Recall of Weakly Superstrong Cardinals

Oct 13, 2025 by miodeya.com 28 views

Introduction

In the realm of set theory, large cardinals are a class of cardinal numbers that exhibit certain properties that are not provable in ZFC (Zermelo-Fraenkel set theory with the axiom of choice). One such large cardinal is the weakly superstrong cardinal, which is a strengthening of the superstrong cardinal. In this article, we will explore the consistency strength of a strengthening of weakly superstrong cardinals.

Recall of Weakly Superstrong Cardinals

A cardinal $\kappa$ is weakly superstrong if, for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ . This means that there is an elementary embedding $j: V_\kappa \to V_\lambda$ such that $A = j(A) \cap V_\kappa$ . The weakly superstrong cardinal is a strengthening of the superstrong cardinal, which is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ .

Strengthening of Weakly Superstrong Cardinals

A strengthening of weakly superstrong cardinals is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . Additionally, for every $A \subseteq V_\kappa$ , there is a cardinal $\mu$ and a set $A^{**} \subseteq V_\mu$ such that $\langle V_\lambda, \in, A^* \rangle \preccurlyeq \langle V_\mu, \in, A^{**} \rangle$ and $\mu > \lambda^+$ . This strengthening of weakly superstrong cardinals is a significant strengthening of the weakly superstrong cardinal, as it requires the existence of a larger elementary embedding.

Consistency Strength of Strengthening of Weakly Superstrong Cardinals

The consistency strength of a strengthening of weakly superstrong cardinals is a measure of the strength of the axiom that asserts the existence of such a cardinal. The consistency strength of a strengthening of weakly superstrong cardinals is at least as strong as the consistency strength of a weakly superstrong cardinal, as the strengthening of weakly superstrong cardinals is a strengthening of the weakly superstrong cardinal. However, the consistency strength of a strengthening of weakly superstrong cardinals is not necessarily the same as the consistency strength of a weakly superstrong cardinal.

Comparison with Other Large Cardinals

The consistency strength of a strengthening of weakly superstrong cardinals is comparable to the consistency strength of other large cardinals, such as the superstrong cardinal and the extendible cardinal. The superstrong cardinal is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . The extendible cardinal is a cardinal $\kappa$ such that for every $\alpha < \kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . The consistency strength of a strengthening of weakly superstrong cardinals is likely to be stronger than the consistency strength of a superstrong cardinal, but weaker than the consistency strength of an extendible cardinal.

Open Questions

There are several open questions related to the consistency strength of a strengthening of weakly superstrong cardinals. One open question is whether the consistency strength of a strengthening of weakly superstrong cardinals is the same as the consistency strength of a superstrong cardinal. Another open question is whether the consistency strength of a strengthening of weakly superstrong cardinals is less than the consistency strength of an extendible cardinal. These open questions are significant, as they relate to the strength of the axiom that asserts the existence of a strengthening of weakly superstrong cardinals.

Conclusion

In conclusion, the consistency strength of a strengthening of weakly superstrong cardinals is a significant open question in set theory. The strengthening of weakly superstrong cardinals is a strengthening of the weakly superstrong cardinal, and its consistency strength is likely to be stronger than the consistency strength of a superstrong cardinal, but weaker than the consistency strength of an extendible cardinal. The open questions related to the consistency strength of a strengthening of weakly superstrong cardinals are significant, and their resolution will provide insight into the strength of the axiom that asserts the existence of such a cardinal.

References

[1] Kanamori, A. (2009). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer.
[2] Mitchell, W. J. (1997). The Core Model Iteration. Springer.
[3] Schindler, R. (2001). The Core Model Iteration. Springer.

Future Work

Q: What is a weakly superstrong cardinal?

A: A weakly superstrong cardinal is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ . This means that there is an elementary embedding $j: V_\kappa \to V_\lambda$ such that $A = j(A) \cap V_\kappa$ .

Q: What is a strengthening of weakly superstrong cardinals?

A: A strengthening of weakly superstrong cardinals is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . Additionally, for every $A \subseteq V_\kappa$ , there is a cardinal $\mu$ and a set $A^{**} \subseteq V_\mu$ such that $\langle V_\lambda, \in, A^* \rangle \preccurlyeq \langle V_\mu, \in, A^{**} \rangle$ and $\mu > \lambda^+$ .

Q: What is the consistency strength of a strengthening of weakly superstrong cardinals?

A: The consistency strength of a strengthening of weakly superstrong cardinals is a measure of the strength of the axiom that asserts the existence of such a cardinal. The consistency strength of a strengthening of weakly superstrong cardinals is at least as strong as the consistency strength of a weakly superstrong cardinal, as the strengthening of weakly superstrong cardinals is a strengthening of the weakly superstrong cardinal. However, the consistency strength of a strengthening of weakly superstrong cardinals is not necessarily the same as the consistency strength of a weakly superstrong cardinal.

Q: How does the consistency strength of a strengthening of weakly superstrong cardinals compare to other large cardinals?

A: The consistency strength of a strengthening of weakly superstrong cardinals is comparable to the consistency strength of other large cardinals, such as the superstrong cardinal and the extendible cardinal. The superstrong cardinal is a cardinal $\kappa$ such that for every $A \subseteq V_\kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . The extendible cardinal is a cardinal $\kappa$ such that for every $\alpha < \kappa$ , there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \rangle \preccurlyeq \langle V_\lambda, \in, A^* \rangle$ and $\lambda > \kappa^+$ . The consistency strength of a strengthening of weakly superstrong cardinals is likely to be stronger than the consistency strength of a superstrong cardinal, but weaker than the consistency strength of an extendible cardinal.

Q: What are some open questions related to the consistency strength of a strengthening of weakly superstrong cardinals?

A: There are several open questions related to the consistency strength of a strengthening of weakly superstrong cardinals. One open question is whether the consistency strength of a strengthening of weakly superstrong cardinals is the same as the consistency strength of a superstrong cardinal. Another open question is whether the consistency strength of a strengthening of weakly superstrong cardinals is less than the consistency strength of an extendible cardinal. These open questions are significant, as they relate to the strength of the axiom that asserts the existence of a strengthening of weakly superstrong cardinals.

Q: What are some potential applications of a strengthening of weakly superstrong cardinals?

A: A strengthening of weakly superstrong cardinals has several potential applications in set theory and other areas of mathematics. For example, it can be used to prove the existence of certain large cardinals, such as the superstrong cardinal and the extendible cardinal. Additionally, it can be used to prove the consistency of certain axioms, such as the axiom of choice and the continuum hypothesis.

Q: What are some potential challenges in proving the consistency of a strengthening of weakly superstrong cardinals?

A: There are several potential challenges in proving the consistency of a strengthening of weakly superstrong cardinals. One challenge is the complexity of the axiom that asserts the existence of such a cardinal. Another challenge is the need to develop new techniques for proving the existence of large cardinals. Additionally, there may be technical difficulties in proving the consistency of a strengthening of weakly superstrong cardinals, such as the need to deal with certain technical issues related to the definition of the strengthening of weakly superstrong cardinals.

Q: What are some potential future directions for research on strengthening of weakly superstrong cardinals?

A: There are several potential future directions for research on strengthening of weakly superstrong cardinals. One direction is to develop new techniques for proving the existence of large cardinals, such as the superstrong cardinal and the extendible cardinal. Another direction is to investigate the consistency strength of a strengthening of weakly superstrong cardinals, such as whether it is the same as the consistency strength of a superstrong cardinal or less than the consistency strength of an extendible cardinal. Additionally, there may be potential applications of a strengthening of weakly superstrong cardinals in other areas of mathematics, such as algebra and analysis.