Completing Kleene's Second Model: Investigating Embeddability and Completions

Core Concepts

Investigating completions of Kleene's second model and its generalizations.

Abstract

This content delves into the investigation of completions of partial combinatory algebras, focusing on Kleene's second model (K2) and its extensions. It explores weak and strong notions of embeddability and completion, highlighting the differences between them. The study reveals that not every pca has a strong completion, contrasting with the consistency that every countable pca has a weak completion. Generalizations of K2 for larger cardinals are considered to show the consistency that every pca has a weak completion. The content also discusses unique head normal forms in K2 and provides arguments for its strong completability.
1. Introduction

Combinatory algebra models computation.
Kleene's second model K2 defined by an application operator on reals.
Application operator described using Turing functionals.
2. Preliminaries

Definition of a partial combinatory algebra (pca).
Properties of combinators k and s in pcas.
Notions of embedding for pcas.
3. Unique head normal forms

Criteria for unique head normal forms in pcas.
Verification that K2 satisfies these criteria.
Arguments showing K2 is strongly completable.
4. Weak versus strong embeddings

Strong completability implies differences between weak and strong embeddings.
Example demonstrating weakly completable but not strongly completable pca.
5. K2 generalized

Generalization of K2 to larger cardinals like Kκ2.
Coding functions for partial continuous functions in large cardinals.
6. Weak completability of all pcas

Consistency result showing all pcas have a weak completion under GCH assumption.
References
List of references cited in the content.

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Completions of Kleene's second model

by Sebastiaan A... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2312.14656.pdf

Deeper Inquiries

Is it possible to prove Theorem 6.2 absolutely?

The proof of Theorem 6.2 relies on assuming the Generalized Continuum Hypothesis (GCH). Since GCH is an assumption based on set theory, the proof of Theorem 6.2 is contingent upon this assumption. To prove Theorem 6.2 absolutely would require establishing its validity without relying on any assumptions like GCH or other axioms that are not universally accepted as true in all contexts.

What implications does GCH have on the existence of large pcas?

The Generalized Continuum Hypothesis (GCH) plays a crucial role in determining the cardinality and structure of large partial combinatory algebras (pcas). Under GCH, which posits a specific relationship between different infinite cardinal numbers, such as $\omega$ and $2^{\omega}$, we can ensure that certain cardinals satisfy conditions necessary for constructing pcas with desired properties.
In particular, when considering large pcas like Kκ_2 where κ<κ = κ, assuming GCH ensures that these structures can be well-defined and consistent within set theory frameworks. Without GCH or similar assumptions about cardinal arithmetic, proving results related to large pcas may become more challenging due to uncertainties about their cardinalities and internal structures.

How do unique head normal forms impact the study of realizability?

Unique head normal forms play a significant role in realizability theory by providing a structured way to represent computations within partial combinatory algebras (pcas). In realizability theory, having unique hnfs allows for clear distinctions between different types of computations and facilitates reasoning about computable functions and their properties.
By ensuring that each element in a pca has a distinct hnf representation, researchers can analyze computability aspects more effectively. Unique hnfs also contribute to defining embeddings between pcas accurately since they provide unambiguous representations for elements within these algebraic structures.
Overall, unique head normal forms enhance the precision and clarity of realizability studies by offering standardized formats for expressing computational processes within partial combinatory algebras.