Russell's Paradox

• Difficult to understand
  • It seems that there is a lack of training in visualizing complex sets in the brain.

$R=\{x|x\notin x\}$

• 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.
  • $R=\{x|x\in \mathbb{U}\wedge x\notin x\}$
    • 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?
                • $S=\{x|x\in \mathbb{U}\wedge x\notin x\}$
                  • In this case, R is equal to S and that’s it.
                • $T=\{x|x\in \mathbb{U}\wedge x\notin x\wedge x\ne 1\}$
                  • In this case, since R includes T, it seems that R is the universal set anyway.
    • $\mathbb{U}$ can be any set, not necessarily the universal set (takker)
      • cf. Axiom of extensionality
      • For example, $R=\{x\in \mathbb{R}|x\notin x\}$, etc.
        • In this case, $R\in R⟺R\in \mathbb{R}\wedge R\notin R$ 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.