# Tetration Forum

Full Version: New thinking about math infinity
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(My understanding about some historical ideas in math infinity and my contributions to the subject)

For basic hyperoperation awareness, try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well. However, this paper is mostly philosophical.
So far as I am aware I am the first to define Nopt structures.
Maybe there are several reasons for this: (1) Recursive structures can be defined by computer programs, functional powers and related fast-growing hierarchies, recurrence relations and transfinite ordinal numbers. (2) There has up to now, been no call for a geometric representation of numbers related to the Ackermann numbers. The idea of Minimal Symbolic Notation and using MSN as a sequential abstract data type, each term derived from previous terms is a new idea. Summarising my work, I can outline some of the new ideas: (1) Mixed hyperoperation numbers form interesting pattern numbers. (2) I described a new method (butdj) for coloring Catalan number trees
the butdj coloring method has standard tree-representation and an original block-diagram visualisation method. (3) I gave two, original, complicated formulae for the first couple of non-trivial terms of the well-known standard FGH (fast-growing hierarchy). (4) I gave a new method (CSD) for representing these kinds of complicated formulae and clarified some technical difficulties with the standard FGH with the help of CSD notation. (5) I discovered and described a “substitution paradox” that occurs in natural examples from the FGH, and an appropriate resolution to the paradox. This substitution paradox is well-known in computer science (object oriented programming) but not so well-known in mathematics, although related to the theory of types. (6) I described the original concept of minimal symbolic notation and invented the seed-theta notation for notating hyperoperations. (7) I gave an original geometric representation of base(m) Ackermann function using seed-theta nopt structures with CSD visualisation technique, and noted an important exponential property of this geometry. ( I generalised the recursion patterns from nested exponential power towers into NOPT form and clarified some technical issues concerning correct definition of the ordertype of a NOPT structure. (9) I accurately described a meaningful number realm (called naropt structures) where the famous Graham’s number resides. This realm relates to a specific range of Conway numbers or Bowers’ operators. (10) The well-known repdigit (repeated digit) SPN numbers can be extended into the NOPT hierarchy. This means that, in theory, SPN numbers can be extended in standard binary tick-tock fashion into the NOPT hierarchy. In reality, it is only relatives of repdigit numbers with low information content that can be described in this way. It’s not possible to see how to do this using the standard well-known fast-growing hierarchy. This has ramifications for set theorists in terms of what has been described by various authors as “right-sizing the infinite” or in other words, figuring out how to talk better about infinite ordinals such as omega and epsilon-zero. Even the low-information repdigit SPN numbers can’t be described beyond the hyperoperation realm into naropt structures (where Graham’s number resides). This means we can’t even be sure where the first infinite ordinal number, omega resides, which contradicts the standard set-theoretical wisdom that says that omega is well-defined as the first infinite number greater than all the finite numbers. (11) This has the amazing corollary that numbers described (by finite methods) either beyond g2 or beyond Bowers’ {{{1}}} operator are effectively infinite numbers.
Infinite phenomena can be observed in some finite mathematical objects and finite phenomena can be observed in some infinite mathematical objects.
(12) I described the curious looking slow-growing nopt structure. (13) I described a technique of using binary filter codes to define sequences that are projections through the ordinal hierarchy similar to transitional sequences through the hyperoperator hierarchy. (14) There are interesting transitional sequences from exponentiation to tetration and also through the hyperoperator hierarchy. (15) There are 8 folding methods for nopt structures (LU, RU, LD, RD, UL, UR, DL and DR)
(16) I gave some reasons for choosing the “canonical method” as fold-LU so that the “answer value” is the first thing you observe (in the top-left-corner position of the structure) when reading some text from right-left and top-down. Although the 8 fold-URDL operations are well known from computer science they haven’t until now been used for giving a geometric representation of numbers related to the Ackermann numbers (so that computational pathways and ordertype phenomena can be pictured and described). (17) Nopt structures encapsulate the requirements of a minimal notation whether via formula using an equation editor etc or via colored-square diagrams.
(1 By analogy to nepts (via the definition of nopts) we can describe naropts, or nested Knuth arrow towers. A particular subsequence from the Conway numbers corresponds with naropts.

The Theory of Large numbers

There is an artistic side to my math ideas, that makes it a curiosity in the world of maths. What’s different about my work is that the art-side is an essential part of the maths, there’s no way the math could be properly and effectively communicated via formula type setting software. The Color Square Diagrams is the logical way to understand these kinds of numbers and functions. The CSD technique is versatile and can illustrate various ideas in number theory and combinatorics using the concept of “small world examples for presentation, practice and organisation” and is a technique commonly used in the Demonstrations Projects from the Mathematica website.

The 19 math animations listed in the paper Composite Mulanept Patterns function on two levels:
(1) as noptiles (non-standard geometric tiling patterns and associated space-filling curve) and
(2) as Nopt structures (where the various functional interpretations are added).
They can help to understand the basics about the theory of computation. The Nopt structures require many animations to be explained clearly, for example (1) the 8 folding patterns for the pure noptiles, (2) a natural and intuitive computational pathway and (3) the natural, practical and visually smooth transitional sequences.

In the school system we learn addition, multiplication, exponentiation and the natural numbers.
At secondary school we learn about the exponent laws. In university math curricula there’s little said about the patterns, character or properties of big numbers except in the subjects of set theory, computability theory, computational complexity theory and the theory of computation. These are difficult graduate level courses to understand. There’s also number theory, which is another difficult subject but only relatively smaller numbers can be analysed using the methods and techniques of number theory. In set theory, there are the transfinite ordinals and cardinals, the continuum hypothesis (CH) the subject developed by Georg Cantor in the 19th century, and regarded as the first problem from David Hilbert’s famous list of 23 unsolved problems. The CH and GCH (generalised CH) are still considered by set theorists, but the results of Kurt Godel and then Paul Cohen (1963) seem to leave the foundations of maths in an uncertain situation, not that that matters to a lot of set theorists who continue researching undeterred. Some other famous mathematicians are:
Andrzej Grzegorczyk (a Polish logician, Fast growing hierarchy)
Wilhelm Ackermann (defined the famous Ackermann number sequence)
Donald Knuth (wrote important books about the art of computer programming)
John Horton Conway (an accomplished and prolific mathematician)
Ronald Graham (Ramsey theory and the famous Graham’s number)
David Madore (transfinite ordinals represented by trees)
Paul Budnik (came up with a transfinite ordinal calculator)

And some good and useful websites include: