TITLE: What pattern does set theory study?
QUESTION [5 upvotes]: If mathematics is the science of patterns, what pattern does set theory study?
My thoughts so far: set theory studies the pattern of relationships between members of a collection and between collections.
Background: This isn't a homework question, I'm starting on Introduction to Mathematical Thinking and ensuring I have a good grounding in the fundamentals of set theory.

REPLY [3 votes]: Although technically all patterns can be reduced to patterns of set membership, this is rarely the best way to think about them.
I don't think that set theory should be characterized as the study of a single type of pattern, although there are sub-fields that can be approximately characterized in this way.
Here are some examples of patterns studied in set theory, although their definitions may not be very accessible to the lay person.

patterns of consistency strength (mostly just a linear structure, but a very important one)
patterns of extenders (in inner model theory)
patterns of scales (in descriptive set theory)