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private val items: MutableList<T?> = mutableListOf() can be written more succinctly as private val items = mutableListOf<T?>() (the type is inferred and then no longer repeated (mostly) in an explicit type definition and the factory method call).

size can be backed by items.size:

 val size: Int
     get() = items.size

 fun push(item: T) {
     set(size + 1, item)
     siftUp(size - 1)
 }

 ...

 fun pop(): T? {
     ...
     set(0, get(size - 1))
     items.removeAt(items.lastIndex) // Remove original reference to the last item
     ...
 }

siftDown throws an AssertionError assertion when size == 0.