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Add ordinal exponentiation notes to iset.mm #4611

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Jan 31, 2025
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12 changes: 11 additions & 1 deletion iset.mm
Original file line number Diff line number Diff line change
Expand Up @@ -61188,7 +61188,17 @@ defined for all sets (being defined for all ordinals might be enough if
This definition is similar to a conventional definition of
exponentiation except that it defines ` (/) ^oi A ` to be ` 1o ` for all
` A e. On ` , in order to avoid having different cases for whether the
base is ` (/) ` or not. (Contributed by Mario Carneiro, 4-Jul-2019.) $)
base is ` (/) ` or not.

We do not yet have an extensive development of ordinal exponentiation.
For background on ordinal exponentiation without excluded middle, see
Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu
(2025), "Ordinal Exponentiation in Homotopy Type Theory",
arXiv:2501.14542 , ~ https://arxiv.org/abs/2501.14542 which is
formalized in the TypeTopology proof library at
~ https://ordinal-exponentiation-hott.github.io/ .

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Hah! I was just reading that paper today!

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You are a step ahead of me; I'll confess I only got to the abstract.

(Contributed by Mario Carneiro, 4-Jul-2019.) $)
df-oexpi $a |- ^oi = ( x e. On , y e. On |->
( rec ( ( z e. _V |-> ( z .o x ) ) , 1o ) ` y ) ) $.
$}
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5 changes: 5 additions & 0 deletions mmil.raw.html
Original file line number Diff line number Diff line change
Expand Up @@ -3718,6 +3718,11 @@
<TD>~ om0 </TD>
</TR>

<tr>
<td>df-oexp</td>
<td>~ df-oexpi</td>
</tr>

<TR>
<TD>oa0r</TD>
<TD><I>none</I></TD>
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