Array notation is a kind of notation for large numbers. Comparing to combinatoric functions, array notation has more complex definition, but easier to evaluate. We’ve known some array notations, such as Bowers’ exploding array function (BEAF), Bird’s array notation (BAN), extensible-E system (ExE), and hyperfactorial array notation (HAN).

Here I bring a **strong** array notation, and it goes far away from the notations mentioned above. It’s made up of several parts shown below.

- Linear array notation (LAN)
- Extended array notation (exAN)
- Expanding array notation (EAN)
- Multiple expanding array notation (mEAN)
- Primary dropping array notation (pDAN)
- Secondary dropping array notation (sDAN)
- Dropping array notation (DAN)

These 5 parts don’t work.

- Nested dropper array notation (NDAN)
- Dropper-expanding notation (DEN)
- Multiple dropper-expanding notation (mDEN)
- Second-order dropper-expanding notation (soDEN)
- Higher-order dropper-expanding notation (hoDEN)

After fixing the problem in NDAN, we have these parts.

- Nested dropper array notation (NDAN)
- Weak dropper-expanding notation (WDEN)
- Multiple weak dropper-expanding notation (mWDEN)
- Weak dropper-multiexpanding notation (WDmEN)
- Primary dropper-dropping notation (pDDN)

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How do Computable functions relate to the Halting problem?

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I would like to ask a few questions about functions and algorithms. So addition is the function and add or adding is the algorithm, multiplication is the function and multiply or multiplying is the algorithm, exponation is the function and exponate or exponating is the algorithm, tetration is the function and tetrate or tetrating is the algorithm. Is this correct?

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What is the limit of SAN with your plans?

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Does anyone know how many symbols it would take to define (pDDN) in the language of First Order Set Theory (FOST)?

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maybe 200.

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Hypcos how many levels of (pDDN) would it take to reach Rayo’s number?

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Uncomputably many levels. All these notations are actually computable.

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Even BIG FOFT, okay how about the limit of pDAN?

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SAN can be extended: +’ is dropper-droppers for ::::…+ and ++ is dropper-droppers for +::::…, then we can defined nested droppers for +++…., like (N^+)

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Hypcos what are the circles (o) used for in your Strong Array Notation?

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The circle is a dropper in NDAN. Read the NDAN page for more details.

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Hypcos which part of your Strong Array Notation has recursion level psi(OMEGA omega)?

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The limit growth rate of mEAN is ψ(Ω_ω).

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So that is comparable to legion array space?

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BEAF is not well-defined beyond tetrational arrays, and so is legion array space.

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Okay.

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Hypcos which part of your Strong Array Notation has the Large Veblen Ordinal (LVO)?

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Separator {1{1`1`2`}2} has recursion level LVO, and first appears in expanding array notation (EAN).

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Hypcos how would you describe TREE(3) with your Strong Array Notation?

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The value of TREE(3) is too hard to work out. It has a lower bound of s(3, s(3, 262138 {1 {1`1,2 `} 2} 2) {1 {1`1,2 `} 2} 2 {1 {1`1,2 `} 5} 2). However, its upper bounds are much weaker, because we don’t know the growth rate of TREE(n). We just know TREE(3)<SSCG(3), where SSCG(n) grows slower than s(n,n{1,,2,2}2).

So s(3, s(3, 262138 {1 {1`1,2 `} 2} 2) {1 {1`1,2 `} 2} 2 {1 {1`1,2 `} 5} 2) < TREE(3) << s(3,3 {1,,2,2} 2).

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Cool.

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Hypcos how would you describe a Golapulus (10^100 & 10 & 10) with your Strong Array Notation?

Where would a Golapulus be in your Strong Array Notation?

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Impossible

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Hypcos how would you describe a 10^100 array of 10s array of 10s with your Strong Array Notation?

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That would be approximately s(10,s(10,s(10,100){2}2){2}2) in exAN.

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You are wrong. & isn’t defined.

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Hyp cos how would you express Meameamealokkapoowa Oompa with your Strong Array Notation?

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Meameamealokkapoowa Oompa is not well-defined. What’s more, Bowers’ googolisms beyond triakulus are not well-defined.

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Okay.

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So we express BEAF up to {1,,2} separator?

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Indeed. {1,,2} is the same level as tettrational BEAF arrays.

However BEAF and SAN are not equivalent at that point.

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Why you ran out of ideas about extensions?

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Aarex I am a huge fan of your work with large numbers. I have been studying your work with large numbers for years. Could you show me the steps to go from Otoperoogol to Otopersuperoogol? These numbers are listed on your Extensible Up-arrow System part 4 Dimensional/Nested Array Up-arrow webpage.

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My old googology site is closed, silly you.

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The question was asked before you closed it, silly. And I’m sure he’ll find the time to bring back the extended arrow notation some time in the future…

And he’s made primary dropper-dropping notation, part 12. Whether he plans to make a part 13, I don’t know.

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I have the idea on pDDN notation, going beyond than +.

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What’s the definition of the iterator in exAN?

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Iterator is defined as the 2nd entry (not in separators) in the array.

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Cant you like, fix it?

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It seems like he already has. (or at least tried to) The new NDAN is a replacement for the old one, and WDEN and mWDEN are analogues to DEN and mDEN. WDmEN seems to go a different direction from soDEN and hoDEN; perhaps he sees this as a better/stronger way to extend things.

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Can you make the new version of DEN? (sDEN)

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Can you explain in more detail why the latter five parts don’t work?

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In NDAN, there is the problem cause infinite loop.

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