Let's analyze the two statements to determine which one is correct.

Statement I: Infinite Irrational Numbers

The first statement claims that the set of all irrational numbers between \(\sqrt{12}\) and \(\sqrt{15}\) is an infinite set.

• First, let's approximate the values: \(\sqrt{12}\) is approximately \(3.464\) and \(\sqrt{15}\) is approximately \(3.873\).
• We are considering the interval approximately \((3.464, 3.873)\).
• A fundamental property of the real number system is the density property. This property states that between any two distinct real numbers, there exists an infinite number of both rational and irrational numbers.
• Since \(\sqrt{12}\) and \(\sqrt{15}\) are distinct real numbers, there must be infinitely many irrational numbers between them.
• Therefore, statement I is correct.

Statement II: Finite Odd Integers

The second statement claims that the set of all odd integers less than 1000 is a finite set.

• An integer is odd if it can be written in the form \(2k + 1\), where \(k\) is an integer.
• The integers less than 1000 include all negative integers, zero, and positive integers up to 999.
• The set of odd integers less than 1000 is \(\{ n \in \mathbb{Z} \mid n \text{ is odd and } n < 1000 \}\).
• This set includes numbers like \(..., -5, -3, -1, 1, 3, 5, ..., 997, 999\).
• Crucially, this set contains all negative odd integers (\(..., -5, -3, -1\)). There are infinitely many negative odd integers.
• A set is finite if its elements can be counted and the count comes to an end. Since the set includes infinitely many negative odd integers, it cannot be counted completely.
• Therefore, the set of all odd integers less than 1000 is an infinite set.
• Statement II is incorrect.

Conclusion

Based on the analysis of both statements:

• Statement I is correct.
• Statement II is incorrect.

Thus, only statement I is correct.