# Order-isomorphic Morass-definable $\eta_1$-orderings

## TL;DR

This paper demonstrates that under certain set-theoretic assumptions, all morass-definable $ ext{eta}_1$-orderings of size $eth_3$ are order-isomorphic, extending classical results to larger cardinalities using morasses.

## Contribution

It proves the order-isomorphism of morass-definable $ ext{eta}_1$-orderings of size $eth_3$ in a Cohen extension with a simplified gap-2 morass, extending previous results to larger continuum sizes.

## Key findings

- Morass-definable $ ext{eta}_1$-orderings of size $eth_3$ are order-isomorphic.
- Existence of ultrapowers of $ ext{Reals}$ over $ ext{omega}$ that are gap-2 morass-definable.
- Construction techniques extend transfinite back-and-forth to size $eth_3$.

## Abstract

We prove that in the Cohen extension adding $\aleph_3$ generic reals to a model of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, gap-2 morass-definable $\eta_1$-orderings with cardinality $\aleph_3$ are order-isomorphic. Hence it is consistent that the $2^{\aleph_0}=\aleph_3$ and that morass-definable $\eta_1$-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over $\omega$ that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order type $\omega_1$, to a function between objects of cardinality $\aleph_3$.