“$P$ is true” is true

Question

Here is an example of a proof by induction that my teacher gave:

Let's prove by induction over $n\in\mathbb N$ the property $P(n):$«$n+3>2$».

• $P(0)$ is clear
• $\forall n\in\mathbb N$, $(n+3)+1>n+3>2$ so $P(n)\implies P(n+1)$

Thus $\forall n\in\mathbb N$, $P(n)$ is true.

I was wondering why we need to specify “is true” in the conclusion rather than just writing $P(n)$. It seems to me that any argument to that effect would force us to write “$P(n)$ is true” is true; “‘$P(n)$ is true’ is true” is true and so on.

Answer 1

The inductive argument shows that $\forall n \ P(n)$. No need to paste "is true" after that ...

However, you could choose to add the "is true" for reading sake; there is nothing against that. And, just because you choose to do so does not force you to get the infinite regress as indicated.