Set

"Set" is a concept referring to 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 whole.

I will say a word about why I specify that sets must consist of "similar" existents, because this idea is foreign to standard mathematics, wherein a set could consist of any existents whatsoever. First of all, similarity requires a context; things can be similar in one context, but dissimilar in another. And if, in a given context, two things are dissimilar, then there can be no reason---in that context---to consider them together as a single set. One therefore loses nothing by specifying that sets consist of "similar" existents. What one gains by this, on the other hand, is a small reminder about the purpose of sets.

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

Examples

A set of plates.

[TODO more]

Empty set

An empty set (not "the" empty set; see below) is one way of viewing the concept of nothing. The concept of "nothing" may seem impossible or invalid. Concepts need to have referents, and yet everything is something; there is no thing which is nothing. How can this be? ITOE explains

["Nothing"] is strictly a relative concept. It pertains to the absence of some kind of concrete. The concept “nothing” is not possible except in relation to “something.” Therefore, to have the concept “nothing,” you mentally specify—in parenthesis, in effect—the absence of a something, and you conceive of “nothing” only in relation to concretes which no longer exist or which do not exist at present.

You can say “I have nothing in my pocket.” That doesn’t mean you have an entity called “nothing” in your pocket. You do not have any of the objects that could conceivably be there, such as handkerchiefs, money, gloves, or whatever. “Nothing” is strictly a concept relative to some existent [sic] concretes whose absence you denote in this form.

Examples

If one has no books on one's table, it may be said that he has an empty set of books on his table.

If there are no more days left until March 3rd, 2055 (i.e. if it is currently March 3rd, 2055), then it may be said that the set of days until March 3rd is empty.

The traditional concept

In standard mathematics, there is supposed to be a single object called "the" empty set. Most sets in standard mathematics are not unique in this manner. The empty set is called "the" empty set because it is unique up to unique isomorphism. Of course, that imaginary object is not what Objective Mathematics means by the concept of an empty set.

Nevertheless, it is sometimes valid to use the phrase "the empty set" in Objective Mathematics. It is valid in the same sense that it is valid to say "the cat" or "the car." Namely, if one has a specific empty set in mind, and wishes to refer to that empty set and not another one, he may call it "the empty set."

Relations among sets

In this section, I will describe some relations among sets. That is, I will describe ways in which some sets (possibly along with functions between them) can be used to identify other sets. This list is non-exhaustive.

Disjoint union

Given two sets $A=\{a_{1},\cdots ,a_{n}\}$ and $B=\{b_{1},\cdots ,b_{m}\}$, the disjoint union of $A$ and $B$, denoted as $A\sqcup B$, is following set

$$A\sqcup B:=\{a_{1},a_{2},\cdots ,a_{n},b_{1},b_{2},\cdots ,b_{m}\}.$$

[TODO this is an unsophisticated treatment, because it's not uniquely defined. E.g. I could have defined

$$A\sqcup B:=\{(1,a_{1}),(2,a_{2}),\cdots ,(n,a_{n}),(1,b_{1}),(2,b_{2}),\cdots ,(m,b_{m})\}$$

category theory says that what really defines these ideas is their universal properties. Could that have an OM interpretation?]

Examples

I have two piles of books on my table. Equivalently, I could say that I have identified two sets $P_{1}:=\{b_{1},b_{2},b_{3}\}$ and $P_{2}:=\{a_{1},a_{2}\}$ of books. Instead of regarding these as two distinct sets of books, I could regard them as a single set of books; instead of thinking of two piles of books on my table, I could think of all the books on my table. Equivalently, I could say that I have identified the set of all the books on my table,

$$P_{1}\sqcup P_{2}=\{b_{1},b_{2},b_{3},a_{1},a_{2}\}.$$

Cartesian product

Given two sets $A=\{a_{1},\cdots ,a_{n}\}$ and $B=\{b_{1},\cdots ,b_{m}\}$, the cartesian product of $A$ and $B$, denoted as $A\times B$, is following set of pairs

$$A\times B:=\{(a_{i},b_{j}):1\leq i\leq n,1\leq j\leq m\}.$$

Examples

The power socket on my wall has two outlets; in other words, I've identified a set $O:=\{o_{1},o_{2}\}$ of outlets. An outlet has three holes; in other words I've identified (abstractly) a set $H:=\{h_{1},h_{2},h_{3}\}$. Now, I can consider the set of all the holes in the power socket on my wall. It is

$$O\times H=\{(o_{1},h_{1}),(o_{1},h_{2}),(o_{1},h_{3}),(o_{2},h_{1}),(o_{2},h_{2}),(o_{3},h_{3})\}.$$

Subset

Given a set $A=\{a_{1},\cdots ,a_{n}\}$, a subset of $A$ is some of the units of $A$, say $a_{i_{1}},\cdots ,a_{i_{k}}\in A$, considered as a single set $B=\{a_{i_{1}},\cdots ,a_{i_{k}}\}$. We denote this by $B\subset A$.

Powerset

Given a set $A=\{a_{1},\cdots ,a_{n}\}$, the powerset of $A$, denoted $2^{A}$, is the set consisting of all subsets of $A$,

$$2^{A}:=\{B:B\subseteq A\}.$$

The reason for the notation is that the powerset may equivalently be defined as the set of all functions $A\rightarrow 2$ where $2$ denotes the set $2:=\{0,1\}$.

Partition

Given a set $A=\{a_{1},\cdots ,a_{n}\}$, a partition of $A$ is a division of $A$ into (nonempty) disjoint subsets, such that the disjoint union of all the subsets is $A$.

A partition of $A$ is equivalent to an equivalence relation of $A$. [TODO explain]

Infinite sets

Objective Mathematics says that infinite sets are an invalid notion. [TODO I actually don't think I should say this. A a concept is an unbounded or infinite set. If it wasn't, then there would be no such thing as the units of a concept.] 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.

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.

My email to Ray 03/27/25

This essay was prompted by Ray's suggestion that I put my problem with the diagonal argument into writing. My problem is not really with the diagonal argument per se, it is with the more basic concept of sets. Rather than saying something negative about set theory as it stands today, I will offer something positive. Starting from scratch, I will sketch what I think a rational theory of sets would look like.

What are sets(="groups")?

The concept of "set" is getting at something similar to what Ayn Rand is getting at with the concept she calls "group" in ITOE. I think these two concepts are more or less interchangeable, though they have different connotations. For the purposes of this email, however, I will stick with the latter terminology of "group," because I want to emphasize that I am doing something independent of the way that mathematics has developed. (I don't want to stick with this terminology forever, though; I will start calling them "sets" again once I'm finished.)

"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.

Definition. A group is some existents, considered together as a single whole.

Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.

Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

1. "Discrete" groups that are finite, like the group of pencils on my desk, or the group of possible outcomes for the dice that I'm about to roll.
2. "Continuous" groups, like the group of all points on the south-facing wall of my room.
3. Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."

Standard mathematics conceptualizes 2 and 3 as infinite sets. However, I disagree that those groups are actually infinite. I think their finiteness follows from the law of identity (every quantity must have some quantity), but that might not be very convincing as an argument. So to make my case, I will analyze the meaning of 2 and 3 more closely.

How many points are there on my wall?

In some contexts, the concept of a point is used to identify a physical object. For example, one can easily identify a point in the picture below [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, whether it be by literally "pointing" to that place (hence the name), by merely thinking of it, or by using some other means.

We may now see the subtlety with the idea of "the set of all points on O." Those points on O which are physical must necessarily have a finite size (otherwise we couldn't know about them in the first place), and 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 things on which one is focusing, rather than physical objects in reality, they do not actually exist until someone focuses on them. And a man can only ever focus on finitely many points of O.

So regardless of which of those we mean by "point," we see that there are only finitely many points on my wall.

How many natural numbers are there?

I gave two examples in 3: bananas, and natural numbers. I think everyone will agree that bananas

The following definition is adapted from a definition Harry gave:

A natural number is an identification of a quantity, by means of a symbol (a "numeral") whose position in a fixed sequence of those symbols is the same amount as that which is being identified.

I agree with this definition, and I have found it to be very clarifying.

Note the genus of "natural number": it is identification. Natural numbers are products of consciousness. So the

There are not infinitely many referents of "banana."

Things like 2 and 3 are what motivated infinite sets in mathematics. [TODO irrelevant delete]

One problem. Someone could ask: How many numbers are there? And the answer is: in whose mind? [TODO meh]

One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist

Now, with examples like these in mind, let's ask: what fact of reality necessitates the finite / infinite distinction?

Group membership

As the above examples demonstrate, it is not always meaningful to ask "how many elements of this group are there?" But it is always meaningful to ask "is this thing a member of that group?"

To express the statement that $x$ is a member of the group $X$, I will use the shorthand notation $x:X$. (This is very similar to the set-theoretic judgement $x\in X$, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)

Functions

Let $X$ and $Y$ be groups. A function $f:X\rightarrow Y$ is a potential action, which, if performed on the same member $x$ of $X$, would always produce the same member $f(x)$ of $Y$. A function $f:X\rightarrow Y$ is something you could do.

A "problem" with this way of thinking about functions is that sometimes indeed you are doing something with $x$, and producing a new thing $f(x)$, but sometimes you haven't actually produced a new thing, you're just thinking about $x$ in a different way. For example, all men are mortal, so there should be a function from the group of all men to the group of all mortal beings. It is arguable, however, that these functions represent something you could do, but it's something you do in consciousness---they represent a change of perspective. You aren't doing anything to $x$ itself, you're just looking at $x$ differently.

For an uncontroversial example of a function, it's totally fine to consider a function like the one $\mathbb {N} \rightarrow \mathbb {N}$ that takes a natural number to its square.

Another type of function, one without a clear mathematization, is a function which takes a length and doubles it. To be very clear about this, by a length I mean a literal property of an object, and not a number which measures that property. This function could be implemented in real life, e.g. given a line segment on a piece of paper, pull out an amount of string equal to that length, then use the string to extend that line segment by drawing a new line segment of equal length. Mainstream math models the above process as the function $\mathbb {R} \rightarrow \mathbb {R}$ which sends $x\mapsto 2x$, but that's not quite right, because $\mathbb {R}$ is a number (or at least, something akin to a number), and not the length itself. It is very common that mainstream math thinks of $\mathbb {R}$ as "lengths as they actually are in reality." It conflates length with measurement of length.

Is a function $\mathbb {R} \rightarrow \mathbb {R}$ something you could do? It's not clear. What even are the elements of the group $\mathbb {R}$? The elements of $\mathbb {R}$ are equivalence classes of Cauchy sequences, so those are two things we need to understand: equivalence classes, and Cauchy sequences.

Cauchy sequences are functions $\mathbb {N} \rightarrow \mathbb {Q}$. Is a Cauchy sequence something you could do? Some of them are, like $\{1/n\}_{n=1}^{\infty }$, but some of them aren't, like $\{1/h_{n}\}_{n=1}^{\infty }$ where $h_{n}=$ "the number of steps the $n$th Turing machine takes to halt." And some of them are very bizarre, like $\{a_{n}\}_{n=1}^{\infty }$ where $a_{n}=$ "the number of stars in the Milky Way of mass $\leq n$ kilograms."

Is it legitimate to talk about Cauchy sequences that are not something you could do? The answer of mainstream mathematics is a resounding "yes." My answer is a tentative "no": if it's not something which exists in reality, and it's not something which potentially could exist in reality, then your sequence doesn't refer to anything, and so it's meaningless. It's like talking about flying purple cats. The reason why my answer is tentative "no" is that sometimes we can learn things from impossible hypotheticals; sometimes they reveal things about the nature of our legitimate concepts. I don't want to police what sort of Cauchy sequences people can or cannot talk about, I just want to reorient math in the direction of reality.

What is an equivalence class? Well that's actually quite simple. First let's consider fractions: 1/3 and 2/6 are the same fraction, but they are different expressions. A fraction is an equivalence class of expressions $p/q$ involving two integers $p,q\in \mathbb {Z}$ (where $q\neq 0$), and where the equivalence relation is that $p/q=r/s$ if $ps=qr$. The concept of equivalence classes extends far beyond mathematics, to any concept which is a special case of another concept. The two copies of Conformal Field Theory (some physics textbook) in my office are different copies, but they are the same book. Now, given two Cauchy sequences $\{1/n\}_{n=1}^{\infty }$ and $\{1/n^{2}\}_{n=1}^{\infty }$, which mind you are two things---two processes---actually out there in reality, we say they are different qua sequence, but they are the same qua real number. Generally, given two Cauchy sequences $\{a_{n}\}_{n=1}^{\infty }$and $\{b_{n}\}_{n=1}^{\infty }$, we say regard them as equivalent if for any $\epsilon \in Q,$ such that $\epsilon >0$, there exists an $N_{\epsilon }\in \mathbb {N}$ such that whenever $n\geq N$, $|a_{n}-b_{n}|<\epsilon$.

So what real thing does $\mathbb {R}$ refer to? It refers to a particular kind of potential action $\mathbb {N} \rightarrow \mathbb {Q}$ (namely one that is "Cauchy"), from a particular perspective (namely the perspective from which two Cauchy sequences are the same if they are going to the same place).

Now, how about functions $\mathbb {R} \rightarrow \mathbb {R}$? Well it's just something that takes a real number and sends it to a real number. If you think of the real number as a sequence $\{a_{n}\}_{n=1}^{\infty }$, that's fine, but the function needs to be defined for $\{a_{n}\}_{n=1}^{\infty }$ qua real number rather than $\{a_{n}\}_{n=1}^{\infty }$ qua sequence. Again, this point is logical and not mathematical. We could consider a function from the group of all books to the group of all strings, e.g. which sends a book to its author; such a function can be defined on copies of books (like maybe your way of implementing it is: purchase a copy of the book and look at who the author of the copy is), but it must necessarily depend on the properties of the copy qua book rather than its properties qua copy. For example, it can't depend on the physical location of the book.

So, in conclusion, the answer to the question "Is a function $\mathbb {R} \rightarrow \mathbb {R}$ something you could do?" is yes. But what sort of processes does it refer to in real life? Is it really practical?

Group formation rules [analogue of ZF]

A group doesn't have to be specified in a completely unambiguous way. In real life, there is no such thing.

Axiom schema of specification

yeah it's fine to "form a group" according to a predicate. BUT

1) you don't need a predicate in order to "form a group." It's totally fine to just say something like "the books on my bookshelf."

2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]

I actually don't like the term "form a set." It's vague. Really what we're doing is we are identifying sets; we are pointing to some things in reality, and saying "that's a set."

Axiom of extensionality

This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there need not be a practical way to check that all the members of a set are the same.

Quotients / identity are actually what I have to think the most about.

Axiom of choice

Bullshit.

Axiom of infinity

Bullshit.

Powerset axiom

I think this is true. If it wasn't true, then it wouldn't make sense to think of "A is a subset of B" as meaning "A is (a subset of B)." Like the noun phrase in parentheses wouldn't make sense.

Other axioms

????

It's not true that for any rule, we can for

One criticism that someone might have is like

"okay yeah fine infinite sets don't exist. But they're getting at something real, so what's the actual practical problem that arises from using them anyway?" I don't need to answer that criticism in this essay.

That's a reasonable question to ask.

In this section I will argue that

I will say that a set $X$ is infinite if there exists a function $f:X\rightarrow X$ which is injective but not surjective.

Hmm that's actually a very interesting definition.

Practically speaking, there actually are things like that. Like imagine if f is sort of like, a machine that takes in bananas and plants them and makes new bananas

Like imagine we're homesteading the Wild West. The Wild West is finite, but it's so unfathomably large that we don't actually have to worry about running out of room, even though we aren't building any houses in the exact same spot.