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.

Explore the mathematical logic problem of determining whether a maximum natural number exists for which all other natural numbers are smaller. This problem involves set theory, natural numbers, and logical quantifiers. Learn why the statement is false and deepen your understanding of infinite sets.

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