This system is also sensibly equivalent to the earlier (and much more difficult to grasp) “ordinal diagrams” of Takeuti and formula_2 functions of Feferman: their range is the same (formula_684, which could be called the Takeuti-Feferman–Buchholz ordinal, and which describes the strength of formula_685-comprehension plus bar induction). Most definitions of ordinal collapsing functions found in the recent literature differ from the ones we have given in one technical but important way which makes them technically more convenient although intuitively less transparent. We now explain this. The following definition (by induction on formula_8) is completely equivalent to that of the function formula_1 above:  We can now make a change to the definition which makes it subtly different:  The first values of formula_717 coincide with those of formula_1: namely, for all formula_719 where formula_720, we have formula_721 because the additional clause formula_716 is always satisfied. But at this point the functions start to differ: while the function formula_1 gets “stuck” at formula_89 for all formula_96, the function formula_717 satisfies formula_727 because the new condition formula_716 imposes formula_729. On the other hand, we still have formula_730 (because formula_731 for all formula_419 so the extra condition does not come in play). Note in particular that formula_717, unlike formula_1, is not monotonic, nor is it continuous. Despite these changes, the formula_717 function also defines a system of ordinal notations up to the Bachmann–Howard ordinal: the notations, and the conditions for canonicalness, are slightly different (for example, formula_736 for all formula_8 less than the common value formula_738). As noted in the introduction, the use and definition of ordinal collapsing functions is strongly connected with the theory of ordinal analysis, so the collapse of this or that large cardinal must be mentioned simultaneously with the theory for which it provides a proof-theoretic analysis.