Journal of Symbolic Logic 85 (1):300-324 (2020)

Abstract
We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets $\mathbb{P}_{emb} $ equipped with the order induced by homomorphisms is embedded into the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain. We then show that $\mathbb{P}_{emb} $ admits both infinite strictly decreasing chains and infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the $\Delta _2^0 $-degrees of the Scott domain.
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DOI 10.1017/jsl.2019.51
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References found in this work BETA

Continuous Reducibility and Dimension of Metric Spaces.Philipp Schlicht - 2018 - Archive for Mathematical Logic 57 (3-4):329-359.
Quasi-Polish Spaces.Matthew de Brecht - 2013 - Annals of Pure and Applied Logic 164 (3):356-381.
A Wadge Hierarchy for Second Countable Spaces.Yann Pequignot - 2015 - Archive for Mathematical Logic 54 (5-6):659-683.
Hierarchies in Φ‐Spaces and Applications.Victor L. Selivanov - 2005 - Mathematical Logic Quarterly 51 (1):45-61.
Some Notes on the Universality of Three-Orders on Finite Labeled Posets.Anton V. Zhukov - 2014 - In Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies. De Gruyter. pp. 393-410.

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