Reviews : DON naming misnomer

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Ly DeSandaru...

I think my naming these numbers as DON or Dimensional Ordinal Numbers, I have created / open doors to a misnomer / mis- interpretations. These numbers are actually not Ordinal numbers but more likely transfinite numbers. Because they use the digits just like normal decimal numbers, but we can have infinite digits both the the left and right of the decimal point.

[So if DONs involve infinite strings of digits indexed by ordinals (say ω, ω+1, etc.), you are essentially walking in similar territory — building numbers that reach “beyond” the reals using ordinal structure.]

No I do not intend to use Ordinal structures.

ChatGPT...

Thank you for the clarification — that helps significantly.

You're right: if your Dimensional Ordinal Numbers (DONs) do not actually use ordinal indexing or depend on ordinal arithmetic/ logics (like successor functions or well-orders), then they are not truly "ordinal" numbers in the classical set-theoretic sense.

Instead, what you're describing seems to be a number system that generalizes real numbers — possibly into transfinite-length digit sequences — but retains digit-based representations like decimal, binary, or hexadecimal. Let’s reframe your system based on what you’ve said:

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What DONs Actually Are (as per your clarification):

1. Digit-Based System: Uses a finite digit set D, such as {0,1} (binary), {0..9} (decimal), {0..F} (hex), etc.
2. Bi-Infinite Strings: Allows infinite digits to the left and right of the decimal point.
3. Not Ordinal-Indexed: You do not use ordinal numbers like ω, ω+1 to index positions.
4. Digit Generation Engine: You allow a component like [∞↻D] to represent a digit-generating process, such as:
   • A fixed algorithm (e.g., π, √2 digits),
   • A rule (e.g., even/odd alternation),
   • A random generator,
   • Etc.
5. Arithmetic Defined Where Precision Is Finite or Bounded: For infinite expansions, addition/multiplication may not always terminate or may depend on constraints.

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Lexicographic vs Value-Based Comparison

You mentioned:

“For the decimal part, left to right digit by digit up to the requisite precision... For the part left of the decimal place, comparison is done right to left...”

This hybrid lexicographic+value-based comparison can work, but you’ll need to define rigorously:

• How “precision” is agreed upon or bounded.
• What happens when two infinite strings never differ (e.g., identical algorithms).
• Whether digit-generating engines are allowed to be compared directly or only by output.

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What Makes DONs Unique?

You may want to emphasize that DONs are:

• Digit-sequence-based transfinite numbers,
• Mechanically generative (through rules/ algorithms),
• Potentially non-computable or partially computable,
• Bi-infinite, unlike real numbers, which only allow finite digits to the left.

This makes DONs similar in spirit to:

• Infinite precision floats (but with greater generality),
• Transfinite continued fractions or decimal expansions,
• Certain non-Archimedean number systems,
• And to some degree, surreal numbers (which I can explain below if you like).

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Bonus: Terminology Suggestion

You may want to rename "Dimensional Ordinal Numbers" to reduce confusion.

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Ly DeSandaru...

How about DTN for dimensional transfinite numbers...

Gemini...

That's a solid name — Dimensional Transfinite Numbers (DTN) has a much clearer connotation:

Dimensional hints at the structured expansion (left/right of the radix point, perhaps even higher "dimensions" of representation later).

Transfinite accurately signals that you're venturing beyond finite digit-length.

And Numbers keeps the door open for arithmetic, comparison, algebra, etc.

It’s also abbreviation-friendly (DTN sounds clean), and avoids the misdirection that “Ordinal” created.

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