User blog:Edwin Shade/A Slight Change to Knuth's Up Arrow Notation

This is a way to combine traditional up-arrow notation and down-arrow notation. Up arrows denote a right-associative operation, (what we normally use), and down arrows denote a left-associative operation.

The number of arrows between the two numbers a and b still determines what order operator is being used, but you may use up and down arrows at the same time. For instance, 3↑↓↑↓4 has four arrows, so it refers to the operation of hexation. There is an up arrow in the fourth arrow from the left's place, which means the operation of hexation is right associative, and so 3↑↓↑↓4 would be expanded to 3↓↑↓(3↓↑↓(3↓↑↓3)). Next though, since the first arrow from the right in 3↓↑↓3 refers to pentation and is a down arrow, pentation is to be evaluated in a left associative way, so that 3↓↑↓3 would expand to (3↑↓3)↑↓3 instead of 3↑↓(3↑↓3). This continues until we reach 3↓3, which is the same as 3↑3, due to the commutative property of multiplication. We work our way out along these rules and eventually calculate a value for 3↑↓↑↓4.

The usefulness of this is that you may define ordinals such as $$\epsilon_\omega$$ in terms of omega. For the case of $$\epsilon_\omega$$ you would rewrite it as ω↓↑↑ω, which would expand to ((((ω↑↑ω)↑↑ω)↑↑ω)↑↑ω)↑↑...