The question asks whether it is consistent that there exists an inaccessible cardinal κ which is not weakly compact, yet has no κ-Souslin trees.

Key Observations:

1. Tree Property and Weak Compactness: A weakly compact cardinal κ has the tree property (i.e., no κ-Souslin trees exist), but the converse is not necessarily true. The tree property is a combinatorial condition weaker than wea...

Yes. In fact there are well–known sufficient reflection principles at κ which rule out κ–Souslin trees, and these can be made to hold at some inaccessible κ while the tree property still fails there (so κ is not weakly compact).

Two ingredients:

• From a κ–Souslin tree to a nonreflecting stationary set. For a regular uncountable κ, a κ–Souslin tree T yields a nonreflecting stationary subset of E...

Yes – in fact nothing “mystical’’ prevents an inaccessible κ from carrying Aronszajn‐trees (so κ fails the tree–property) while at the same time failing every one of Jensen’s square–principles so that Jensen’s usual construction of a κ–Souslin tree cannot be carried out. Equivalently one can arrange:

1. κ is inaccessible,
2. there is a κ‐Aronszajn tree (so κ is not weakly compact),
3. ...

Yes.
Starting from a weakly compact cardinal one can force a model in which

 • κ is still inaccessible,
 • κ is no longer weakly compact, but
 • the tree property holds at κ, i.e. there are no κ-Souslin (κ-Aronszajn) trees.

Thus the consistency strength needed is no more than that of a weakly compact cardinal.

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1. The idea...

<think> Okay, I need to tackle this question about Souslin trees and weakly compact cardinals. The user is asking if it's consistent that there's an inaccessible cardinal κ that isn't weakly compact but still has no κ-Souslin trees. Let me break this down.

First, recalling some basics: A κ-Souslin tree is a tree of height κ with no branches or antichains of size κ. The tree property at κ means ev...

Yes.

Start with a model V of ZFC in which κ is weakly compact.
Force with the standard club-shooting poset

  P = { c ⊂ κ : c is closed, bounded, ⟨c, <⟩ is increasing
           and every γ∈c has cofinality ω } ,

ordered by end–extension.
Facts about P:

• Size and closure |P| = κ and P is <κ-closed (indeed κ-closed).
• Cardinal preservation Because of κ-closure, no cardinals <κ are coll...

This is an excellent question that gets to the heart of the relationship between different large cardinal properties. The answer is yes, it is consistent.

Short Answer:

Yes, assuming the consistency of a strongly compact cardinal, it is consistent that there exists an inaccessible cardinal that is not weakly compact but has the tree property (and thus no -Souslin trees).

Th...

Souslin Trees and Weakly Compact Cardinals

This question explores the relationship between inaccessible cardinals, weakly compact cardinals, and the existence of κ-Souslin trees.

Understanding the Question

The question asks whether it's consistent with ZFC that there exists an inaccessible cardinal κ such that:

1. κ is not weakly compact
2. There are no κ-Souslin trees

To properly underst...

This is an excellent question about the relationship between Souslin trees and large cardinal properties. Let me provide a comprehensive answer.

Answer: Yes, it is consistent.

The consistency of having an inaccessible cardinal that is not weakly compact but has no -Souslin trees has been established through several different approaches:

1. Lévy Collapse Method (Kunen)

O...