We investigate the eliminability of the absoluteness operator Delta in Goedel logics. While Delta is not definable from the standard connectives and disrupts important proof-theoretic properties, we show that it becomes eliminable at the propositional level under a restricted semantics in which all propositional atoms (except the truth constant 'True') are interpreted strictly below 1. Under this semantics, every formula containing Delta is equivalent to a disjunction of chain formulas, yielding a Delta-free normal form (standard and restricted semantics coincide w.r.t. valid formulas without Delta). We further analyze the situation in the first-order setting, where Delta-elimination fails in general due to recursion-theoretic and topological constraints, but can be recovered under witnessed semantics.

翻译：我们探讨了哥德尔逻辑中绝对性算子Δ的可消去性。虽然Δ无法从标准连接词中定义，并且破坏了重要的证明论性质，但我们证明，在一种受限语义下——其中所有命题原子（除真值常量“真”外）均被解释为严格小于1——Δ在命题层次上变得可消去。在此语义下，每个包含Δ的公式都等价于一个链公式的析取，从而得到一个无Δ的范式（标准和受限语义在无Δ有效公式方面是一致的）。我们进一步分析了在第一个有序层次中的情况，在那里，由于递归论和拓扑约束，Δ消去在一般情况下不成立，但在有见证语义下可以恢复。