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Qiaochu_Yuan comments on You only need faith in two things - Less Wrong Discussion
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Comments (86)
This question seems to confuse mathematical induction with inductive reasoning.
So I have. Mathematical induction is, so I see, actually a form of deductive reasoning because its conclusions necessarily follow from its premises.
Mathematical induction is more properly regarded as an axiom. It is accepted by a vast majority of mathematicians, but not all.
Utter pedantry: or rather an axiom schema, in first order languages.
How should I think about the terminologies "faith" and "axiom" in this context? Is this "faith in two things" more fundamental than belief in some or all mathematical axioms?
For example, if I understand correctly, mathematical induction is equivalent to the well-ordering principle (pertaining to subsets of the natural numbers, which have a quite low ordinal). Does this mean that this axiom is subsumed by the second faith, which deals with the well-ordering of a single much higher ordinal?
Or, as above, did Eliezer mean "well-founded?" In which case, is he taking well-ordering as an axiom to prove that his faiths are enough to believe all that is worth believing?
It may be better to just point me to resources to read up on here than to answer my questions. I suspect I may still be missing the mark.
I'm not sure how to answer your specific question; I'm not familiar with proof-theoretic ordinals, but I think that's the keyword you want. I'm not sure what your general question means.