I missed this week's lecture. I really dislike missing lectures, as it takes me extra time to catchup to the material I missed. Looking at the notes it seems I missed the introduction to the Principle of Well Ordering. I can sort of see how knowing that there exists a smallest value can induce a proof for some problems. I found it quite interesting to know that simple induction, principle of well ordering, and complete induction are all correlated. In the proofs of the relations between the principles I see that we use the anticedant for each proof, however is there a way to not use the anticedant?
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Each principle has the structure antecedent implies consequent:
PWO(A) implies PWO(C)
PCI(A) implies PCI(C)
PSI(A) implies PSI(C)
When we say they are equivalent, we mean that one implication implies the other, for example:
[PCI(A) implies PCI(C)] implies [PSI(A) implies PSI(C)]. So, if you assume all of the PCI, then you need to show that when you assume PSI(A), you derive PSI(C).
Hope that's not too cryptic.
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