes. Even with this multiple-layer-deep set of parenthetical statements though, we must also assign the parenthesis different colors, to make clear which pair are to be evaluated at each level. Granted, this does utilize more than three-symbols, but as it only consists of three fundamental concepts, (the parenthesis, which is an operator; the colors and types of parenthesis, which are labels; and the unit | itself), I consider it to be composed of only three-symbols. And now, on to the examples !

$$|||\equiv 3$$

$$(|)\equiv\omega$$

$$(|)(|)\equiv\omega\cdot 2$$

$$\big( (|)\big)\equiv\omega^2$$

$$\bigg(\big( (|)\big) \bigg)\equiv\omega^3$$

$$\big( (\big)_1 | \big\big)_1\equiv\omega^{\omega}$$

$$\bigg(\big( (\big)_1 | \big\big)_1 \bigg)\equiv\omega^{\omega+1}$$

$$\big( (\big)_2 \big( (\big_1 | \big\big)_1 \big\big)_2\equiv\omega^{\omega\cdot 2}$$

$$\bigg( \big( (\big)\bigg)_1 | \bigg( \big\big)\bigg)_1\equiv\omega^{\omega^2}$$

Examples aside, what is the limit of this notation ? Personally, I think it to be $$\epsilon_0$$, though finding a general form for $$\omega,\omega^{\omega},\omega^{\omega^{\omega}},...$$ may mean we can diagonolize over this within the system and create ordinals larger than $$\epsilon_0$$ So as to clarify also, the sub-scripted numbers in the prior examples refer to whether a pair of parenthesis is linked or not. It may have created confusion, and for that I apologize. Lastly, if you have recommendations or examples of your own, by all means write a comment about it and I'll add it to the list. Ordinals are fascinating to me, and I hope this notation proves to be useful in some aspect of Googology, whether tangentially or directly. Note what a tangled web we weave, when the symbols of our system is but 3 !.

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