= 27d567bc2d48d9f40e9ccc20ea370b15)
redxaxder comments on You only need faith in two things - Less Wrong Discussion
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (86)
This phrase confuses me:
Every definition I've seen of ordinal either includes well-ordered or has that as a theorem. I'm having trouble imagining a situation where it's necessary to use the well-orderedness of a larger ordinal to prove it for a smaller one.
*edit- Did you mean well-founded instead of well-ordered?
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