- #1
Hill
- 823
- 630
- TL;DR
- How does axiom of foundation prevent the infinite descending sequence of elements?
Axiom of foundation says, ##\forall x (x\neq \emptyset \to \exists y(y\in x\wedge (y\cap x=\emptyset )))##.
The book says,
"As a consequence of the Axiom of Foundation, we see that there is no
infinite descending sequence x0 ∋ x1 ∋ x2 ∋ ···, since otherwise, the
set {x0,x1,x2,...} would contradict the Axiom of Foundation."
How does it contradict the Axiom of Foundation? If y=x0, it seems to obey the axiom.
The book says,
"As a consequence of the Axiom of Foundation, we see that there is no
infinite descending sequence x0 ∋ x1 ∋ x2 ∋ ···, since otherwise, the
set {x0,x1,x2,...} would contradict the Axiom of Foundation."
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