An infinite set of things called Al

I love that you can use generators to represent infinite lists. There is something about representing things that it feels like you shouldn’t be able to represent in a finite machine that excites me.

Language has incredible representative power, so large that it is easy to create contradictory sentences such as the sentence.

“This sentence is false”

You can use language to talk about things you can’t imagine such as when talking about infinity. You can define it by saying:

“The number that is bigger than any other number.”

Or a more accurate description of the aleph-null infinity, (because there are different kinds of infinity!)

“The size of the set of natural numbers”

You can go further and say that there is this thing which is:

“The thing you can’t represent using words”

Which similarly to “This sentence is false” is contradictory. If the thing truly can’t be represented using words then the sentence “The things that can’t be represented using words” can’t represent it.

A lot of these issues are similar in vain to the set of sets which contain all sets which don’t contain themselves.

It is this case where you make up some starting axioms and traverse through using those axioms until you end up with a contradiction.

The sets containing themselves contradiction was used by Russell to show that naive set theory led to a contradiction.

What I like about language and what I would love to write more programs to explore is the idea that you can make something up and then reason about it even in the absence of any specific seeing of the thing you are talking about.

The ability to talk about infinities fascinates me because the underlying item of discussion is so incomprehensible.

This ability to reason about things in such an abstract way is one of the main reasons I feel there is something in the representation of knowledge as a graph could be so powerful.

The power of natural language is unfortunately plagued with the problem of ambiguity.

That lack of specificity causes many problems when trying to encode anything useful computationally in it. The specificity of the nodes of a knowledge graph alongside the generality of what those nodes can represent, I believe, could be a key component in the production of an automated mathematician.

One example of this which I enjoyed learning more about recently is the W function.

This function is defined such that . I love the idea that in maths if you don’t know the answer to something, you can just create it and see if it has interesting properties.

This is similar to have imaginary numbers came about. For a long time people rejected the notion that there could be an answer to the square root of minus one. However, if you simply accept that an answer exists a lot of mathematics can be done atop.

This axiomatic production of answers can sometimes lead to contradictions, what that happens you know you couldn’t assume what you thought you could.

This has got me interested in what might happen if you wrote a program to simply assume the answer to a series of functions for which there is no closed form solution which is an unfactorisable function of order 5 or more.

I wondered if you took some of these if some of their results would relate to each other. Could the answer to one polynomial be stated in terms of the answer to another?

If many could be stated in terms of a single inverse, does that make that inverse significant in some way? Much like how the significance of the W function lies in the fact that it allows for a whole collection of exponential functions to be solved algebraically.

As part of producing a program that would search for these, I needed a function that would generate all the integer coefficients of the polynomials.

All of them? Isn’t there an infinite amount, that's where the generator comes in.