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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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    $\begingroup$ Your order-raising operation doesn't actually raise the order when $j$ is a limit ordinal. $\endgroup$
    – Joel David Hamkins
    Commented Nov 9, 2012 at 14:42
  • $\begingroup$ Yes. In fact I want it for finite order arithmetic. $\endgroup$
    – Colin McLarty
    Commented Nov 9, 2012 at 14:48
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    $\begingroup$ 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... $\endgroup$
    – François G. Dorais
    Commented Nov 9, 2012 at 17:05

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