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Universal wide Aronszajn tree

Siiri Kivimäki

TL;DR

The paper proves, under the consistency of a weakly compact cardinal, that there exists a universal wide $\aleph_1$-Aronszajn tree with respect to strong embeddings. It achieves this via a two-phase forcing construction: collapse the weakly compact cardinal to $\aleph_1$ and introduce a name for a wide $\kappa$-Aronszajn tree $\dot T$, then perform a finite-support iteration with side conditions to embed every wide $\kappa$-Aronszajn tree into $\dot T$. The work builds a robust framework of residues and quotient-residue maps to preserve the Aronszajn property throughout the iteration and to guarantee no new branches arise, yielding universality at $\aleph_1$. It complements and extends prior results on universality in non-elementary classes and suggests potential paths to reduce large-cardinal assumptions via Diamond principles. The result provides a concrete, forcing-based method to obtain a master tree that contains isomorphic copies of all wide Aronszajn trees under strong embeddings.

Abstract

A wide Aronszajn tree is a tree of size $\aleph_1$ with no uncountable branches. Assuming the consistency of the existence of a weakly compact cardinal, we show the consistency of the existence of a wide Aronszajn tree that is \textit{universal} in the sense that it contains an isomorphic copy of every wide Aronszajn tree.

Universal wide Aronszajn tree

TL;DR

The paper proves, under the consistency of a weakly compact cardinal, that there exists a universal wide -Aronszajn tree with respect to strong embeddings. It achieves this via a two-phase forcing construction: collapse the weakly compact cardinal to and introduce a name for a wide -Aronszajn tree , then perform a finite-support iteration with side conditions to embed every wide -Aronszajn tree into . The work builds a robust framework of residues and quotient-residue maps to preserve the Aronszajn property throughout the iteration and to guarantee no new branches arise, yielding universality at . It complements and extends prior results on universality in non-elementary classes and suggests potential paths to reduce large-cardinal assumptions via Diamond principles. The result provides a concrete, forcing-based method to obtain a master tree that contains isomorphic copies of all wide Aronszajn trees under strong embeddings.

Abstract

A wide Aronszajn tree is a tree of size with no uncountable branches. Assuming the consistency of the existence of a weakly compact cardinal, we show the consistency of the existence of a wide Aronszajn tree that is \textit{universal} in the sense that it contains an isomorphic copy of every wide Aronszajn tree.

Paper Structure

This paper contains 13 sections, 33 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Wide Aronszajn trees
  3. Preliminaries
  4. The poset
  5. Adding a flexible tree
  6. Adding embeddings
  7. Properties of the poset
  8. Node density
  9. Strong Properness
  10. Keeping $\dot T$ Aronszajn
  11. Residue systems in quotients
  12. No new branches
  13. Conclusion

Key Result

Theorem 1.4

Two countable models $A$ and $B$ are non-isomorphic if and only if there is a countable ordinal $\beta<\omega_1$ such that player $\mathop{\mathrm{\mathsf{II}}}\nolimits$ has a winning stategy in $\mathop{\mathrm{\mathsf{EF}}}\nolimits^\alpha(A,B)$ for $\alpha\leqslant\beta$ and player $\mathop{\mat

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Scott scott2014logic
  • Theorem 1.5: Karp Karp1965-KARFQE
  • Theorem 1.6: Scott's isomorphism theorem scott2014logic
  • Definition 1.7
  • Theorem 1.8: Hyttinen, Väänänen hyttinen1990scott
  • Definition 1.9
  • Lemma 1.10
  • ...and 77 more