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I'm working in computability theory and need to use partial functions with finite domain from $\omega$ to 2 as approximations in my current paper. Normally this is simply done using $2^{< \omega}$ but I need partial functions whose domain isn't an initial segment. Is there any vaguely standard notation for this?

Note that specifically, I need domains that include arbitrary finite initial segments of each column so if you want to suggest a notation that works for that but not all finite partial functions from $\omega$ to 2 that's fine as well but I don't need to enforce that constraint so I'm fine with a general notation for a finite partial function.

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    $\begingroup$ $\mathrm{Add}(\omega,1)$ $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 5, 2021 at 12:46
  • $\begingroup$ @AsafKaragila What is Add abbreviating here? and shouldn't that be a 2 since the range is {0,1} = 2? Or am I misunderstanding? $\endgroup$
    – Peter Gerdes
    Commented Feb 5, 2021 at 14:25
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    $\begingroup$ @PeterGerdes The Add$(\omega,1)$ notation refers to "the" forcing notion that adds $1$ Cohen-generic subset of $\omega$. Unfortunately, $2^{<\omega}$ and the larger set that you want are equivalent as forcing notions, so the notation Add$(\omega,1)$ can be used for either of them. $\endgroup$
    – Andreas Blass
    Commented Feb 5, 2021 at 14:32
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    $\begingroup$ Kunen uses $Fn(\omega,2,\omega)$. $\endgroup$
    – Monroe Eskew
    Commented Feb 5, 2021 at 17:03
  • $\begingroup$ @AndreasBlass Ahh, that's not that helpful then but thanks. $\endgroup$
    – Peter Gerdes
    Commented Feb 5, 2021 at 18:17

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