The statement you're asking about is:

∃m ∈ N ∀n ∈ N: n ≤ m
which translates to:

"There exists an $m$ in the set of natural numbers $N$ such that for every $n$ in $N$, $n \leq m$."

Understanding the Statement

• $∃m ∈ N$: There exists some natural number $m$.
• $∀n ∈ N$: For every natural number $n$.
• $n \leq m$: $n$ is less than or equal to $m$.

Is this True?

In the set of natural numbers $N = \{1, 2, 3, \dots \}$, for any number $m$ that you might choose, there will always be a number $n$ larger than $m$. For example, if you choose $m = 3$, then $n = 4$ would not satisfy $n \leq m$. In fact, the set of natural numbers is infinite, so for any natural number $m$, there will always be numbers larger than $m$.

Thus, the statement is false, because there is no maximum natural number $m$ that can satisfy $n \leq m$ for all $n \in N$.

Conclusion

The statement is false because no such natural number $m$ exists that is greater than or equal to all other natural numbers.

Would you like a detailed explanation of why this is the case, or any clarification on specific parts?

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Here are five related questions to deepen your understanding:

1. What is the formal definition of the natural numbers $N$?
2. Can there be a largest number in a set that is infinite?
3. How does this statement change if we consider a different number set, like the integers?
4. What happens to the logic if we introduce inequalities like $n \geq m$?
5. How do we prove a statement is false in mathematical logic?

Tip: In infinite sets, the concept of "maximum" is often non-existent unless specifically bounded.