This is the principle of complete induction, which establishes the property of vicious circle.
2.
Another proof by complete induction uses the hypothesis that the statement holds for " all " smaller " n " more thoroughly.
3.
Complete induction is equivalent to ordinary mathematical induction as described above, in the sense that a proof by one method can be transformed into a proof by the other.
4.
Becker was correct that complete induction was needed for assertions of consistency in the form of universally quantified sentences, as opposed to claiming that a predicate holds for each individual natural number.
5.
If, on the other hand, P ( " n " ) had been proven by ordinary induction, the proof would already effectively be one by complete induction : P ( 0 ) is proved in the base case, using no assumptions, and is proved in the inductive step, in which one may assume all earlier cases but need only use the case P ( " n " ).
6.
These thinkers seem to have maintained a modified observational standpoint for the "'introduction of natural numbers "', for "'the principle of complete induction "'[ . . . ] For these, even for such theorems as were deduced by means of classical logic, they postulated an existence and exactness independent of language and logic and regarded its non-contradictority as certain, even without logical proof.
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