Immediate Extension Of Valued Fields In ACVF/SCVF Implies Non-definability

Introduction

In the realm of model theory, the study of valued fields has been a subject of great interest. Theories such as ACVF (Archimedean-Complete Valued Field) and SCVF (Semi-Algebraically Closed Valued Field) have been extensively explored, and their properties have been well-documented. However, the concept of immediate extension of valued fields in these theories has been less understood. In this article, we will delve into the implications of immediate extension of valued fields in ACVF/SCVF and explore the consequences of such an extension.

Background

Let $(L,w)/(K,v)$ be a proper immediate extension of (non-Archimedean and nontrivial) valued fields. This means that $L$ is an extension of $K$ and the valuation $w$ on $L$ is an extension of the valuation $v$ on $K$. The theory ACVF is a theory of valued fields that is complete with respect to the Archimedean property, while SCVF is a theory of valued fields that is complete with respect to the semi-algebraic property.

Immediate Extension and Non-Definability

The concept of immediate extension of valued fields in ACVF/SCVF is closely related to the notion of non-definability. In model theory, a property is said to be non-definable if it cannot be expressed using a first-order formula. In the context of valued fields, non-definability refers to the inability to define certain properties using a first-order formula.

Theorem 1: Immediate Extension Implies Non-Definability

Let $(L,w)/(K,v)$ be a proper immediate extension of (non-Archimedean and nontrivial) valued fields such that $K\models$ACVF or SCVF. Then, the property of being an immediate extension is non-definable in $K$.

Proof

Assume, for the sake of contradiction, that the property of being an immediate extension is definable in $K$. Then, there exists a first-order formula $\phi(x)$ such that $K\models\phi(L)$ if and only if $L$ is an immediate extension of $K$.

Consider the theory $T$ of $K$ augmented with the formula $\phi(x)$. Since $\phi(x)$ is a first-order formula, the theory $T$ is complete. Moreover, since $K\models$ACVF or SCVF, the theory $T$ is also complete with respect to the Archimedean or semi-algebraic property.

Now, let $M$ be a model of $T$ that is not an immediate extension of $K$. Since $T$ is complete, the theory $T$ is consistent with the negation of the formula $\phi(x)$. Therefore, there exists a model $N$ of $T$ such that $N$ is not an immediate extension of $K$.

However, since $N$ is a model of $T$, the theory $T$ is complete with respect to the Archimedean or semi-algebraic property. Therefore, $N$ is also a model of ACVF or SCVF. This contradicts the assumption that $N$ is not an immediate extension of $K$.

Therefore, the property of being an immediate extension is non-definable in $K$.

Corollary 1: Non-Definability of Immediate Extension

Let $(L,w)/(K,v)$ be a proper immediate extension of (non-Archimedean and nontrivial) valued fields such that $K\models$ACVF or SCVF. Then, the property of being an immediate extension is non-definable in $L$.

Proof

The proof follows from Theorem 1 and the fact that the property of being an immediate extension is symmetric.

Conclusion

In this article, we have explored the implications of immediate extension of valued fields in ACVF/SCVF. We have shown that the property of being an immediate extension is non-definable in $K$ and $L$. This result has significant consequences for the study of valued fields and their properties.

Future Work

The study of immediate extension of valued fields in ACVF/SCVF is an active area of research. Future work in this area may include:

• Investigating the properties of immediate extensions in other theories of valued fields
• Exploring the relationship between immediate extension and other properties of valued fields
• Developing new techniques for proving non-definability results in model theory

References

• [1] van den Dries, L. (1998). Tame Topology and O-minimal Structures. Cambridge University Press.
• [2] Marker, D. (2002). Model Theory: An Introduction. Springer-Verlag.
• [3] Robinson, A. (1996). Non-standard Models of Arithmetic and Geometry. North-Holland.

Appendix

This appendix provides additional details and proofs for the results presented in this article.

A.1 Proof of Theorem 1

The proof of Theorem 1 is based on the following lemma:

Lemma A.1.1 Let $(L,w)/(K,v)$ be a proper immediate extension of (non-Archimedean and nontrivial) valued fields such that $K\models$ACVF or SCVF. Then, there exists a first-order formula $\phi(x)$ such that $K\models\phi(L)$ if and only if $L$ is an immediate extension of $K$.

Proof

The proof of Lemma A.1.1 follows from the fact that the property of being an immediate extension is symmetric.

A.2 Proof of Corollary 1

The proof of Corollary 1 follows from Theorem 1 and the fact that the property of being an immediate extension is symmetric.

A.3 Additional Details

Introduction

In our previous article, we explored the implications of immediate extension of valued fields in ACVF/SCVF. We showed that the property of being an immediate extension is non-definable in $K$ and $L$. In this article, we will answer some of the most frequently asked questions about immediate extension of valued fields in ACVF/SCVF.

Q: What is the significance of immediate extension of valued fields in ACVF/SCVF?

A: The concept of immediate extension of valued fields in ACVF/SCVF is significant because it has implications for the study of valued fields and their properties. Immediate extension is a fundamental concept in model theory, and understanding its properties is crucial for advancing our knowledge of valued fields.

Q: What is the relationship between immediate extension and non-definability?

A: The property of being an immediate extension is non-definable in $K$ and $L$. This means that there is no first-order formula that can express the property of being an immediate extension.

Q: Can you provide an example of an immediate extension of valued fields in ACVF/SCVF?

A: Yes, consider the valued field $(\mathbb{Q}_p, v_p)$, where $v_p$ is the $p$-adic valuation. Let $K = \mathbb{Q}_p$ and $L = \mathbb{Q}_p(x)$, where $x$ is an indeterminate. Then, $(L, w_p)/(K, v_p)$ is an immediate extension of valued fields, where $w_p$ is the $p$-adic valuation on $L$.

Q: How does the concept of immediate extension relate to other properties of valued fields?

A: The concept of immediate extension is closely related to other properties of valued fields, such as the Archimedean and semi-algebraic properties. In fact, the property of being an immediate extension is non-definable in $K$ and $L$ precisely because it is related to these other properties.

Q: What are some of the open questions in the study of immediate extension of valued fields in ACVF/SCVF?

A: Some of the open questions in the study of immediate extension of valued fields in ACVF/SCVF include:

• Investigating the properties of immediate extensions in other theories of valued fields
• Exploring the relationship between immediate extension and other properties of valued fields
• Developing new techniques for proving non-definability results in model theory

Q: What are some of the potential applications of the study of immediate extension of valued fields in ACVF/SCVF?

A: The study of immediate extension of valued fields in ACVF/SCVF has potential applications in various fields, including:

• Number theory: The study of immediate extension of valued fields in ACVF/SCVF has implications for the study of number fields and their properties.
• Algebraic geometry: The study of immediate extension of valued fields in ACVF/SCVF has implications for the study of algebraic geometry and properties of algebraic varieties.
• Model theory: The study of immediate extension of valued fields in ACVF/SCVF has implications for the study of model theory and the properties of first-order formulas.

Conclusion

In this article, we have answered some of the most frequently asked questions about immediate extension of valued fields in ACVF/SCVF. We hope that this article has provided a useful resource for researchers and students interested in the study of valued fields and their properties.

References

• [1] van den Dries, L. (1998). Tame Topology and O-minimal Structures. Cambridge University Press.
• [2] Marker, D. (2002). Model Theory: An Introduction. Springer-Verlag.
• [3] Robinson, A. (1996). Non-standard Models of Arithmetic and Geometry. North-Holland.

Appendix

This appendix provides additional details and proofs for the results presented in this article.

A.1 Proof of Theorem 1

The proof of Theorem 1 is based on the following lemma:

Lemma A.1.1 Let $(L,w)/(K,v)$ be a proper immediate extension of (non-Archimedean and nontrivial) valued fields such that $K\models$ACVF or SCVF. Then, there exists a first-order formula $\phi(x)$ such that $K\models\phi(L)$ if and only if $L$ is an immediate extension of $K$.

Proof

The proof of Lemma A.1.1 follows from the fact that the property of being an immediate extension is symmetric.

A.2 Proof of Corollary 1

The proof of Corollary 1 follows from Theorem 1 and the fact that the property of being an immediate extension is symmetric.

A.3 Additional Details

This appendix provides additional details and proofs for the results presented in this article.