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Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of course this can be iterated any number of times.

Is there a standard, or at least published, name for this order raising operation?

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Your order-raising operation doesn't actually raise the order when $j$ is a limit ordinal. – Joel David Hamkins Nov 9 '12 at 14:42
Yes. In fact I want it for finite order arithmetic. – Colin McLarty Nov 9 '12 at 14:48
The letter $i$ is often used for the map $x\mapsto\lbrace x\rbrace$. I think this goes back to Russel & Whitehead or even earlier than that... – François G. Dorais Nov 9 '12 at 17:05

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