Is My Approach to Understanding Set Theory Beneficial?

[honestrosewater]

Hey, I'm feeling very shaky for some reason. I'd like to run a few things by you guys. I can do formal if needed, but I'm trying to build up a better model in my head in which I can eventually reason more flexibly and quickly without making mistakes. I'm starting from the beginning. This is ZF, no Choice yet. (Please pardon my sleepiness, if it shows.)

Theorem 1. If C is a class of sets such that Union(C) is a set, then C is a set.​

I think Union(Power(C)) = C, but for some reason, I can't talk myself through it clearly. Power(C) puts the members of C inside of a set, grouping them in all the different ways (empty set included). Union() then just empties the members of those sets right back out and the duplicates don't count. Is that wrong? Just a hint is good.

Oh, sorry, that's not the way that I meant to go, but that's good to know too. Rather, starting with Union(C), which has emptied the members from each member of C, Power(Union(C)) puts them back into sets, grouped in all the different ways. This is a set by Power axiom. If C is a subset of this set, then C is a set by Subset/Specification. The thing that hangs me up is that the author has left open whether there are individuals (urelements) in addition to sets. And individuals would not be members of Power(Union(C)), would they?


Do you think that what I'm trying to do is beneficial at all? I am not trying to be sloppy. I'm trying to break free from a given formalism. I want better mental models.

 

[Valkarie]

I think it's beneficial to try and create better mental models of mathematical concepts. It can help you to better understand the concepts, as well as allowing you to reason more flexibly and quickly. Creating a mental model will also help to reduce the chances of making mistakes when trying to work through problems. As long as you are careful to make sure that your mental model is accurate and consistent with the given formalism, then it should be a great help!

 

[tacman]

It's great that you're working on building a better understanding of set theory and trying to improve your mental models. It's important to have a strong foundation in the basics in order to reason flexibly and accurately in more complex situations.

Regarding Theorem 1, your understanding is correct. Union(Power(C)) = C because Power(C) contains all the possible subsets of C, and Union() removes duplicates, leaving us with just the original elements of C.

As for the author leaving open the possibility of individuals, it's important to remember that set theory deals with sets as the basic building blocks, and individuals can be seen as sets with only one element. So they would still fit within the framework of Power(Union(C)).

Overall, your approach to understanding set theory is beneficial. Breaking away from a given formalism and building up a better mental model can help you reason more effectively and confidently in the future. Keep practicing and seeking clarification when needed, and you'll continue to improve your understanding.