Induction and Universal Generalization

Question

Let $(\mathbb N,<)$ be the set of natural numbers as defined in the book Introduction to Set Theory by Hrbacek and Jech.

Suppose you are asked to show that $n<m$ implies $n+1\leq m$ for all $m,n\in\mathbb N$. Consider the following proof :

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Let $P(n,m)$ denote the property that $n<m$ implies $n+1\leq m$.

Fix some arbitrary $n\in \mathbb N$. We will show that $P(n,m)$ holds for all $m\in \mathbb N$ by induction on $m$.

$P(n,0)$ holds vacuously since $n<0$ is false. Suppose $P(n,m)$ holds for some $m\in\mathbb N$ and suppose that $n<m+1$. Then $n\leq m$. If $n<m$ then $n+1\leq m $ by the induction hypothesis, and if $n=m$ then $n+1=m+1$. In either case $n+1\leq m+1$. By the induction principle we conclude that $P(n,m)$ holds for all $m\in \mathbb N$. As $n$ is arbitrary the conclusion follows.

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It seems to me that this proof is ok. However the answer here seems to suggest that it is incorrect.

Am I missing something?

Answer 1

I don’t see how the answer you linked to suggests that your proof is incorrect. It seems fine to me (though somewhat unnecessary, since $n\lt m$ trivially implies $n+1\le m$).

The linked answer doesn’t say that you can’t ever prove something for all $m,n\in\mathbb N$ by fixing arbitrary $n$ and performing induction only over $m$. I don’t fully agree with its analysis of what was wrong in the question, but as far as it’s correct, it just says, in the context of a question that was full of quantifiers and should have had a universal quantifier for $n$, that this quantifier was missing and that this was related to a flaw in the proof.

The differences from your case are that a) your proof is in a less formalized style, it doesn’t contain a single quantifier, so it’s OK to just imply the universal quantifier over $n$ by “fixing some arbitary $n\in\mathbb N$”, and b) there’s no flaw in your proof related to this informality.