Hello everyone! Apologies for skipping a week – Naz’s Spring Break was a week ago, and I’m still catching up after accompanying a group of students to Hungary. Fortunately, the topic for this week’s Monday Morning Math fell into my lap (well, inbox – thanks Mark!) Last Monday (March 20), the BBC posted posted “The numbers that are too big to imagine” about infinity. Here’s a quote from the original article.
Some infinities, [Cantor] showed, are bigger than others.
How so? One of the simplest ways to understand why is to imagine the set of all the even numbers. This would be infinite, right? But it must be smaller than the set of all whole numbers, because it does not contain the odd numbers. Cantor proved that when you compare such sets, they contain numbers that do not match up, therefore there must be multiple sizes of infinity.
If you’ve taken an introduction to Proof class, you might remember the idea of sizes of infinity, but you might also remember that the whole numbers and even numbers are actually the same size of infinity. The Cantor proof referenced in the paragraph above doesn’t exist.
The reason for the error? Maybe Richard Fisher, the author, just made a mistake. Or maybe the mistake was deeper than that. At the end of the article was the quote:
The author used ChatGPT to research trusted sources and calculate parts of this story.
The error was found quickly, and corrected within a day. The article now reads:
Some infinities, [Cantor] showed, are bigger than others.
How so? To understand why, consider the numbers as ‘sets’. If you were to compare all natural numbers (1, 2, 3, 4, and so on) in one set, and all the even numbers in another set, then every natural number could in principle be paired with a corresponding even number. This pairing suggests the two sets – both infinite – are the same size. They are ‘countably infinite’.
However, Cantor showed that you can’t do the same with the natural numbers and the ‘real’ numbers – the continuum of numbers with decimal places between 1, 2, 3, 4 (0.123, 0.1234, 0.12345 and so on.)
If you attempted to pair up the numbers within each set, you could always find a real number that does not match up with a natural number. Real numbers are uncountably infinite. Therefore, there must be multiple sizes of infinity.
Yay! A nice explanation of the different sizes of infinity. The explanation about ChatGPT also states, “For the sake of clarity, BBC Future does not use generative AI as a primary source or to replace the journalism needed for our articles.“
Indeed – as useful as Artificial Intelligence can be, it doesn’t replace the need for understanding and evaluating what it generates.
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