Dropper Transfinite Leveling

Dropper Transfinite Leveling is a hydra notation that is heavily based off of Zenith. It can be considered a simplifcation of R function, SAN, and Zenith, and it is pretty similar to Hypcos's Dropping Hydra. It is made by Solarzone, Varyana, and Eliza.

DTL is revolutionary as it can provide an extremely useful as a normal form for an ordinal notation that can be used to semi-formalize many notations which had painful histories of attempts of formalization such as DTLOCF and UNOCF.

## Primitive DTL

### Definition

**Basic Notation**

[base][insert terms here]

EX. |base|{[[1]]}

**Terminology:**

**Base:** |n| is the base of this notation. A lot of function copy and expand a certain amount of times based on n.

**Term:** A pair of curly braces surrounding another expression {##}

**Base Successor term:** A term that is {0}. The basic term that increases the base. It is also equivalent to 1. n copies of {0} = n. Empty term {} is interchangeable with 0

**Successor term:** A term that is {n} where n is a positive integer, or {{0}#}. Basically it is a successor, an ordinal that you can subtract one from to get a smaller ordinal, to a smaller term and you can subtract one from a successor term.

**Non-Successor term:** A term that is {0#} r, or {{m}#} where m is not 0. Basically it is not a successor to a smaller term; you can't subtract one from a non-successor term.

**Rules:**

* Scan from LEFT to RIGHT
* Base Successor Rule: Case |n|{0}# (where we encounter the leftmost term to be a {0} base successor term):
	+ Delete {0} leaving |n| #
	+ Increase n by 1

* Successor Rule: Case |n|##{m#}# (where we counter a non-successor term):
	+ Scan inside the nonsuccessor term {m#}

	+ ##{m#}# = ##|n copies of {m-1#}|#
	+ ##[m#]# = ##|n copies of [m-1#]|#

* Layer Search and Expand Rule: Case |n|##[0]# where you encounter a [0]:
	+ Search for a {\_\_} layer, let that layer be {##[0]#}
	+ If you can't find a {\_\_} layer, enclose [...[0]...] with {} to make {##[0]#}
	+ Search outward for a layer that is smaller than {##[0]#}, let that layer be {@@\_\_@} (It is inevitable that \_\_ will consist of {\_\_}s and @/@@ would consist of [\_\_]s, copy \_\_ but not @@ and @.
	+ If you can't find that layer, copy everything
	+ {@\_...{##[0]#}...\_@} = {@@\_...{##\_...{##\_...{##\_...{##0#}...\_#}...\_#}...\_#}...\_@} with n nestings

### DTL Normal Form for Ordinal Notation

This notation hits

Π

n

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{\displaystyle \Pi \_{n}-reflection}

so now we have the luxury of being able to convert it into a

Π

n

−
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{\displaystyle \Pi \_{n}-reflection}

 ordinal notation. Since we have DTL formalized and we know how it precisely works, we can semi-formalize Dropper Ordinal Notation/DON and UNOCF. The problem is that we need an ordinal that is huge enough to collapse. We can't use a

Π

n

{\displaystyle \Pi \_{n}}

 reflecting cause it will restrict the notation's strength in the future. But we can pull a DOR and use "Bad Ordinals".

Let

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b
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{\displaystyle \Omega \_{1,0}=T=First\ bad\ ordinal}

Now we just need to do something similar to Buchholz. (I am retarded when it comes to set theory so this might epicly fail, please forgive me - Varyana)

SOMEONE HELP ME WITH THIS

C

n

0

(
α
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n

{\displaystyle C\_{n}^{0}(\alpha )={n}}

C

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{\displaystyle C\_{n}^{a+1}(\alpha )=\{\beta +\gamma ,\delta |\beta ,\psi \_{\epsilon }(\delta ),\delta \in C\_{n}^{a}\land \delta <\alpha \}}

C

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⋃

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{\displaystyle C\_{n}(\alpha )=\bigcup \_{m<\omega }C\_{n}^{m}(\alpha )}

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{\displaystyle \psi \gamma (\alpha )}

Still A WIP!!! (This is really hard help)

[n] can be written as

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{\displaystyle \psi \_{\Omega \_{1,0}}(n)=\psi \_{T}(n)}

{m[n]} can be written as

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{\displaystyle \psi \_{\psi \_{\Omega \_{1,0}}(n)}(m)=\psi \_{\psi \_{T}(n)}(m)}

## DTL/Strong Binary Zenith Notation

### Definition

**Basic Notation**

[base][insert terms here]

EX. |base|{{{0,2},1}}

**Terminology:**

**Base:** |n| is the base of this notation. A lot of function copy and expand a certain amount of times based on n.

**Term:** A pair of curly braces surrounding another expression {##}

**Base Successor term:** A term that is {0}. The basic term that increases the base. It is also equivalent to 1. n copies of {0} = n. Empty term {} is interchangeable with 0

**Successor term:** A term that is {n} where n is a positive integer, or {{0}#}. Basically it is a successor, an ordinal that you can subtract one from to get a smaller ordinal, to a smaller term and you can subtract one from a successor term.

**Non-Successor term:** A term that is {0#} r, or {{m}#} where m is not 0. Basically it is not a successor to a smaller term; you can't subtract one from a non-successor term.

**Rules:**

Scan from left to right, evaluate the first rule applicable to the first applicable term you see.

Rule 1 (number rule): DTL[n] = n

Rule 2 (base rule): DTL[n]{0}# = DTL[n+1]##

Rule 3 (expansion rule): #{m+1,#}## = #n copies of {m,#}##

Rule 4 (dropping rule): #{0,m}##

* If #{0,m}## isn’t in the form of {#{0,m}##,m-1} add {\_\_,m-1} within the first layer that is smaller than it.
* The idea here is that {0,m} is our initial diagonalization term, our ascending layer is initially in the form of {#{0,m}##,m-1} that contains our diagonalization term and our descending layer is initially in the form of {#\_\_##,m-1} where it contains one less diagonalization term than our ascending layer. Once we finish this process once, we set our diagonalization term to our current ascending layer and set our current ascending layer to our current descending layer and repeat this process once again. We want to repeat this m times to create a m-dropper notation. [n + 1 [diagonalization term]s ascending layer] is the ascending layer for our descending layer [n [diagonalization term]s ascending layer]
* Repeat m times:
	+ Step 1: Search outward for a layer that is smaller than our current ascending layer
	+ Step 2: If the smaller layer isn’t the descending layer of our current ascending layer within the smaller layer
	+ Step 3: Our ascending layer diagonalizes over our descending layer so set our current diagonalization term to our current ascending layer
* Replace our current ascending layer with n nestings of the our current descending layer

## Extended DTL

### Definition

**Basic Notation**

[base][insert terms here]

EX. |base|{{{0,2},1}}

**Terminology:**

**Base:** |n| is the base of this notation. A lot of function copy and expand a certain amount of times based on n.

**Term:** A pair of curly braces surrounding another expression {##}

**Base Successor term:** A term that is {0}. The basic term that increases the base. It is also equivalent to 1. n copies of {0} = n. Empty term {} is interchangeable with 0

**Successor term:** A term that is {n} where n is a positive integer, or {{0}#}. Basically it is a successor, an ordinal that you can subtract one from to get a smaller ordinal, to a smaller term and you can subtract one from a successor term.

**Non-Successor term:** A term that is {0#} r, or {{m}#} where m is not 0. Basically it is not a successor to a smaller term; you can't subtract one from a non-successor term.

**Dropper:** (n){##} means that {##} is a n-dropper. All droppers in the form of ({greater than 0}{0}xn...){##} drops down n times to ({greater than 0}){##}.

**Rules:**

Scan from left to right, evaluate the first rule applicable to the first applicable term you see.

Rule 1 (number rule): DTL[n] = n

Rule 2 (base rule): DTL[n]{0}# = DTL[n+1]##

Rule 3 (expansion rule): #{m+1,#}## = #n copies of {m,#}##

Rule 4 (dropping rule): #(){0,m}##

* If #{0,m}## isn’t in the form of {#{0,m}##,m-1} add {\_\_,m-1} within the first layer that is smaller than it.
* If
* The idea here is that {0,m} is our initial diagonalization term, our ascending layer is initially in the form of {#{0,m}##,m-1} that contains our diagonalization term and our descending layer is initially in the form of {#\_\_##,m-1} where it contains one less diagonalization term than our ascending layer. Once we finish this process once, we set our diagonalization term to our current ascending layer and set our current ascending layer to our current descending layer and repeat this process once again. We want to repeat this m times to create a m-dropper notation. [n + 1 [diagonalization term]s ascending layer] is the ascending layer for our descending layer [n [diagonalization term]s ascending layer]
* Repeat m times:
	+ Step 1: Search outward for a layer that is smaller than our current ascending layer
	+ Step 2: If the smaller layer isn’t the descending layer of our current ascending layer within the smaller layer
	+ Step 3: Our ascending layer diagonalizes over our descending layer so set our current diagonalization term to our current ascending layer
* Replace our current ascending layer with n nestings of the our current descending layer