What Infinity Means to Me


I have a habit of reading no matter where I am or what else I am doing. I read at my computer. I read on my phone. I read while waiting at the doctor’s office. I read in the bathroom. Lately I’ve been reading a book by Stanley Fish called How to Write a Sentence: And How to Read One. Several times he makes the claim that the number of sentences one could write is infinite. This is demonstrably untrue. The number of unique sentences one could write is certainly very large, but no mathematician would ever mistake a very large number for infinity. The fact is you can’t get an infinite collection of things using only finite collections for building blocks. For example, no matter how many molecules remain to be discovered, the total number of possible molecules cannot be infinite because the number of atoms is finite, and molecules are made up of atoms. In the same way, the collection of words is finite, so the collection of sentences (which are made up of words) must also be finite, even if we place no particular limits on the length of the sentences except that they must terminate.

The concept of infinity arises in set theory. A set is a collection of things together with a rule that tells us whether a thing belongs to the set. A set can be an explicit list: {coffee, butter, flour, sugar, tomatoes, eggs}. In this case the rule is: “If the thing is on the list, it is part of the set. Otherwise it is not.” Or a set can be defined by a rule: the set of all cats in the world. To see if something is in this set, we need only answer two questions: 1) Is it a cat? 2) Is it in the world? (We might also need to decide whether “cat” refers only to “small, domestic cats” or also to “big cats” such as lions, tigers, ocelots, pumas, bobcats, cheetahs, or other animals of the cat family, but we could in principle make such a decision ahead of time and use it to determine which things belong to the set and which do not.) One question we could ask about a set is how big it is. How many members does a set have? In the case of my shopping list, we can just count the items in the list and see that there are 6. In the case of cats in the world, we could in principle count the cats, but doing so raises a number of practical considerations. For example, how do we keep from counting the same cat more than once? How do we account for cats that are born or die while we are doing the counting? There may be 600 million small cats in the world. Counting them is going to take some time. Nevertheless, we know that having counted them, the number we obtain will be a natural number. We can say whether it is larger or smaller than the number of people in the world or the number of dogs in the world. We may not know exactly what the number is, but we know that it is a number we could count to if we counted long enough.

One of the things we know about the universe is that it is finite. It’s not hard to see why. If the universe were infinite, then the number of stars would be infinite; the number of galaxies would be infinite; the gravitational pull would be infinite; the light energy produced by all those infinite stars would be infinite. The night sky would be white if by some anomaly in the physics of the universe the earth were still pretty much as it is now. All of our experience of the universe is with finite things. Nothing is infinite. Everything is enumerable—at least in principle.

Infinity is an acknowledgement that some sets have no boundary on the number of elements they can have. However, those sets are themselves ideas only; they do not represent material objects.

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