Hey
Is structural induction the same as complete induction? That is, does it begin with the induction beginning, assumption, and step?
And could someone solve this or at least one of these so I can see what it's like.
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A complete induction is a special case of structural induction, which is built up above the amount of natural numbers.
Let us consider the natural figures. With these, I can say:
1) 0 is in natural numbers
2) if x is in the natural numbers, is also succ(x) in the natural numbers.
This means that we have structured this amount. We therefore know (way 1)) that 0 is in it and (way 2)) also succ(0)=1, and then succ(1)=2, etc.
An induction works ONLY because of these rules. If we have a statement to show, then we show them for the 0 (IA), and then we assume that there is an x, for that the statement applies (IV) and show that it must also apply to (x+1) (IS). By accepting these IV, we can (and must) use them in IS.
We can also do the same for other structurally constructed quantities. For example, in your example, or in general in different formula systems.
You have the atomic formulas (like the “0” in N), for which you show the statement in IS. Then you have different rules that build more formulas from atomic formulas. For example, if atomic formulae A and B are given, ‘A implies B’ is a formula’ etc. You therefore assume in the IV that the statement applies to atomic formulas (for example A and B), and then in the IS that it also applies to “not A”, “A and B”, ….