To solve this problem, we need to find the greatest number of consecutive natural numbers greater than 25 that can each be represented as the sum of two different prime numbers, which we refer to as semi-prime.

First, let's identify a few prime numbers as potential candidates: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.

To check if a number $n$ is semi-prime, we look for two distinct primes $p_1$ and $p_2$ such that:
$$n = p_1 + p_2$$

We systematically check consecutive numbers starting from 26 (since the problem states the number must be greater than 25) to form pairs that sum to these numbers.

Let's test some consecutive numbers:

- 26: $26 = 23 + 3$ (both are primes)
- 27: $27 = 19 + 7$ (both are primes)
- 28: $28 = 23 + 5$ (both are primes)
- 29: $29 = 23 + 3$ (both are primes)
- 30: $30 = 23 + 7$ (both are primes)
- 31: $31 = 29 + 2$ (both are primes)

We observe that from 26 to 31, all numbers are semi-prime, providing a total of 6 consecutive numbers.

Let's check the next sequence:

- 32: $32 = 29 + 3$ (both are primes)
- 33: $33 = 31 + 2$ (both are primes)
- ... continue this checking process

After further inspection, we notice that the sequence length where each number is semi-prime stabilizes at 5 for most sets of consecutive numbers checked beyond 26 to 31.

Therefore, the greatest number of consecutive natural numbers greater than 25 that can be semi-prime is:
$$\boxed{5}$$