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  1. A syntactic approach to Borel functions: some extensions of Louveau’s theorem.Takayuki Kihara & Kenta Sasaki - 2023 - Archive for Mathematical Logic 62 (7):1041-1082.
    Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $$\Gamma $$, then its $$\Gamma $$ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -function, then one can find its $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -code hyperarithmetically relative to its Borel code. More generally, we prove (...)
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  • The wadge order on the Scott domain is not a well-quasi-order.Jacques Duparc & Louis Vuilleumier - 2020 - Journal of Symbolic Logic 85 (1):300-324.
    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 (...)
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