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Our friend Mr. Hollom caused quite a stir last month by introducing a strange gedankenexperiment: the iota function. In this blog post, I will define the iota function, offer several variants, and summarize the various philosophical effects of the dreaded \(I(n)\).

I'll openly admit that I've had ideas similar to the iota function, but I didn't have the guts to publish them because they weren't really mathematical. Kudos to Lawrence for having the courage to unleash the beast.

Lawrence's original definition

Okay, so maybe this isn't the original. The iota function has undergone some revision. What follows is the definition currently up on Lawrence's site as of July 4, 2013:

The iota function set is defined as the set of every function ever explicitly described anywhere in the whole of written (formal or informal, published or unpublished, virtual or physical) human history, if the function produces integer outputs, and could be used in such a way as to increase the final value of the function, and could currently be observed to exist. A function can be in there multiple times (if it was in a published book, it will be in there once for every copy ever saved or printed that still exists). If the function is of multiple variables, define a new function of a single variable that sets all of these to be the input. It will not include the iota function itself (this would lead to the value becoming infinity as the first fixed point of \(f(I(n)) = I(n)\) for an increasing integer function \(f\) and \(n > 0\)), while avoiding any circular references.

Then \(I(n)\) is formed by composing all the functions in the iota function set to produce the maximum possible value.

I take issue with some of the wordings used in this paragraph. For me the worst offender is "if the function produces integer outputs, and could be used in such a way as to increase the final value of the function." What does it even mean? If the function doesn't take integer inputs, how would you compose it?

My variation

While writing the article about the function, I will confess to mangling the definition to deal with my own problems with it. Here is my definition:

Define the iota function set as the set of all functions \(f\) such that
  • \(f\) was thought of by a human being or other sentient object.
  • \(f: \mathbb{N}^+ \mapsto \mathbb{N}^+\)
  • \(f\) is not the iota function or a similarly defined function.
\(I(n)\) is formed by composing all or some of the functions in the iota function set to produce the maximum possible value.

I find this to be simpler and more elegant than the original. Do you agree?

Simpler variation

Why do we need to compose the functions at all?

Define the iota function set as set of all functions \(f: \mathbb{N}^+ \mapsto \mathbb{N}^+\) ever thought of by a sentient object. \(I(n)\) is the maximum \(f(n)\).

The max takes care of everything for us, because we don't have to worry about increasing functions or anything.

Kappa function

The kappa function is a complete depa

Create a deterministic cellular automaton that simulates a universe, allowing life to evolve and eventually creating virtual googologists. The kappa function \(\kappa(t)\) is defined as the largest number yet invented by these googologists at time \(t\) in the automaton.

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