Place potatoes, carrots, celery, onion, and salt into a large pot. Add water to cover and bring to a boil over medium-high heat. Reduce the heat and simmer until potatoes are tender, about 20 minutes; do not drain.

When the vegetables are almost finished, combine milk, butter, and flour in a small saucepan over medium-low heat. Whisk constantly until butter melts and mixture is smooth and thick, 5 to 10 minutes.

Pour hot milk mixture into cooked vegetables. Add corn and Cheddar; cook and stir until Cheddar melts, about 3 minutes.