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Claim: digital and countable infinity are the same size; or, \(|\mathscr{D}| = |\mathbb{N}|\).

Countable infinity is the size of the set \(\mathbb{N} = \{0, 1, 2, 3, …\}\).

Remember that by digital infinity we mean the collection of all digital information ever. All the books, films, digitized music. All the content on the internet, and on every hard drive. Imagine our civilization living forever, digitizing all of our art and knowledge, our communications, all our documentations of life, generation after generation, with finer and finer detail. Call this set \(\mathscr{D}\).

The claim is that digital and countable infinity are the same size, in other words that \(|\mathscr{D}| = |\mathbb{N}|\). Before we can explain why this is true, we need to make digital infinity more mathematically precise.

First, consider how much any one person contributes to digital infinity. Digital implies discrete. We take all our art and knowledge, and turn it into ones and zeros. Each human's digital life is a mass of discrete bits of information. These discrete bits accumulate just as the discrete natural numbers \(\{0, 1, 2, 3, …\}\) accumulate.

Over time, as digitization technology develops, we gather finer and finer detail as well -- our files grow from megabytes, to gigabytes, to terabytes. We envision, generation after generation, an average person’s digital footprint growing, and, far enough into the future, accumulating beyond any finite discrete limit. In this sense, we imagine every individual contributing nothing more, and perhaps nothing less, than \(\mathbb{N}\) -- the discrete infinite -- to the larger set \(\mathscr{D}\).

And now we imagine an ever-growing population doing the same. The population grows and grows, surpassing every finite discrete number, up towards \(\mathbb{N}\). For our infinite set \(\mathscr{D}\), we imagine an infinite population; a population of \(\mathbb{N}\). And each individual, in this limit, may contribute as much as \(\mathbb{N}\) to \(\mathscr{D}\).

"We envision, generation after generation, an average person’s digital footprint growing, and, far enough into the future, accumulating beyond any finite discrete limit."

Digital infinity, then, is at most an infinite population, with each individual contributing an infinite digital accumulation. But, as we’ve just argued, these infinities are countable infinities, like \(\mathbb{N}\). Thus digital infinity, \(\mathscr{D}\), is no larger than \(\mathbb{N}\) copies of \(\mathbb{N}\) -- one copy for each individual. And, assuming an infinite population, it is at least as large as \(\mathbb{N}\). Mathematically, we can state these last two sentences as the following: \(|\mathscr{D}| ≤ |\mathbb{N}\times\mathbb{N}|\) and \(|\mathscr{D}| ≥ |\mathbb{N}|\).

(In Basics of Set Theory, we explain how the set \(\mathbb{N}\times\mathbb{N}\) is like \(\mathbb{N}\) copies of \(\mathbb{N}\).) All that is left is to show that \(|\mathbb{N}\times\mathbb{N}| = |\mathbb{N}|\).

For, once we have shown that \(|\mathbb{N}\times\mathbb{N}| = |\mathbb{N}|\), we will have \(|\mathscr{D}| ≤ |\mathbb{N}|\) and \(|\mathscr{D}| ≥ |\mathbb{N}|\), which forces \(|\mathscr{D}| = |\mathbb{N}|\). In other words, we will have shown that digital infinity is the same size as countable infinity.

We have reduced the claim that digital and countable infinity are the same size, to the claim that \(|\mathbb{N}\times\mathbb{N}| = |\mathbb{N}|\).

You can read a short explanation of why \(|\mathbb{N}\times\mathbb{N}|=|\mathbb{N}|\), or an in-depth explanation (a rigorous proof), or you can take our word for it, and go read the story or the meta-discussion.

Short Version Meta Discussion In-Depth Version True Story

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