1. **State the problem:** We have a totally ordered set $S$ with $n=7$ elements, each element $s_k$ corresponds to the $k$-th successor of zero in Peano arithmetic, i.e., $s_k = k$ for $k=1,2,\ldots,7$. 2. **Understand the summation:** The problem asks for the scalar $\Sigma$ which is the sum of all elements $s_k$ in $S$: $$\Sigma = \bigoplus_{k=1}^7 s_k = \sum_{k=1}^7 k$$ 3. **Formula for sum of first $n$ natural numbers:** $$\sum_{k=1}^n k = \frac{n(n+1)}{2}$$ 4. **Apply the formula for $n=7$:** $$\Sigma = \frac{7 \times (7+1)}{2} = \frac{7 \times 8}{2} = \frac{56}{2} = 28$$ 5. **Interpretation:** The scalar $\Sigma$ is the sum of the first 7 natural numbers, which equals 28. **Final answer:** $$\boxed{28}$$