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redxaxder comments on You only need faith in two things - Less Wrong Discussion
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Every ordinal (in the sense I use the word[1]) is both well-founded and well-ordered.
If I assume what you wrote makes sense, then you're talking about a different sort of ordinal. I've found a paper[2] that talks about proof theoretic ordinals, but it doesn't talk about this in the same language you're using. Their definition of ordinal matches mine, and there is no mention of an ordinal that might not be well-ordered.
Also, I'm not sure I should care about the consistency of some model of set theory. The parts of math that interact with reality and the parts of math that interact with irreplaceable set theoretic plumbing seem very far apart.
[1] An ordinal is a transitive set well-ordered by "is an element of".
[2] www.icm2006.org/proceedings/VolII/contents/ICMVol203.pdf