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It is a principle and proof from Introduction to Set Theory, Hrbacek and Jech.

In the proof, line 1 and 2, I couldn't understand why $Q(0)$ is true.

$Q(0)$ means that "$P(k)$ holds for all $k<0$".

I understood there are no $k<0$.

And then I couldn't proceed.

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2 Answers 2

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$Q(n)$ is the predicate "$\forall k\in{\Bbb N}_0 (k<n\Rightarrow P(k))$." Then $Q(0)$ is vacuously true, since the premise is false and so the implication is true.

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A sentence like "for all $x$, if $x$ has the property $A$, then $x$ has the property $B$" is false if (and only if) there exists a counterexample, that is, if there exists some $x$ with the property $A$ but without the property $B$.

If there is no $x$ that has the property $A$, then there is no counterexample, so the sentence is true.

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