Does induction state a fact about the territory or the map? Is it more akin to “The information processing influencing my sensory inputs actually has to a processor in which P(0) & [P(0) & P (1) & … & P(n) → P(n+1)] for all propositions P and natural n?” Or is it “my own information processor is one for which P(0) & [P(0) & P (1) & … & P(n) → P(n+1)] for all propositions P and natural n?”
It seems like the second option is true by definition (by the authoring of the AI, we simply make it so because we suppose that is the way to author an AI to map territories). This supposition itself would be more like the first option.
I’m guessing I’m probably just confused here. Feel free to dissolve the question.
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.
Does induction state a fact about the territory or the map? Is it more akin to “The information processing influencing my sensory inputs actually has to a processor in which P(0) & [P(0) & P (1) & … & P(n) → P(n+1)] for all propositions P and natural n?” Or is it “my own information processor is one for which P(0) & [P(0) & P (1) & … & P(n) → P(n+1)] for all propositions P and natural n?”
It seems like the second option is true by definition (by the authoring of the AI, we simply make it so because we suppose that is the way to author an AI to map territories). This supposition itself would be more like the first option.
I’m guessing I’m probably just confused here. Feel free to dissolve the question.
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.
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.
Utter pedantry: or rather an axiom schema, in first order languages.