User blog:King2218/Infinite Extensibility

In this blog post, I'll be extending the mighty arrow notation 'infinite'ly! (not really)

Arrows
Basically, there is no last extension.

When extending and generalizing arrow notation, you'll find out that you can still extend it even more. You can even create a function for the nth generalization and still be able to extend.

Up up up
When using arrow notation, you'll find out that you can use arrow notation to define the number of arrows and you can also use the large number generated using that large number of arrows to define the number of arrows!

$$3=3$$

$$3\uparrow^33=\text{tritri}$$

$$3\uparrow^{3\uparrow^33}3=...$$

$$3\uparrow^{3\uparrow^{3\uparrow^33}3}3$$

At this point, increasing the number of layers is getting impractical.

Let's denote the symbol for the generalization of the sequence above as a $$\rightarrow$$.

$$3\rightarrow1=3$$

$$3\rightarrow3=3\uparrow^{3\uparrow^33}3$$

We can do some things here:

$$3\rightarrow^23=3\rightarrow\rightarrow3=3\rightarrow(3\rightarrow3)$$

$$3\rightarrow^33$$

$$3\rightarrow^{3\rightarrow^33}3$$

Again:

$$3\downarrow3=$$

And so on...

In general
At this point, the arrows are colliding with other notations so now let's denote $$\uparrow_n$$ as the nth generalization shown above. Then,

$$3\uparrow_13=3\uparrow3$$

$$3\uparrow_23=3\rightarrow3$$

$$3\uparrow_23=3\downarrow3$$

Now, how are we going to extend this?

Easy:

$$3\leftarrow1=3$$

$$3\leftarrow2=3\uparrow_33$$

$$3\leftarrow3=3\uparrow_{3\uparrow_33}3$$

Although we have extended it, the arrows are very confusing.

Operator Space (vague)
PENDING TOMORROW