vacauously true, logic question when there's no matching element

Question

if $P(m)$ is true for all natural numbers $m_0 \le m \lt n$ then $P(n)$ is also true

(Given above statement, we are to prove $P(n)$ is true for all $n \ge m_0$ https://math.stackexchange.com/a/3127180/291153)

The proof begins with $P(m_0)$ is vacously true.
And I wonder what "$P(m_0)$ is vacously true" means?

If we define $Q(n) := P(m)$ is true for all $m_0 \le m \lt m_0+n$

We can rephrase the original statement $Q(n) \implies P(m_0 + n)$

When $n=0$,
$P(m)$ is true for all $m_0 \le m \lt m_0+0$ because there's no such m to check if $P(m)$ is true

So we say $P(m_0)$ is true.

https://en.wikipedia.org/wiki/Vacuous_truth#Scope_of_the_concept gives example where

$\forall x P(x) \implies Q(x) $ If there's no x that satisfies $P(x)$ then $Q(x)$ is true.

When there's no cell phones in a room, you can say cell phones in the room is turned off.

I can see, if there's nothing to talk about, there's no harm saying Q(x) is true because Q has also nothing to talk about.

Here we have something $m_0$ so that we say $P(m_0)$ is true.

So I can construct a better example with the cellphone.

If every cell phone in a room with serial number $m_0 \le m \lt m_0+n$ is mine then cell phone with serial number $m_0+n$ is mine

Since there's no phone in a room with serial number $m_0 \le m \lt m_0+0$, cell phone with serial number $m_0$ is mine .

So $m_0$ is mine is vacuously true.
What does it say?
How should I interpret it?
I can't go to a cellphone store and claim hey the phone with $m_0$ serial number is mine.

Another attempt...

Look mathematicians said, statement of the form: for all x, $P(x) \implies Q(x)$
is always true when there's no x such that $P(x)$ is true.

Say $P(x)$ means x pays for a cellphone, and $Q(x)$ means you give x a cellphone

If any x who pays for a cellphone, you give him a cellphone.
If there's no one who pays for a cellphone, the statement above is still true.
That's good, you can make a manual and put that statement into the manual.

Now, a given day, if sale is good so that you sell every m-th person $m_0 \le m \lt m_0+n$ , you are gonna give a cellphone for free to $m_0 +n$ th person

We can summarize the policy
$P(m)$ is true for all m in $m_0 \le m \lt m_0+n$ then $R(m_0+n)$.
Sounds like a good plan? deal? ok.

So $P(m)$ is true for all m in $m_0 \le m \lt m_0+0$, so $R(m_0)$
Now I know I'm the $m_0$-th customer today, so give me the free cellphone.

seller: what? I didn't say that?
me: no that's the deal, we have the mathematical proof?

What has gone wrong?
Or is it just the convention to interprete the above statement as I did?

I think I can tighten the policy $P(m)$ and $!P(m+1)$ is true for all m in $m_0 \le m \lt m_0+n$ then $R(m_0+n)$, but the argument still seems to hold.

edit)

I moved the following example down, because the above cellphone example is clearer.

if $q(t)$ is true for all t st $t_0 \leq t \lt t_0+n$ then $q(t_0+n)$ is true

what does it mean to say $q(t_0+0)$ is vacuously true?

Suppose $q(t)$ was defined by "there is no earth in time t"

Then we can say "there is no earth in time at $t_0$?
What does it mean when "there is no earth in time at $t_0$ is vacuously true?

Answer 1

How is the "formal" expression of Strong Induction (see Tao I, Proposition 2.2.14 (Strong principle of induction), page 28-29)?

The logical form is:

$\forall m [\forall k (k < m \to P(k)) \to P(m)] \to \forall m P(m)$.

I've considered for simplicity the special case with $m_0=0$.

It is equivalent to The Principle of Mathematical induction (Tao's Axiom 2.5, page 19): this needs several steps to be verified.

What happens to the antecedent of the formula above when $m=0$?

We have that the formula amounts to:

(1) $\forall k (k < 0 \to P(k)) \to P(0)$.

We know, i.e. we prove it from arithmetical axioms, that $\lnot (k < 0)$.

Thus, using tautology: $\lnot P \to (P \to Q)$ in the form: $\lnot (k < 0) \to (k < 0 \to P(k))$ we get (by Modus Ponens and rules for removing and adding the universal quantifier):

(2) $\forall k (k < 0 \to P(k))$.

From (1) and (2), using again MP, we get: $P(0)$.

This is the reason behind Tao's comment [with $m_0$ in place of my $0$]:

In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous."

Having said that, what we have proved is:

$\forall m [\forall k (k < m \to P(k)) \to P(m)] \vdash P(0)$.

For a full proof of SI form MI, see my answer to this post.

Answer 2

Here's my take on this.

We are to prove strong induction

$\forall m [\forall k (k < m \to P(k)) \to P(m)] \to \forall m P(m)$.

s1:

$\forall k (k < m \to P(k)) \to P(m)$

In the problem we are given the above conditional as true statement.

for m = 0, we have the following form

s2: (vacuous truth)

$A(m)$ is an empty set $\to P(m)$
$m=0 \to A(m)$ is empty set

So with that proof structure, I can prove P(0) and ~P(0)
This seems weird, and I wanna know how I should interpret this.

You know $P(0)$ is true because you have the "s1" as a true statement. (Although how you got to know s1 is true is a mystery to me, I think using regular induction is the only way you got there)

So then, next question would be the following.
Let's say I accept the s3, s2 as truth.

s3: (strong induction)

$\forall m [\forall k (k < m \to P(k)) \to P(m)] \to \forall m P(m)$.

Would there be no possibility of me proving s4, where W(m) is inconsistent with P(m)?

s4:

$\forall m [\forall k (k < m \to W(k)) \to W(m)] \to \forall m W(m)$.

The question can be rephrased as,

s5: (regular induction)

$P(0)$.
whenever $P(n)$ is true, $P(n+1)$ is True

s6:

$W(0)$.
whenever $W(n)$ is true, $W(n+1)$ is True

s5 (regular induction), s2 (vacuous truth) implies s3 (strong induction).

conversly, s3 (strong induction), s2 (vacuous truth) can imply s5 (regular induction)

Is it possible for s3, s2 to imply s6? where the proposition(is it the right term? P(x)) to prove is inconsistent with s5 we proved earlier?

For a $P(x)$ statement we know how to construct, we can check the validity of it, but for some $P(x)$ can't be constructed so that we can check the validity.

Well in that case, I guess whether we can prove s5, s6 at the same time doesn't matter, since we have no way to know which one is true.

But there can be times where we just don't know how to construct P(x) yet?

And yet another question is, is this question a meaningful question?