The image contains five mathematical problems related to ordinals and set theory, primarily involving the application of the axiom of choice. Here’s a quick breakdown of each problem:

Problem 1:

Prove that every element of an ordinal is an ordinal.

• Any ordinal coincides with the set of all smaller ordinals.
• If $\alpha \in \beta$, then $(\alpha \in \beta) \equiv (\beta \in \alpha) \iff \alpha = \beta$.

Problem 2:

Prove that any two ordinals are comparable under $\leq$:

• If $\alpha \subseteq \beta$, then $\alpha \leq \beta$.
• $\alpha + 1 = \alpha \cup \{\alpha\}$ is the smallest ordinal greater than $\alpha$.

Problem 3:

Prove that every partial order on a set is contained in some linear order.

• This is a classic application of Zorn’s Lemma, which is a form of the Axiom of Choice.
• The question also asks whether “linear” can be replaced with “well-ordered.”

Problem 4:

In a real vector space, a set of vectors is linearly independent if every finite subset is linearly independent.

• The basis of a vector space is a linearly independent set $B$ such that any vector can be written as a linear combination of vectors from $B$.
• Prove that every linearly independent set of vectors is contained in some basis.

Problem 5:

Prove that the Cartesian product of a family of non-empty sets is non-empty.

• The Cartesian product $\prod_{i \in I} A_i$, where each $A_i$ is non-empty, is non-empty.
• This is a direct application of the Axiom of Choice.

These are foundational problems in set theory and abstract algebra, especially related to ordinals, vector spaces, and the axiom of choice. Would you like a detailed explanation for any specific problem, or help with any particular part?

Here are 5 related questions to deepen your understanding:

1. How does Zorn's Lemma relate to the Axiom of Choice?
2. Why is the property of comparability crucial in ordinals?
3. What is the significance of proving that every vector space has a basis?
4. Can the proof of Problem 3 work without the Axiom of Choice?
5. How does Problem 5 connect to the general concept of infinite sets?

Tip: Understanding Zorn's Lemma is key to grasping the deeper implications of the Axiom of Choice in many fields of mathematics.