![]() ![]() When doing mathematics though, we need to make sure we’re being rigorous, so let’s examine this definition a little closer. This is much better than Definition Attempt #1, and in fact this is exactly how “finite set” is often defined. This might lead us to formulate the following attempted definition:ĭefinition Attempt #2: A finite set is a set whose elements can be placed into a one-to-one correspondence with some natural number. ) A finite set should thus be a set whose size is the same as some natural number. What do you use to count the elements of the set? The natural numbers! (Recall that natural numbers are whole numbers of the form. Now, when you think about a finite set, you probably think about the act of counting the elements in the set, and eventually completing this count. We say that two sets and have the same size if their elements can be put into a one-to-one correspondence with one another. Saying that a set is finite is making an assertion about its size, so let’s recall how we think about the size of a set. Let’s think about things a bit more carefully. When we’re defining the term “finite set”, what we’re really trying to do is define the word “finite”, so it’s unacceptable to use the word “finite” or “finitely” in its definition. This is obviously a bad definition, though. When asked to define the term “finite set”, an immediate first attempt might be something like this:ĭefinition Attempt #1: A finite set is a set with only finitely many elements. Can you think of multiple approaches to the task?) (Before continuing, take a moment to think about how you might try formally to define “finite set”. And, considering that this is a concept that we are all quite familiar with, this turns out to be surprisingly tricky. This intuitive understanding of “finite” and “infinite” works fine in everyday practice, but it does not amount to a definition, and if we want to study sets generally rather than specific instances of sets, or if we want to study the notion of “infinity”, then it will be important to have a solid definition of the concept “finite set”. For example, the set of all real numbers is infinite, whereas the set is finite, as it contains only three elements. Most of you reading this probably think that you understand this distinction quite well and, when presented with a relatively simply defined set, you could quite reliably and accurately classify it as either finite or infinite. A set can, of course, either be finite or infinite. We’ve talked about sets quite a bit here all you need to remember today is that a set is simply a collection of objects. Four years into a blog about infinity and we’re only now rigorously defining what it is we’ve been talking about. (And hence, by negation, the definition of “infinite set”. Today, we dig a little bit deeper by considering what should be a rather straightforward mathematical notion, and certainly a foundational issue for this blog: the definition of the term “finite set”. In our previous post, we began a discussion of circularity, first in dictionary definitions and then in mathematics. Mark van Hoeij on Circles 2: Defining Infin…Īshby Neterer on Zeno, Russell, and Borges on… ![]() Greifswald and the M… on Ultrafilters VII: Large C… Circles 3: Building Everything out of Nothing.Greifswald and the Mathematical Sublime.Follow Point at Infinity on Search for: Pages ![]()
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