Decius comments on You only need faith in two things - Less Wrong

22 Post author: Eliezer_Yudkowsky 10 March 2013 11:45PM

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Comment author: Decius 11 March 2013 01:39:46AM 0 points [-]

Isn't is possible to trivially generate an order of arbitrary size that is well-ordered?

Comment author: DanielLC 11 March 2013 02:50:11AM 1 point [-]

How?

You can do it with the axiom of choice, but beyond that I'm pretty sure you can't.

Comment author: Qiaochu_Yuan 11 March 2013 04:18:19AM * 2 points [-]

If "arbitrary size" means "arbitrarily large size," see Hartogs numbers. On the other hand, the well-ordering principle is equivalent to AC.

Comment author: Decius 11 March 2013 08:07:33PM 0 points [-]

Take the empty set. Add an element. Preserving the order of existing elements, add a greatest element. Repeat.

Comment author: DanielLC 12 March 2013 01:11:30AM 0 points [-]

That sounds like it would only work for countable sets.

Comment author: Decius 14 March 2013 02:10:20AM 0 points [-]

Is the single large ordinal which must be well-ordered uncountable? I had figured that simply unbounded was good enough for this application.