Set - Revision history

Lfox: /* My email to Ray 03/27/25 */

2025-08-18T23:23:12Z

My email to Ray 03/27/25

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	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.)		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.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.		"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed in a particular way by a particular man.

	Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:		Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:
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	# Groups of the units of concepts, like the group of all referents of "banana," or the group of all referents of "natural number."		# 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.		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 to the uninitiated. So to make my case, I will analyze the meaning of 2 and 3 more closely.

	=== How many points are there on my wall? ===		=== How many points are there on my wall? ===

Lfox: /* Functions */

2025-05-13T03:25:47Z

Functions

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

	so ~~note~~ that ~~uh if~~		A "problem" with this way of thinking about functions is that sometimes indeed you are doing something with <math>x</math>, and producing a new thing <math>f(x)</math>, but sometimes you haven't actually produced a new thing, you're just thinking about <math>x</math> 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 <math>x</math> itself, you're just looking at <math>x</math> differently.

	A function <math>~~f : X~~ \rightarrow Y</math> ~~is ''something you could do''. [TODO I expect this~~ to ~~be one of the most controversial parts~~.]		For an uncontroversial example of a function, it's totally fine to consider a function like the one <math>\mathbb{N} \rightarrow \mathbb{N}</math> that takes a natural number to its square.

	So it~~'s totally fine~~ to ~~consider~~ a function ~~like the one~~ <math>\mathbb{N} \rightarrow \mathbb{N}</math> 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 <math>\mathbb{R} \rightarrow \mathbb{R}</math> which sends <math>x \mapsto 2x</math>, but that's not quite right, because <math>\mathbb{R}</math> 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 <math>\mathbb{R}</math> as "lengths as they actually are in reality." It conflates length with measurement of length.

	~~Something much more confusing is like,~~ 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.~~ It's not clear ~~that it has a math description, but mainstream math models it as a function~~ <math>\mathbb{R} ~~\rightarrow~~ \mathbb{R}</math>.		Is a function <math>\mathbb{R} \rightarrow \mathbb{R}</math> something you could do? It's not clear. What even are the elements of the group <math>\mathbb{R}</math>? The elements of <math>\mathbb{R}</math> are equivalence classes of Cauchy sequences, so those are two things we need to understand: equivalence classes, and Cauchy sequences.

	~~A function~~ <math>\mathbb{R} \rightarrow \mathbb{R}</math> ~~is not~~ something you could do~~. Like oh yeah ok I~~'~~ll take an equivalence class~~ of ~~Cauchy sequences and then do something with it~~. No, ~~I'm not being fair~~.		Cauchy sequences are functions <math>\mathbb{N} \rightarrow \mathbb{Q}</math>. Is a Cauchy sequence something you could do? Some of them are, like <math>\{ 1/n \}_{n=1}^\infty </math>, but some of them aren't, like <math>\{ 1/h_n \}_{n=1}^\infty</math> where <math>h_n = </math> "the number of steps the <math>n</math>th Turing machine takes to halt." And some of them are very bizarre, like <math>\{ a_n \}_{n=1}^\infty </math> where <math>a_n = </math> "the number of stars in the Milky Way of mass <math>\leq n</math> kilograms."

	Is ~~a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math>~~ something you could do? ~~Yeah sure~~, ~~with the confusing caveat that you~~'~~re mapping into~~ something ~~with an equivalence relation. Now~~, a sequence ~~<math>a</math> could have the property that~~ it's Cauchy, ~~or not~~.		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.

	A ~~problem with functions~~ (~~at least in~~ the ~~way~~ that ~~I think about them~~) ~~is that sometimes indeed~~ you are ~~doing something with~~ <math>x</math>, ~~and producing~~ a ~~new thing~~ <math>f(x)</math>, but ~~sometimes you haven~~'~~t actually produced~~ a ~~new thing~~, ~~you~~'~~re just thinking about~~ <math>x</math> in ~~a different way.~~		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 <math>p/q</math> involving two integers <math>p,q\in \mathbb{Z}</math> (where <math>q \neq 0</math>), and where the equivalence relation is that <math>p/q = r/s</math> if <math>ps = qr</math>. 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 <math>\{ 1/n \}_{n=1}^\infty </math> and <math>\{ 1/n^2 \}_{n=1}^\infty </math>, 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 <math>\{ a_n \}_{n=1}^\infty </math>and <math>\{ b_n \}_{n=1}^\infty </math>, we say regard them as equivalent if for any <math>\epsilon \in Q, </math> such that <math>\epsilon > 0</math>, there exists an <math>N_\epsilon \in \mathbb{N}</math> such that whenever <math>n \geq N</math>, <math>\| a_n - b_n\| < \epsilon </math>.

			So what real thing does <math>\mathbb{R}</math> refer to? It refers to a particular kind of potential action <math>\mathbb{N} \rightarrow \mathbb{Q}</math> (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 <math>\mathbb{R} \rightarrow \mathbb{R}</math>? 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 <math>\{ a_n \}_{n=1}^\infty </math>, that's fine, but the function needs to be defined for <math>\{ a_n \}_{n=1}^\infty </math> qua real number rather than <math>\{ a_n \}_{n=1}^\infty </math> 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 <math>\mathbb{R} \rightarrow \mathbb{R}</math> 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] ==		== 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 ===		=== Axiom schema of specification ===

Lfox: /* Functions */

2025-04-17T14:20:21Z

Functions

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	Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.		Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

			A problem with functions (at least in the way that I think about them) is that sometimes indeed you are doing something with <math>x</math>, and producing a new thing <math>f(x)</math>, but sometimes you haven't actually produced a new thing, you're just thinking about <math>x</math> in a different way.

	== Group formation rules [analogue of ZF] ==		== Group formation rules [analogue of ZF] ==

Lfox: /* My email to Ray 03/27/25 */

2025-04-03T15:17:38Z

My email to Ray 03/27/25

Lfox: /* Axiom schema of specification */

2025-03-29T16:20:30Z

Axiom schema of specification

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	2) any group formed via a predicate is necessarily a subgroup of another group. [todo maybe it's already like this?]		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 ===		=== Axiom of extensionality ===
	This one needs to be re-worked. Sameness / identity is a relative and contextual thing, not an absolute thing. Also, there~~'s no~~ practical way to check that all the members of a set are the same.		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 ===		=== Axiom of choice ===

Lfox: /* Functions */

2025-03-29T15:17:01Z

Functions

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	=== Functions ===		=== Functions ===
			Let <math>X</math> and <math>Y</math> be groups. A function <math>f : X \rightarrow Y</math> is a potential action, which, if performed on the same member <math>x</math> of <math>X</math>, would always produce the same member <math>f(x)</math> of <math>Y</math>.

			so note that uh if

	A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]		A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]

Lfox: /* Group membership */

2025-03-29T05:52:34Z

Group membership

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 === 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?"
-The question might have a clear answer, or it might not (because e.g. there are borderline cases).
+To express the statement that <math>x</math> is a member of the group <math>X</math>, I will use the shorthand notation <math>x : X</math>. (This is very similar to the set-theoretic judgement <math>x \in X</math>, but I shall use different notation to emphasize the difference between what I'm doing and set theory.)
 === Functions ===
-A function is something you could do.
+A function <math>f : X \rightarrow Y</math> is ''something you could do''. [TODO I expect this to be one of the most controversial parts.]
 It's not true that for any rule, we can for

Lfox at 04:02, 29 March 2025

2025-03-29T04:02:16Z

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	My problem is not really with the diagonal argument per se, it is with the more basic concept of infinite sets.		My problem is not really with the diagonal argument per se, it is with the more basic concept of infinite sets.

	~~=== What do sets refer~~ to ~~in reality? ===~~		I want to go on a quest, where I justify set theory from the ground up. This is inspired by [TODO]
	~~"Set~~,~~" like any potentially valid concept, refers to things in reality. Sets are getting at something similar to what Ayn Rand is getting at with~~ the ~~concept she calls "group" in ITOE~~. ~~For the purposes of this email I will stick with the latter terminology, because I want to emphasize that everything below~~ is ~~only in parallel to mathematics~~ [TODO ~~wut TODO no ur using the terminology set~~]

	"~~Set~~" is ~~a primary~~ concept, ~~which cannot be reduced to more basic concepts~~. ~~However~~, I will ~~provide~~ the ~~following ostensive definition~~.~~<blockquote>'''Definition'''. A ''set~~'' ~~is some existents, considered together as a single whole~~. ~~</blockquote>Here is a (not necessarily exhaustive~~) ~~list of some different types of sets that we might consider:~~		=== 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.)

	# "~~Discrete~~" ~~sets that are finite~~, ~~like the set of pencils on my desk~~, or the ~~set of possible outcomes for the dice that I~~'~~m about to roll~~.		"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.
	~~# "Continuous" sets~~, ~~like the set of all points on the south-facing wall of my room~~.
	~~# Sets coming from~~ concepts, ~~like the set of all referents of "banana~~,~~" or the set of all referents of "natural number~~."

	Standard mathematics ~~tells us that we should conceptualize~~ 2 and 3 as ''infinite sets''. However, I disagree that those ~~sets~~ 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.		Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:

			# "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.
			# "Continuous" groups, like the group of all points on the south-facing wall of my room.
			# Groups coming from 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? ===		=== How many points are there on my wall? ===
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	One problem: sets are being conceptualized as things which exist, rather than things which potentially could exist		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?"

	~~Now~~, ~~with examples like these in mind, let~~'s ~~ask: what fact of reality necessitates the finite / infinite distinction?~~		The question might have a clear answer, or it might not (because e.g. there are borderline cases).

			=== Functions ===
			A function is something you could do.

			=== Group formation rules ===
			It's not true that for any rule, we can for

	~~=== Infinite sets from the perspective of standard mathematics ===~~
	One criticism that someone might have is like		One criticism that someone might have is like

Lfox: /* How many natural numbers are there? */

2025-03-27T05:42:39Z

How many natural numbers are there?

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	One criticism that someone might have is like		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?"		"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.		That's a reasonable question to ask.

Lfox: /* My email to Ray 03/27/25 */

2025-03-27T05:15:36Z

My email to Ray 03/27/25

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	Hmm that's actually a very interesting definition.		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.

	== Notes ==		== Notes ==

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	== My email to Ray 03/27/25 ==		== My email to Ray 03/27/25 ==
	Ray ~~asked me to~~ put my problem with the diagonal argument into writing~~, and I think this is a good idea~~.		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.

	My problem is not really with the diagonal argument per se, it is with the more basic concept of ~~infinite~~ sets.

	I ~~want to go on a quest~~, ~~where~~ I ~~justify set~~ theory ~~from the ground up~~. ~~This is inspired by [TODO]~~

	=== What are sets(="groups")? ===		=== 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.)		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.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>Groups are concepts of consciousness, and so they are relational. Groups are things out there, as viewed particular way by a particular man.		"Group" is a primary concept, which cannot be reduced to more basic concepts. However, I will provide the following ostensive definition.<blockquote>'''Definition'''. A ''group'' is some existents, considered together as a single whole. </blockquote>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:		Here is a (not necessarily exhaustive) list of some different types of groups that we might consider:
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	# "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.		# "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.
	# "Continuous" groups, like the group of all points on the south-facing wall of my room.		# "Continuous" groups, like the group of all points on the south-facing wall of my room.
	# Groups ~~coming from~~ concepts, like the group of all referents of "banana," or the group of all referents of "natural number."		# 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.		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.
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	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.		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 now see the subtlety with the idea of "the set of all points on O." ~~As for those~~ points on O which are ''physical''~~, they~~ 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.		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 ~~whatever it is that~~ we mean by "point," we see that there are only finitely many points on my wall.		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? ===		=== How many natural numbers are there? ===
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	A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.		A function <math>\mathbb{R} \rightarrow \mathbb{R}</math> is not something you could do. Like oh yeah ok I'll take an equivalence class of Cauchy sequences and then do something with it. No, I'm not being fair.

	Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.		Is a sequence <math>a : \mathbb{N} \rightarrow \mathbb{Q}</math> something you could do? Yeah sure, with the confusing caveat that you're mapping into something with an equivalence relation. Now, a sequence <math>a</math> could have the property that it's Cauchy, or not.

	== Group formation rules [analogue of ZF] ==		== Group formation rules [analogue of ZF] ==