# shinoteki comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong

8 25 August 2011 02:17AM

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Comment author: 05 September 2011 09:06:48PM *  3 points [-]

You don't need Z, third-order arithmetic is sufficient. Every set of ordinals is well-ordered by the usual ordering of ordinals.

Comment author: 05 September 2011 09:30:10PM 0 points [-]

Only if you accept excluded middle.

Comment author: 09 September 2011 03:17:59AM *  2 points [-]

That depends on what you mean by "well-ordered". My philosophy of doing constructive mathematics (mathematics without excluded middle, and often with other restrictions) is that one should define terms as much as possible so that the usual theorems (including the theorems that the motivating examples are examples) become true, so long as the definitions are classically (that is using the usually accepted axioms) equivalent to the usual definitions.

As the motivating example of a well-ordered set is the set of natural numbers, we should use a definition that makes this an example. Such a definition may be found at a math wiki where I contribute my research (such as it is). Then (adopting a parallel definition of "ordinal") it remains a theorem that every set of ordinals is well-ordered.