Philosophically speaking, every algorithmically derived set is a recursive set, and the only sets that aren’t recursive are those derived from biological means. One might ask, “How?” as this deviates from current and historical knowledge.
First, let’s establish what a recursive set is: A subset A of a set 𝐵 is said to be recursive, relative to 𝐵, if there exists an algorithm or effective procedure that, given an element 𝑏 in 𝐵, will output “yes” if 𝑏 is an element of A and “no” if 𝑏 is not an element of 𝐴. The set 𝐴 is thus also said to be decidable or computable.
Recursive vs. Recursively Enumerable Sets
While all recursive sets are indeed algorithmically definable, not all algorithmically derived sets are recursive. Some are recursively enumerable but not recursive. For instance, the set of all factual statements in arithmetic is recursively enumerable but not recursive, aligning with Gödel’s Incompleteness Theorems.
Halting Problem and Rice’s Theorem
The Halting Problem and Rice’s Theorem highlight inherent limitations in algorithmically determining specific properties of sets and functions, suggesting that not all algorithmically derived sets can be fully captured by recursive definitions. The claim that every algorithmically derived set is recursive raises essential questions about randomness. Can randomness truly be manufactured algorithmically? While this question remains somewhat elusive in the philosophy of mathematics, I believe everything derived algorithmically can be reverse-engineered. Whether we possess the skills and resources to do so doesn’t change the fundamental possibility.
This raises a more profound question: “Can a function that follows the rules of math and logic truly create randomness?” World War II provides a compelling historical example. Alan Turing helped reduce the war’s timeframe by cracking the Enigma code. The Germans believed they had created an unbreakable code that could safely transmit through any channel. However, Turing — one of history’s greatest mathematicians — cracked the code because, ultimately, it was developed by an algorithm and wasn’t truly random.
This historical example supports why every algorithmically derived set is recursive: even when we cannot immediately derive an algorithm to compute whether 𝑎 a or 𝑏 b is a member of the set, such an algorithm must exist somewhere — just as Turing found the algorithm to compute the Enigma code. An algorithm exists that can compute membership in finite time. Without Turing, we might have believed Enigma wasn’t computable in finite time. As depicted in The Imitation Game, those early computers searched endlessly for the Enigma algorithm.
The only non-recursive sets are those derived from biological beings; even this classification isn’t definitive. I would argue they are recursive sets for now, as I believe biological beings are fundamentally deterministic creatures. Too many variables and contingencies exist for us to comprehend the underlying processes fully.
In conclusion, mathematics is deterministic, and algorithms follow the rules of math and logic. Therefore, just as software systems can be reverse-engineered, any algorithmic creation can be reverse-engineered in finite time to determine whether 𝑎 a or 𝑏 b is a set member. This writing would be scientifically proven wrong by the Halting Problem and Rice’s Theorem, so I’m posing it in a philosophical manner rather than a scientific statement. I would conclude this by saying philosophically, “All algorithmically derived sets are recursive sets.
rustian ⚡
