Concept of Set Theory

In Section 54, Frege aims to convey the message that the "units" said to compose number are isolated and indivisible. His theory is that the concept to which a number is attributed determines the conditions of congruence for the items falling under it. No one of these items is a part of any other as these items are separated from one another. Using the German word "Zahl", he sets the concept to letters in that word, in which case we would have four distinct parts (none of which is a part of another part). If we take our condition to be "syllables of Zahl", we have only one item (no other syllable is a part of it). However, he goes on to claim that not all concepts work in this manner and concepts exist where that concept does not have a finite number.

An example of the first instance can be a certain bookshelf with a number of books on it. We can attribute a definite number to the number of books in the bookshelf, the number of shelves on the bookshelf, even the number of pages (collectively) on the bookshelf. All these conditions can be defined by a number and are also under the condition. Furthermore, if we look at a part of a certain condition, we can note that these parts do not overlap into other parts. An example of the second instance can be the concept of '‹Å"cotton'. This concept has no finite number associated with it as '‹Å"cotton' can be divided up into various ways, such as cotton shirts or cotton pants.

As far as a criteria for distinguishing between the two: There are concepts that have parts under it, yet has a finite number associated with it. On the other hand, some concepts may be too broad and may be defined in different ways causing us to not be able to associate a number with it.