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Functions in "gaps" in Hardy hierarchy

Recall the definition of Hardy's hierarchy: $H_0(n)=n+1\\ H_{\alpha+1}(n)=H_\alpha(n+1)\\ H_\alpha(n)=H_{\alpha[n]}(n)$, where the last rule applies if $\alpha$ is limit ordinal, and $\alpha[n]$ fo …
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