Set

A set is an irreducible primary, and therefore can only be defined ostensively.

A circular way of defining set, which may nonetheless provide some insight, is that a set is a collection of similar existents, considered together as a unit.

Sets are sometimes called groups (e.g. in ITOE), but Objective Mathematics reserves that terminology for a different concept.

Infinite sets

Objective Mathematics says that infinite sets are an invalid notion. In essence, this is because an infinite set does not refer to anything that can be perceived. Unsurprisingly, this disconnection from reality causes many derivative problems for infinite sets. The concept of Objective Mathematics which is closest to that of an infinite set is that of a concept.

The traditional concept

Standard mathematics says that a set $X$ is infinite if there exist functions $X\rightarrow X$ which are injective but not surjective.

To demonstrate how absurd this is, and how much it conflicts with real-life, consider David Hilbert's thought experiment about what it would be like if a hotel had infinitely many rooms. Suppose that there is an infinite hotel, where each room has a room number $n:\mathbb {N}$. This hotel is completely full; every room is occupied. But unlike real hotels, if a new guest comes in and asks for a hotel room, the concierge can make room for him, even though the hotel is already full. The concierge need merely request that every hotel guest move to the room next door: if a guest's room number is $n$, he should move to room number $n+1$. (This is the key step: here the concierge is applying a function which is injective but not surjective). After everyone changes rooms, the hotel has space, because room number 1 is no longer occupied. Note that all of the guests still have their same room.

But what about extended objects?

There are an unlimited number of points on an extended object. It may therefore seem like there should be, for any extended object O, such a thing as "the set of all points on O." Since there are unlimited number of points on O, doesn't that mean that there is an infinite set?

To see why that is wrong, we must examine more carefully what is meant by a point. In some contexts, the concept of a point is used to identify a physical object. For example, the reader can easily identify a point in this picture [TODO]. In other contexts, the concept of a point is used to identify a location whose only distinguishing characteristic is that it is the place where one is focusing one's attention. For an example, the reader should try to focus on one specific point on a blank and featureless area of his wall.

We may now see the subtlety with the idea of "the set of all points on O." There may be some points on O which are physical, i.e. points which can be perceptually distinguished from the rest of O. But in order for such points to be perceived, they must necessarily have a finite size, so there can only be finitely many of them. As for those points on O which are non-physical, it is true that there is no limit on how many could exist. But since the concretes in question are merely objects of one's focus, rather than objects in reality, they do not actually exist until one chooses to focus. And one can only ever focus on finitely many points of O. We have thus refuted the idea that there exist infinitely many points on O.

Note: Objective Mathematics does not use the notion of "the set of all points on O," because of the aforementioned subtlety. However, Objective Mathematics does use the notion of "any point on O." Standard mathematics does not draw a sharp distinction between those two notions. See also the related article on the distinction between "any" and "all."

But what about all the practical infinite sets?

Standard mathematics makes use of infinite sets, and many of those infinite sets appear to be practical. Some examples are the set of all integers, the set of all fractions, infinite sequences, etc. Objective Mathematics accepts that integers, fractions, and sequences are concepts, and useful ones at that. But it denies that integers, fractions, etc. are infinite sets.