(Bonus Chapter, Today is my birthday)
On Saturday, Richard arrived at his tutor's lessons in a good mood. He expected another round of levitation practice.
Madam Marchbanks, as usual, was waiting for the young man in a chair by the fireplace. Richard's wand lay on the table.
- Good morning, mentor.
The first thing Richard did was pick up his practice wand. He was about to begin performing the levitation spell when the old woman emerged from her reverie.
- Good morning, Richard. I noticed that you are bored with our classes.
- Not that they are boring, but they are too monotonous.
- Well, I suggest we diversify them. Look, my boy.
A large crystal platter of fruit sat on the table in front of Madam Marchbanks. The sorceress waved her wand and levitated an apple from the platter onto the table.
The old woman pointed her wand at the apple, twirled it and said clearly:
- Geminio.
After this, a miracle occurred before Richard's eyes: there were now two apples. They were indistinguishable in appearance.
"Is this an illusion?" the boy asked incredulously.
"No, Richard, it's a copy. Using the Geminio spell, you can create a duplicate of any object, food, or drink. But copies of magical items will be unstable and will not possess their magical properties."
- Can't be!
"Here's an apple," Madame Marchbanks said, pointing to the table. "You can eat it. It's perfectly safe. It will never disappear unless it rots."
"No!" Richard took a step back. "It can't be!" he said with incredible shock. "How is that possible?!"
"Heh-heh-heh!" a croaking laugh erupted from the elderly sorceress's chest. "Oh, Richard, you should have seen your eyes. I've never seen such astonishment. Of course, this isn't the easiest of charms-they're taught in sixth-year Charms-but the spell isn't as complicated as you might think. But the benefits to a wizard from this charm are enormous."
"I don't believe it..." Richard muttered quietly. "Am I really witnessing the Banach-Tarski paradox with my own eyes?! Thousands of students around the world racked their brains trying to figure out how, in theory, something could be created from nothing, and then, with a simple wave of a wand, a replica of the object emerges. An apple from nothing!"
- Heh-heh! Richard, I've never seen such a violent reaction to a doubling spell. What paradox were you mentioning?
"Ma'am..." Richard, stunned, couldn't take his eyes off the apples. "Mathematicians have theoretically proven that it's possible to create a copy of an object from nothing. If you take a sphere and divide it into five parts with non-intersecting points, you can double the sphere. Any two bounded subsets of three-dimensional Euclidean space with a non-empty interior are congruent. One plus one equals one. Infinity plus one equals infinity. Infinity minus one equals infinity. There are an infinite number of numbers from zero to one..."
"Oh, Richard, I see you haven't wasted any time and have taken up numerology!" Madam Marchbanks said enthusiastically. "Most commendable. And Muggle mathematicians are quite the innovators, being able to prove the existence of doubling. Funny, but you guessed it. To cast this spell, you need to concentrate on an object and choose one point on it, then twirl your wand to the left and mentally mark five more points anywhere on the object. Then, mentally divide the object into five parts, and then reassemble them into two complete objects."
"I'm calm... I'm calm... I'm completely calm..." Richard muttered under his breath. "So what if magicians just use the Hausdorff-Banach-Tarski paradox? Magic can do anything... We really need to not just remember it, but accept it..."
- So, Richard, would you like to learn this spell?
- Yes, of course, ma'am!
Richard remembered that he was a Lord and always had to keep up appearances. Although he could be forgiven, since the transmigrator had actually been a Lord for just over a year.
"But, mentor, if this spell can copy objects, then how come wizards haven't collapsed their economy? The most banal thing that comes to mind is to make copies of money and spend it. This would inevitably lead first to rampant inflation, then collapse the entire economy, and wizards would have to return to barter, and even then, only sparingly."
"Richard, first of all, all magical coins are enchanted," Madam Marchbanks explained. "After creation, they are soaked in a cheap potion, which transforms the coins into weak magical items. This means that the duplicate Knut will inevitably be destroyed. There's a simple way to detect such counterfeits-the Finite Reversal Charm. Using counterfeit money will result in punishment."
- Uh-huh. Got it.
"Also, young man, in your first year at Hogwarts, during Transfiguration class, you'll be given five exceptions to Gamp's Law of Transfiguration. One of them states that food cannot be created from nothing, but it can be copied or summoned with a Summoning Charm. However, water or other liquids can be created in this way."
"It fits into the mathematical models of paradoxes," Richard agreed.
"There are still exceptions, but they will tell you about them at school," the old woman continued.
- And now there is no way?
"The second exception is that it is impossible to transfigure magical objects. The third is that it is impossible to transform an intangible object, such as a thought or a ghost. The fourth is that it is impossible to create life through transfiguration. The fifth is that it is impossible to create money, precious stones, and precious metals."
"Impossible or impossible?! Ma'am, while the magical money thing is more or less clear, ordinary people use colored paper. It's probably easier to copy them than an apple. And people have learned to create jewelry, so a wizard could do the same."