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Stream: theory: mathematics

Topic: Exploration into Sets

Julius Hamilton (Jul 01 2024 at 19:00):

This is a sketch of some thoughts which I hope will be a warm-up to some more rigorous questions about set theory that I can write up hopefully later today. Let me know if you have any thoughts or critiques on what I say. Thank you.

Exploration Into Sets.pdf

Nathan Corbyn (Jul 02 2024 at 05:47):

The latter part of this seems to be hinting at the idea of the constructible hierarchy and Gödel’s constructible universe

Jack Jia (Jul 22 2024 at 16:30):

Very interestong. One thing that I always wondered is that when we think about "things", do we implicitly assume that we can "reach", or "interact" with these "things"? (like with morphisms going in or out, and as you mentioned we can also see these interactions as "things".)

I think human behaviors are based on perception, so it probably doesn't make much sense to talk about "just a thing" without refering to specific kinds of interactions.

Another question I always have is that monad in Latin means "an elementary individual substance which reflects the order of the world and from which material properties are derived", maybe the discoverer/inventor of monad thinks "things" should automatically come with a binary operation? I don't quite understand why though.

Julius Hamilton (Jul 26 2024 at 01:20):

I'd like to hear your thoughts on all of those questions, they also interest me as well!

Julius Hamilton (Jul 26 2024 at 01:48):

This is just my next step forward in trying to basically define set theory in terms of free algebra, I think. It is in rough condition and I had help from an AI, but it is a step in the direction I want to go in.

settheorycategorytheory3.pdf

Julius Hamilton (Jul 26 2024 at 15:45):

I'd really like to discuss the formal definition of the constructible universe if anyone would like to.

https://en.wikipedia.org/wiki/Constructible_universe

Screenshot-2024-07-26-at-9.11.41AM.png

It appears that $X$ is a collection of sets, and $z_1, \ldots, z_n$ elements of $X$. I'm not sure if $\Phi(y, z_1, \ldots, z_n)$ means a formula where the only constants are $y, z_1, \ldots, z_n$, or that it must have at least all of those constants? I think it says that this formula must hold true for set $X$, and that $\text{Def}(X)$ is therefore the set of all elements of $x$ which satisfy any first-order formula! So maybe this is an example:

If $X$ is $\emptyset$, then I think $\text{Def}(X)$ should be all sets which can be obtained from an application of an axiom of ZFC to $\emptyset$. I am pretty sure all you can define is $\{\emptyset\}$ and the set stated to exist by the axiom of infinity:

Screenshot-2024-07-26-at-9.23.36AM.png

For the next stage, I believe we can create a lot more sets, because of the axiom of specification. I think we can specify subsets of $\omega$ using well-formed formulae. I think that we can order these formulae using something like a Gödel numbering. However, I think I am looking for a simpler numbering system than that.

I am now thinking of something which will help me in many situations regarding term algebras and free algebras, I think. For any algebraic signature with $m$ $n$-ary relations, we can order the relations by their arity. If two relations have the same arity, we can choose an arbitrary order for them. Next, impose an arbitrary order for the generator elements for the algebra. For each arity $n$, create the $n$-fold product of the generators, which I believe has an inherent ordering. For each tuple in the product, apply the operation to create a new term.

This would allow you to biject the natural numbers to the term algebra, I believe. And it would allow you to define a "generation rank" for sets of terms - those constructed in stage 0, stage 1, stage 2, etc.

I look forward to exploring this idea further and making use of it!

James E Hanson (Aug 14 2024 at 02:36):

@Julius Hamilton You have the behavior of $\mathrm{Def}(X)$ a little bit wrong. $\mathrm{Def}(X)$ is the set of all subsets of $X$ that are first-order definable in the structure $(X,\in)$ with parameters. In particular, this means that $\mathrm{Def}(\{\varnothing\})$ is just going to be $\{\varnothing,\{\varnothing\}\}$.

Julius Hamilton (Sep 02 2024 at 17:10):

Thanks. Sorry for the delayed response, as I had turned my attention to others topics.