Ordinals aren't much worse than Quaternions
The article explores the computability of ordinals up to ε₀, comparing them to quaternions and complex numbers, and discusses their unique properties, Python implementations, and recursive nature in mathematics.
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The article discusses the utility and computability of ordinals, particularly those up to ε₀, comparing them to other mathematical constructs like quaternions, complex numbers, and polynomials. It highlights that ordinals, while seemingly complex, can be represented and manipulated in a computable manner similar to these other number systems. The author explains that quaternions are useful for 3D rotations and have non-commutative multiplication, while complex numbers are well-accepted and can be easily implemented in programming languages like Python. The article also touches on the representation of infinity in floating-point arithmetic and how ordinals can be viewed as a cohesive package that combines various mathematical concepts. The author provides examples of how to implement polynomials and ordinals in Python, emphasizing the unique properties of ordinal arithmetic, such as non-commutativity and specific distributive laws. The discussion culminates in the recursive nature of ordinals, particularly in reaching ε₀, and the potential for further exploration in ordinal exponentiation and surreal numbers. Overall, the piece aims to demystify ordinals and illustrate their practical applications in computation.
- Ordinals can be represented and computed similarly to quaternions and complex numbers.
- The article emphasizes the non-commutative nature of ordinal arithmetic.
- Python implementations of polynomials and ordinals are provided as examples.
- The concept of infinity in ordinals is linked to existing mathematical constructs like floating-point numbers.
- The discussion includes the recursive nature of ordinals, particularly in relation to ε₀.
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7 commentsWhat a shame... I think the dude is overcomplicating stuff just by not understanding this.
As an analogy, the naturals are the closure of 0 being a natural number, and to each natural number, having a function that gives you another number not in the set, as the "next" natural number (called the successor of that original number).
The class of ordinals are just the closure of having 0 be an ordinal, and to each *set* of ordinals being able to get the "next" ordinal, (i.e. the smallest ordinal that comes (not necessarily immediately) after all elements in the set). Then omega is the "next" ordinal wrt the natural numbers (aka finite ordinals), and that's what's meant by a limit ordinal, defined by exclusion, as not the "next" ordinal of any set of ordinals that has a maximum element (the natural numbers have no maximum element, but the natural numbers and omega together have omega as the maximum element); the "next" ordinal of a set that has a maximum element is equally the "next" ordinal of the set containing just that maximum element, and we can do a +1, just as with natural numbers, hence we call it successor ordinal.
It’s only mentioned briefly in a footnote in the article, but one of the most mind-blowing concepts related to ordinals is called the Goldstein sequences.
These are basically sequences that start at an arbitrary natural number n and follow some simple rules (like the Collatz conjecture) to go from one item in a Goodstein sequence to the next. Much like the sequences in the Collatz conjecture, the Goodstein sequences can reach incredibly (extraordinarily) high numbers before eventually coming back down and terminating at zero. However, unlike the Collatz conjecture, the statement that all Goodstein sequences eventually terminate at zero for any starting value n has been proven.
But here’s the remarkable part: the proof that all sequences eventually terminate requires axioms “beyond” what you think of as involved in standard arithmetic. Specifically, we require the ordinals and transfinite induction to prove that all sequences terminate (i.e., there is a proof that Peano Arithmetic cannot prove that all sequences terminate; ZFC can however).
What this means is that you can create an extraordinarily tiny Turing machine (or a Python script that’s about 10 lines of code) whose halting behavior requires these bizarre axioms that most non-mathematicians have never heard of to prove. That is just nuts to me for some reason.
The only library I know of that supports infinite surreal numbers is CGSuite, you can see its hierarchy here: https://github.com/aaron-siegel/cgsuite/blob/e909ab8bc330e39...
"Order summation. Disjoint sum of underling sets and the sayeverything in Right is greater than everything in Left. This shows the composition of two prcoesses is terminating.
Lexiocgraphic product. Tuple of underlying sets. Nesting of two terminating properties is terminating. How do I square that this feels like two nested for loops with lex being part of ackermann termination also?"
I thought this was an article, and then it turned into a bunch of word salad esque personal notes...?
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