Conceptions Of Infinity

Our goal, with this essay, is to clarify three conceptions of infinity.

Each one of these conceptions serves as a way of visualizing, and engaging with, infinity.

Or, at least, one form — perhaps, a set of forms, even — of infinity.

None of this directly pertains to the other concepts and themes within this Medium Blog. But, writing this essay was a great deal of fun, and infinity is one of my favorite subjects!

Conception 01: A Neverending Growth

Right before we go over our first conception, we must first define what our “growth” function is.

And, with that, our growth function is as follows: taking a single number and, then, allowing that number to grow by producing another number that is that number, followed by the number of zeroes it denotes.

Just to start off, your first number might be “1.”

And, then, when you use our growth function, the number that follows “1” would be “10.”

But, what about the number after “10?”

Well, that number would be 100,000,000,000.

And, then, the next number would, of course, be the number above, followed by 100 billion zeros.

You can probably see where this is going.

But, here’s the funny thing about this function: no matter how many growths you allow to take place, the end result of this function will always be a finite number.

A finite number that is no closer to infinity than, well, any inherently finite thing can be.

Just as an example, let’s say you run this growth function for a period of 100 trillion millenia.

And, let’s say that a single growth occurs within every Planck time, across this vast period.

For clarity’s sake, a Planck time is to one second, what one second is to a period of 316,887,385,068,114,309,645,621,034,629,706,950 years.

And, with this in mind, a growth is occurring within every Planck time, across a period of 100 trillion millenia.

Right after this period comes to an end, and the function stops, you will have a number.

A number that we cannot imagine. A number that we cannot conceive.

But, in the end, this number will exist as a finite quantity.

The same is true for any number produced with this function, for this function can only produce finite numbers.

And, for this reason, it can never truly encroach on the territory that is infinity.

A territory that we can imagine and, on some level, feel, but never truly reach or access.

Conception 02: A Vast Network Of Books

The title of this conception is somewhat inaccurate.

Rather than engaging with a vast network of books — and only a vast network of books — we are, instead, engaging with a vast network of folders that contain a vast network of books.

And, well, as you will soon see, it just keeps getting more and more vast.

To begin, though, imagine a book.

A book that is infinite; there is a beginning, but there is no end.

Right within this book, there exists an infinite number of unique volumes, sections, chapters; and so on and so forth, endlessly and infinitely; ad infinitum.

To go along with the above, there also exists, as one might expect, an infinite number of pages.

But, these are no ordinary pages: every single page, within this book, is infinite.

Every single one of these infinite pages comprises the book, as well as the volumes, sections, chapters — and so on and so forth — that make up the book.

Right within these pages, though, there exists an infinite number of items, components, facets; and so on and so forth, endlessly and infinitely; ad infinitum.

Every single one of what is mentioned in the previous sentence serves as a part of something.

A collection of possibilities, for something that offers infinite possibilities.

That’s just one example, but it works.

Returning to the infinite pages, though, within, and outside, of those unique items, there exists infinite sections, chapters, spaces; and so on and so forth, endlessly and infinitely; ad infinitum.

And, every single one of these…compartments, as it were, is infinite and contains an infinite number of items.

Right within these compartments, there exists an infinite number of other compartments, each of which is also infinite.

And so on and so forth, infinitely and endlessly; ad infinitum.

You can spend eternity after eternity, going through a single section within a section within a section — and so on and so forth, for it would take 100 trillion millenia to write out the recursions present within this example — of a single page and, in doing so, you would never reach the end of the items within this compartment.

But, remember, this single page, and what it contains, is part of a single book.

A single book that is, of course, infinite.

And, this book is only one book, within an infinite folder that contains an infinite number of infinite books.

Right outside of this infinite folder, there are an infinite number of other infinite folders, each of which contains an infinite number of books.

And, then, right outside of the body that contains these infinite folders, there are an infinite number of other folders that contain infinite folders; and so on and so forth, endlessly and infinitely; ad infinitum.

You can probably see where this is going.

Rather than sticking with this recursive visualization, though, let’s go outside of it.

You can go up and up, infinitely and endlessly, to find other folders.

Other folders and other bodies; ad infinitum.

But, what if you go left, right, or down? And, furthermore, what if you go in an infinite number of other directions?

You can even go outside of this framework, transcending all forms of dimensionality and spatiality; ad infinitum.

And, in doing so, what will you find?

Right now, that’s hard to say. But, it’s certainly a fun thing to think about!

Speaking of going down, though, let’s say that every folder, within this vast, vast entity of ours, contains something more than just other folders and books.

Rather, let’s say that, on some level, there exists an infinite number of sets within each folder.

And, let’s also say that, on a similar level, every book contains an infinite number of sets.

Let’s also say that every book and folder contains not just an infinite number of sets, within itself, but an infinite number of series, lists, arrays, networks; and so on and so forth, endlessly and infinitely; ad infinitum.

All of this brings me to the “set conception.”

Conception 03: The Set Conception

Returning to what we outlined in the previous section, let’s say that every infinite book and infinite folder contains an infinite number of sets, series, lists, arrays, networks; and so on and so forth, endlessly and infinitely; ad infinitum.

And, let’s say that every single one of these sets — among other things, of course — is infinite.

But, not just “infinite.”

Rather, each set is infinite and contains an infinite number of other folders and books, depending on which item we’re talking about.

Right within every folder or book, within a set, there are an infinite number of other sets.

And so on and so forth, infinitely and endlessly; ad infinitum.

But, what if, within these sets, there were an infinite number of different sections, volumes, spaces, passages; and so on and so forth, infinitely and endlessly; ad infinitum.

And, what if these, much like the previous conception, contained an infinite number of other such things, each one within itself, endlessly and infinitely?

You can even extrapolate this in another way.

Every single categorical form — a “set” or “series,” to name two examples — contains, within itself, an infinite number of sets and series and lists; ad infinitum.

And, each one of these contains other categorical forms, similar to the others, but entirely unique.

But, well, this is just one way of engaging with this particular conception and, in turn, the infinite.

Going back to the basic conception, you can go into a single set for an infinite number of eternities within bodies that contain infinite eternities within even greater bodies of infinite eternal bodies — ad infinitum — and, in doing so, you will never reach the end.

Conclusion

The infinite is one of my greatest passions. And, as such, sharing these three conceptions with you was an absolute gift.

Even if you didn’t particularly enjoy this essay, thank you so much for reading!

As always, if you wish to reach me, you can do so at “maxwellcakin@gmail.com.”

Best wishes and have a lovely day!