Notes - Set Theory HT25, Foundation


Flashcards

@State the axiom of foundation.


\[\forall x (x \ne \emptyset \to \exists y \in x \text{ }y \cap x = \emptyset)\]

Every non-empty set $x$ has an $\in$-minimal element, i.e. an element $y \in x$ such that no element of $x$ is an element of $y$.

@Justify that the axiom of foundation makes sure that certain pathological sets don’t exist:

  1. There is no $x$ with $x \in x$.
  2. There are no $x$ and $y$ with $x \in y \in x$.
  3. There is no $x$ such that $\mathcal P(x) \in x$.
  4. There is no $x$ such that $\langle x, y\rangle \in x$ for any set $y$.

  1. If $x \in x$, then $\{x\}$ violates the axiom of foundation, since the only element of $\{x\}$ is $x$, but $x \cap \{x\} = \{x\} \ne \emptyset$.
  2. If $x \in y \in x$, then $\{x, y\}$ violates foundation, since $y \in x \cap \{x, y\}$ and $x \in y \cap \{x, y\}$.
  3. If $\mathcal P(x) \in x$, then $y = \{x, \mathcal P(x)\}$ violates foundation, since $x \cap y \ne \emptyset$ and $\mathcal P(x) \cap y \ne \emptyset$.
  4. If $\langle x, y \rangle \in x$ for some $y$, then $x \in \{x\} \in \langle x, y \rangle \in x$ and hence $Y = \{x, \{x\}, \langle x, y \rangle\}$ has no $\in$-minimal element.

For reference, the axiom of foundation states:

\[\forall x (x \ne \emptyset \to \exists y \in x \text{ }y \cap x = \emptyset)\]

Every non-empty set $x$ has an $\in$-minimal element, i.e. an element $y \in x$ such that no element of $x$ is an element of $y$.




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