- Difficult to understand
- It seems that there is a lack of training in visualizing complex sets in the brain.
R={x∣x∈/x}
- What is R?
- Is it the “set of sets that do not contain themselves”?
- Now, let’s consider if R belongs to R.
- If it does,
- According to the definition of R, “R does not belong to R” would be true.
- This is a contradiction.
- If it doesn’t,
- According to the definition of R, “R does not belong to R” would be false.
- This is also a contradiction.
- I see~ (blu3mo)
- It has a mathematical bug feeling and is interesting.
- As a countermeasure, let’s define the universal set U and impose a constraint.
- R={x∣x∈U∧x∈/x}
- Why is this okay? (blu3mo)
- R becomes the “set of sets that do not contain themselves and belong to U.”
- Now, let’s consider if R belongs to R.
- If we assume that it does,
- According to the definition of R, “R does not belong to R” and “R belongs to U” would be true.
- This is a contradiction.
- If we assume that it doesn’t,
- According to the definition of R, “R does not belong to R” and “R belongs to U” would be false.
- Taking the contrapositive, “R belongs to R” or “R does not belong to U” would be true.
- Here, if we consider “R does not belong to U” as true, it doesn’t lead to a contradiction (blu3mo)(blu3mo)
- It doesn’t feel very intuitive, but I understand the logic.
- So, does this definition mean that R is excluded from the universal set?
- If that’s the case, does R become equal to U?
- What if we consider something similar to R but not R?
- S={x∣x∈U∧x∈/x}
- In this case, R is equal to S and that’s it.
- T={x∣x∈U∧x∈/x ∧x=1}
- In this case, since R includes T, it seems that R is the universal set anyway.
- U can be any set, not necessarily the universal set (takker)
- cf. Axiom of extensionality
- For example, R={x∈R∣x∈/x}, etc.
- In this case, R∈R⟺R∈R∧R∈/R clearly leads to a contradiction.
- If you interpret the universal set as “a set that contains all elements in the current domain of consideration,” then it is fine as it is.
- However, if you consider the universal set as “a set that contains all sets,” then it is problematic.
- The naive interpretation of “a set that contains all sets” leads to paradoxes.
- What about {R}?
- Let’s consider if {R} belongs to R.
- {R} is no different from ordinary numbers or any other ordinary things.
- So, it simply belongs to R.