How Things Nest

The way that functions nest into themselves and collapse, especially when dealing with transfinite numbers, determines the strength of a notation. There is little difference between TON and Veblen's notation except that one nests in a different manner so one is stronger.

Here a simple notation will be devised to illustrate the different types of nesting. We will be using racks.

Racks are a series of elements separated by comas enclosed by parenthesis. Racks may represent transfinite ordinals.

These are racks:

\((0)\) \((0,0)\)  \((0,0,0)\)  \((0,0,0,0)\)  \(...\)

Slots are the places the elements occupy. Slots are ordered from 1 to any arbitrary value from right to left.

For example, the first slot of

\((0,0,\color{#FFFF00}{0} ,\color{#FF0000}{0} )\)

is where the red zero is, and the second slot is where the yellow zero is.

Oh, and every rack starts out with just zeroes.

Okay, now for the transformations...


 * Placing the current rack inside the 1st slot of a larger rack is called monification.
 * Placing a rack inside the 1st slot of the current rack is called anti-monification.
 * Placing the current rack inside the bubila of a larger rack is called bification.
 * Placing a rack inside the bubila of the current rack is called anti-bification.
 * Placing the current rack inside the corlus of a larger rack is called trification.
 * Placing a rack inside the corlus of the current rack is called anti-trification.
 * Placing the current rack inside the dendum of a larger rack is called quatrification.
 * Placing a rack inside the dendum of the current rack is called anti-quatrification.
 * In general placing the current rack inside the nth slot of a larger rack is called nthification.
 * In general placing a rack inside the nth slot of the current rack is called anti-nthification.
 * In general placing a rack inside the nth slot of the current rack is called anti-nthification.

After one monification the rack \(\color{#FF0000}{(0,0,0,0)}\) will turn into \((0,0,0, \color{#FF0000}{(0,0,0,0)} \color{#ffffff}{)}\) but after one anti-monification the rack \(\color{#FF0000}{(0,0,0,0)}\) would turn into \(\color{#FF0000}{(0,0,0,} \color{#ffffff}{(0,0,0,0)} \color{#FF0000}{)}\)

Note the red color is to keep track of the original rack. Also note that whenever a rack undergoes a transformation of nthification or anti-nthification, the racks on all levels have the same number of slots.

Now a double monification of \(\color{#FF0000}{(0,0,0,0)}\) would be \(\color{#ffffff}{(0,0,0,(0,0,0,} \color{#ff0000}{(0,0,0,0)} \color{#ffffff}{))} \), and the triple monification of \(\color{#FF0000}{(0,0,0,0)}\) would be \(\color{#ffffff}{(0,0,0,(0,0,0,(0,0,0,} \color{#ff0000}{(0,0,0,0)} \color{#ffffff}{)))} \)

Of course we can do this an infinite number of times - and this is where we introduce the other types of nthification and anti-nthification (because so far we've only been talking about monification.

We can say that after the effect of monification is negligible (i.e. when you encounter the first fixed point of \(\gamma\rightarrow (0,0,0,\gamma)\) that we will go back to the original rack and perform one bification, which will be equal to an infinitely recursed monificaton).

Bification of \(\color{#FF0000}{(0,0,0,0)}\) would be \(\color{#ffffff}{(0,0,} \color{#ff0000}{(0,0,0,0)} \color{#ffffff}{,0)}\).

Now let nthification m-times upon a rack \(\mathbb{R}\) be denoted \(\Sigma_{n}^m\mathbb{R}\). \(\Sigma_{2}^1\color{#ff0000}{(0,0,0,0)}=(0,0,\color{#ff0000}{(0,0,0,0)},0)\). Anti-nthfication can be denoted \(\dot{\Sigma}_{n}^m\mathbb{R}\). We will be using this Sigma notation from now on for brevity.

Now just as there is a Sigma notation there's a meta-Sigma notation, denoted with \(\Pi\). Sigma notation is for consistently correct expressions within a given system, but the system that tells us how Sigma notation functions is Pi notation, and Pi notation is a meta-ordinal notation since Sigma notation is an ordinal notation. Note that Sigma notation has no fixed rules unless we make those rules with Pi Notation.

For instance, with Pi notation we can say that the simple rack progression \(\color{#ff0000}{(0,0,0,0)}\), \((0,0,0,\color{#ff0000}{(0,0,0,0)})\), \((0,0,0,(0,0,0,\color{#ff0000}{(0,0,0,0)}))\), ... or recursed monification leads to bification with the relation \(\Pi(\Sigma_1^{\omega}\equiv \Sigma_2^{1})\). The presence of the Pi denotes a meta-operation, and in this case it is the equation of two Sigma operators. Of course, infinite monification of a rack can also be equal to a single trification, in which the Pi-Sigma relation would be \(\Pi(\Sigma_1^{\omega}\equiv \Sigma_3^{1})\).

To set up an ordinal notation we first set up a list of acceptable Sigma conversions according to Pi notation.

The first, and simple, notation we will be using takes the form of a dual-slot rack, (0,0). This is about as simple as it gets, but, let's make up the Sigma conversions for this anyways. The first is \(\Pi(\Sigma_1^{\omega}\equiv\Sigma_2^2)\). So according to this, it means the supremum of \(\color{#ff0000}{(0,0)}\), \((0,\color{#ff0000}{(0,0)})\), \((0,(0,\color{#ff0000}{(0,0)}))\), ... is \(\Sigma_2^1\color{#ff0000}{(0,0)}\) or \((0,\color{#ff0000}{(0,0)})\).