axiom of choice meaning in Hindi

Examples

1. There are models of ZF in which the axiom of choice fails.
2. If the axiom of choice holds, then every successor cardinal is regular.
3. Without the axiom of choice, there are cardinals which cannot be well-ordered.
4. The construction of " f " relies on the axiom of choice.
5. However, these may all be different if the axiom of choice fails.
6. Indeed, their existence is a non-trivial consequence of the axiom of choice.
7. The Axiom of Choice can be proven from the well-ordering theorem as follows.
8. One consequence of Shoenfield's theorem relates to the axiom of choice.
9. Meanwhile, the negation of the axiom of choice is, curiously, an NF theorem.
10. But none of these theorems actually proves that the axiom of choice holds.