Logic is our alphabet, and sets will be the words in our language of mathematics. Some foundation has to be laid down, though, before any discussion in this language can proceed. Our foundation is Set Theory and its functions, and we will discuss these notions in the next two chapters.
There is one assumption that we will use implicitly and explicitly throughout this book. Our underlying assumption is that all mathematical objects considered in this book are from a Mathematical Universe in which these objects exist. Thus, when we say that is a cardinal, it is to be understood that this cardinal lives in a Mathematical Universe and that we can examine it there. This Mathematical Universe is a classical idea attributed to the Greek philosopher and mathematician Plato. Therefore in making our universal assumption, we are following in a good classic tradition. The intent here is clear.
Our definition of Set requires us to know when an object is given. We will write given x if we wish to examine an object in our assumed Mathematical Universe. As you can see, our universal assumption goes to work right away. Whatever else you might believe, let us agree that this our assumed Mathematical Universe exists and that we can study the elements in it. Said assumption will not change as we work our way through this book.
Mathematical statements are explicitly ...