@Zanna It probably is related, but I was only thinking of special relativity.

@Zanna Did you ever figure out what was going on with your system glitching?

@EliahKagan so... we don't really need a higher order logic for this because we can use set theory?

Yes.

@EliahKagan I realised that I misread that message that would intermittently appear at the end of a session - it was actually warning me that unsaved work would be lost! :)

that's in the nature of unsaved work, so...

but, still, I would like to know why my desktop background sometimes disappears

I must investigate as you suggested

@EliahKagan Btw, I wish I had not said "sort" there, since as you may recall, that has a related but different meaning... and it was not the meaning I intended. It does also happen to be true that we do not need more than one sort of object, and for this reason, but this is not what I wanted to say and it is not as conceptually central.

also the function keys don't work, which is not an unusual thing when using Linux, and is probably fixable

The function keys don't work, or their special "Fn" functionality doesn't work?

there are probably some other things I haven't investigated yet. I haven't tried hibernation or Bluetooth

@EliahKagan That is, the things I'm saying there that set theory makes it so we don't need are things that are not present in first-order logic:

second-order predicates, i.e., predicates whose arguments denote predicates rather than objects, which second- (and higher) order logic has but first-order logic does not; third-order predicates, i.e., predicates whose arguments denote second-order predicates rather than bjects, which third- (and higher) order logic has but first- and second-order logic do not; and so on.

@EliahKagan actually, what I said was just wildly incorrect, because the function keys do all sorts of things they are supposed to do, like even adjust the volume and brightness, turn on airplane mode and I know not what, only I used to press Alt+F4 to close a window and that doesn't work. So, I need to adjust the keyboard shortcuts hahaha

the worst problem I'm having is that I can't find a keyboard layout for Tamil

Do they do those things without pressing a special "Fn" key?

I should ask an Ask Ubuntu question about that

@EliahKagan yes

because a few versions ago all the Indian language layouts were removed because someone decided they were not useful

When you do press the "Fn" key, do they then do the other things they're not doing, e.g., does Alt+Fn+F4 quit programs?

@EliahKagan it does not

Then yes, posting a question sounds like a good idea. :)

@Zanna and then they were put back in

@EliahKagan about the function keys?

Yes, but I now realize that this is not what you were saying you should post a question about.

I can post more than one question...

Indeed.

but it may be that there is some other keyboard shortcut for doing what my muscle memory thinks Alt+F4 should do

and I haven't even bothered to look at the keyboard shortcuts yet

@EliahKagan that seems good!

@EliahKagan In first-order logic, there are only first-order predicates. "Predicate" means "first-order predicate" when talking about first-order logic. In second- (and higher) order logic, one can quantify over (first-order) predicates, there are variables that reign over (first-order) predicates, and there are metapredicates (i.e., second-order predicates, per above) that take (first-order) predicates as arguments.

But one can also, as I mentioned a while ago, have systems with more than one sort of object. Most commonly first-order logic is one-sorted, and I gave ZFC as an example of a one-sorted first-order theory; there are two-sorted theories, such as NBG; and there is no limit to how many sorts of objects one may have (so long as one's underlying system of first-order logic permits it).

That is, one can have a three-sorted theory, a two-sorted theory, etc., and one can have a theory with an unlimited number of sorts (many-sorted). These can all be first-order theories.

@EliahKagan that the pair of things (x, y) is one member of R... since the things in a set have to be unordered... Like an unordered pile of trousers (one leg being useless without its partner)?

@Zanna I'm not sure I know what you mean. Can you elaborate?

@EliahKagan I do remember that, but I took the 'sort' you wrote in the message you wish you hadn't said it in to be the more general one

Excellent!

:)

@EliahKagan I mean it isn't that x and y are members of R but that (x, y) is a member of R

Oh, yes, that's correct.

Or more precisely, that if x or y is a member of R, that's mere coincidence.

I think, this is very obvious and not worth mentioning (or, maybe I am totally confused and wrong) but this compound thing (x, y) looks new and interesting

Go on.

I mean what I already said was very obvious and not worth mentioning

sorry

Huh?

The semantics, construction, and practical significance of ordered pairs are extremely deep and important. I don't know quite what you're thinking about them, but you are definitely right to mention your interest. I don't recall that I covered them in any detail.

I was hoping you might share your specific thoughts.

I mean, I said that it, since members of a set have to be unordered, for this clever thing to work, the thing that is the member of the set has to be the pair and not the two things. This was an obvious thing to say

but I said it because, I don't think we had met any things like that before

in this, sort of, logic walk

so I wasn't about to say any new thing

I was just uselessly commentating on the thing I had already said

I guess I will have to take your word for it that this commentary is useless. :)

...I have things to say in reply to it that I think (or at least hope) are not useless.

awesome!

then it will not have been useless at all!

@Zanna There are, and I say this with a smile on my face, two things you could mean by "two things" there.

hahaha

If we consider the less-than relation "<" on integers, constructed set-theoretically as its graph, then:

0 < 1

So:

(0, 1) ∈ <

But 1 is not less than 0:

¬ 1 < 0

So:

¬ (1, 0) ∈ <

(Usually I would just say "(1, 0) ∉ <". Or "1 ≮ 0".)

this is very nice and clear

Cool. :)

So if what you meant by "the two things" is the unordered pair of the two things, i.e., the set of just them, then what you were saying was correct -- though I caution that referring to sets as the things they contain is sometimes ambiguous (and that this is one of those cases).

That is, {0, 1} = {1, 0} so we can't have something be true of one of them but false of the other. In contrast, (0, 1) ≠ (1, 0) (because 0 ≠ 1) and more generally ∀x ∀y [(x, y) = (y, x) → x = y], so something can be true of (0, 1) but not of (1, 0). One thing that is true of (0, 1) but not of (1, 0) is that its first entry is 0. Another is that, with the meaning of "<" given above, (0, 1) ∈ < but (1, 0) ∉ <.

I guess I had not really thought about the pair having to be ordered within itself, i.e. to be (x, y) and not (y, x), only about them being stuck together

@EliahKagan makes sense

@Zanna In that case, can you elaborate about what you meant by, "since members of a set have to be unordered, for this clever thing to work, the thing that is the member of the set has to be the pair and not the two things"?

I meant that the pair of things x and y have to be counted together as one object, like a pair of trousers

Right. Specifically, another object, which is the pair of them, is used, rather than directly using them.

yes

The reason I was asking for clarification about what you were saying, though, is that there are two related but different issues here, and I want to avoid conflating them.

Our inability to use {x, y} in place of (x, y) when the order matters -- as it does in relations, or at least in non-symmetric relations -- is due to {x, y} = {y, x}, i.e., due to sets being unordered. But even if we were only interested in symmetric relations R, i.e., relations where ∀x ∀y (xRy → yRx), and thus could accept a construction that only worked for symmetric relations, we still could not dispense with packing packing each related x and y into pairs.

it wouldn't mean anything for them to individually be members of this set

Correct, it would not.

But I'm making a stronger claim. I'm not just saying that it's not meaningful with the particular construction for binary relations that we've chosen. I'm saying that there there is no way to construct relations as sets, even if you're only interested in being able to represent symmetric relations, where for all relations R we have xRy ↔ (x ∈ R ∧ y ∈ R).

Because then if R = {a, b, c, d}, you don't know which of a, b, c, and d are related to which other ones.

I see that

Cool. :)

In contrast, if we were only interested in symmetric relations, we could choose to construct them as sets of unordered pairs, i.e., as sets of two elements.

yes

We don't do this, because it is extremely valuable to construct symmetric relations the same way we construct non-symmetric relations. But we could.

That's all. That's the distinction I wanted to make sure was clear.

But since we're on the topic of ordered pairs...

@EliahKagan ok, got it :)

...I want to point out that the notion of ordered pairs used in set theory is the same notion of ordered pairs that you already know.

And that a graph is the abstract analogue of what you already know as a graph.

that's helpful

To clarify, since there are two things that are called graphs in math: a graph, in the sense of a graph of a relation or function, is the abstract analogue of what you already know as a graph, in the sense of a graph of a relation or function, or of the relationship between variables.

ok

So, consider a relation, call it C, where xCy if and only if x and y are real numbers and x² + y² = 1.

This relation is the unit circle in the Cartesian coordinate plane.

If you plot C, you have drawn a circle.

The points you plot are the members of the graph of C, and since we construct relations as their graphs, they are the members of C.

Thus C is the set of the points that your plot represent pictorially.

The other meaning of graph in math is the graph-theoretic meaning.

I bring this up because I've been talking about that kind of graph before.

I used graphviz to draw graphs ("graph-theory graphs") of the "⊆" predicate.

That is, in graph theory, a graph consists of vertices and edges.

I'm not familiar with this thing, though I do remember you talking about it and demonstrating it earlier with the example of tasks that depend on each other

Yes.

@EliahKagan :) nice

@EliahKagan incidentally, and very irrelevantly, I think I have exactly one paintbrush at the moment and I'm hoping I can find it so I can decorate the single unique clay jar I have also got

Graphs, in the sense of that term used in graph theory, are not geometric in nature. They are not closely related to plots. They can often be represented pictorially, but the choices one makes in representing them pictorially are quite different from the choices one makes in representing graphs, in the sense of a set of ordered pairs (or triples, quadruples, etc.).

Graphs, in the sense of the graph of a relation or function, i.e., the set of tuples of things that correspond to one another under the relation or function, are often represented pictorially through a choice of coordinate system and scaling. In contrast, graphs, in the sense of graph theory, are, if represented pictorially, only rarely represented pictorially in a manner that has anything to do with that.

It seems from glancing at this article that the scale of these things is not significant

when you were talking about it before, I thought of another kind of handy picture - flow diagrams

@EliahKagan I should say that graphs, in the sense of a graph of a relation or function, are not strictly speaking geometric in nature. Even when I say, "This relation is the unit circle in the Cartesian coordinate plane," that is algebra, rather than geometry.

But it borrows terminology from geometry (and to such an extent that is arguably not merely pedantic but actually wrong to claim it is non-geometric) and borrows for very good reason--the Cartesian coordinate plane can be used to provide, and is very often used to provide, coordinate representations for points on a Euclidean plane. Then graphs become coordinate representations of figures on the Euclidean plane.

@Zanna Excellent. Then "paintbrush(zanna)" succeeds! :)

:D

@Zanna Yes, I think flow diagrams can often be represented by weighted directed graphs. (A graph is weighted if it has weights associated with its edges. The weights can represent anything.)

@Zanna Had you not previously been familiar with graph theory?

Graphs, in the sense of graphs in graph theory, are sometimes called "node graphs" for clarity, since "node" is another word for "vertex." But for some reason I dislike the phrase "node graph." It is also not common enough that it is a sure-fire way to avoid ambiguity.

no I had not, although it seems to me I have met these kinds of objects here and there in the past without any particular introduction... for example there are tests set by selective secondary schools for 10 year olds to take, and some of the schools deliberately create questions with unfamiliar things in them

And that has included graphs, in the graph-theoretic sense?

(related)

I suspect that would be the context in which I've seen them. I would often get asked to tutor kids for these tests... if possible I would avoid that work, but sometimes I would be helping with homework and they'd be practising for these tests, or maybe just using them as novel homework materials (I often found these tests fun and interesting)

I don't want to over state the degree to which graphs -- as in the graph of a relation or function -- is geometric. Many graphs have no reasonable geometric interpretation whatsoever.

@Zanna There are good reasons to want to define relations as their graphs, yes. However, there are also some possible disadvantages. In particular, it means relations don't know what kinds of things they relate.

They know which objects they relate, but they don't know which sets you consider them to be defined on.

hmm

This is a somewhat tricky conceptual point, but although it can be put off, it can't be dispensed with altogether, because it comes up again in force when one gets to the concept of the codomain of a function.

Consider the unit circle (whose center is the origin) again, which I called C.

(0, 0) ∉ C

Suppose no goat is a real number, and that a and b are goats. Then we also have

(a, 0) ∉ C
(0, b) ∉ C
(a, b) ∉ C

Those are all perfectly good sentences, i.e., they are all well-formed. We can even write "aCb". (It's false, of course.) And yet it seems like (0, 0) ∉ C for one reason, but (a, 0) ∉ C, (0, b) ∉ C, and (a, b) ∉ C for another.

A binary relation between sets A and B is a subset of the Cartesian product of A and B. That is, A × B (the Cartesian product of A and B, as detailed above) is the set of all the ordered pairs (x, y) where x ∈ A and y ∈ B, i.e.:

A × B = {(x, y) | x ∈ A ∧ y ∈ B}

We construct relations as their graphs, and the graph of a relation R is the set of (x, y) for which xRy. So if R relates elements of A to elements of B, then for any x, y, where xRy:

x ∈ A ∧ y ∈ B

Note that the converse need not hold. Relations don't have to always be true for all elements of the sets they relate. Of course!

Thus:

R ⊆ A × B

(The claim that the converse does not hold is then the claim that, there's some relation R between sets A and B for which R ≠ A × B. Most interesting relations have this property.)

R is a relation between A and B. We can say that by writing "R ⊆ A × B". However, given any A' ⊇ A and B' ⊇ B, it is also the case that R ⊆ A' × B'.

I will give an example of how this is sometimes nonintuitive and potentially undesirable. Note first that typically we write "ℝ", called "blackboard-bold R," to denote the set of real numbers. This should not be confused with R, which I've been using as an arbitrary example of a binary relation.

(Similarly, typically we write "ℂ" to denote the set of complex numbers. This should not be confused with C, which I've been using to denote the unit circle centered at the origin on the 2-dimensional Cartesian coordinate plane, which since we construct relations as their graph is {(x, y) | x² + y² = 1}.)

C is a relation between ℝ and ℝ (i.e., between ℝ and itself), which we can express by saying C ⊆ ℝ × ℝ (or C ⊆ ℝ², as ℝ² = ℝ × ℝ). But it is just as good to say "C is a relation between ℝ ∪ G and ℝ ∪ D, i.e., C ⊆ (ℝ ∪ G) × (ℝ ∪ D), regardless of what sets D and G are. I am thinking of D as the set of doorknobs and G as the set of goats.

You can also regard C to be a relation between some proper subsets of ℝ. There are no x and y for which xCy and |x| > 1 or |y| > 1, for instance. So you could regard C as a relation between [-1, 1] and itself. (As you probably know, [-1, 1] means the closed interval from -1 to 1, so in this context, i.e., over the reals, [-1, 1] ⊆ ℝ and, specifically, [-1, 1] = {x ∈ ℝ | -1 ≤ x ≤ 1}.) It is true that C ⊆ [-1, 1] × [-1, 1].

I don't know in the UK, but in the US there is an incessant demand, at least up through high school, that math students draw the arrows on their axes to indicate they go on forever. A student might defend their failure bold choice not do so, on a graph of x² + y² = 1, by saying they were viewing the relation not as between ℝ and ℝ but rather between the bounded subsets of ℝ indicated by the axes that were drawn.

One might imagine another student who draws their x axis as a line (with arrows on the ends, lest it be a line segment) together with some pictures of goats (the elements of G) and their y axis as a line (with arrows on the ends, lest it be a line segment) together with some pictures of doorknobs (the elements of D).

These goat and doorknob pictures wouldn't be separate from the axes, but rather should be thought of as part of the axes, since this hypothetical students is thinking of how C ⊆ (ℝ ∪ G) × (ℝ ∪ D).

In all seriousness, though, whilst I am unconvinced of the wisdom of deducting points for the failure to draw arrows on axes, the way one draws axes does potentially indicate what kinds of things one is claiming that one's plot provides information about.

*this hypothetical student is thinking

Consider a binary relation F where xFy means "x has y as a friend." If this relation is symmetric then we could just say "x and y are friends." Let's go with that.

I'll call Alice, Bob, Cassidy, Derek, and Eve by the short names a, b, c, d, and e, respectively. Also, by "Eve," I actually mean the abstract idea of of an office photocopier that is able to output at least three pages per second and has built-in stapling capabilities. Also, I am not undertaking any trickery here; none of Alice, Bob, Cassidy, or Derek is Eve. Not only that, no two of those names are names of the same person (i.e., Alice, Bob, Cassidy, and Derek are four people).

Suppose Alice is friends with both Bob and Cassidy but, regrettably, Derek has no friends. Then we know:

(a, b) ∈ F
(a, c) ∈ F
(b, a) ∈ F
(c, a) ∈ F
∀x (d, x) ∉ F
∀x (x, d) ∉ F

With usual semantics and assumptions about abstract ideas of office equipment, it is also true that:

∀x (e, x) ∉ F
∀x (x, e) ∉ F

However, it does not seem like the e-sentences, such as "∀x (e, x) ∉ F", are relate to the same kind of facts as the d-sentences, such as "∀x (d, x) ∉ F".

This is not to say that we wish to deny either of them, or even to define F in such a way as to falsify any of them...

...but rather than it might sometimes be of interest to define F in such a way as to contain enough information to distinguish between that which is friendless because it's unclear what it would mean for it to have a friend and F is not intended to cover it, and that which is friendless due to the unfortunate circumstance of not having any friends, which is the sort of thing F is intended to cover.

Note that I have not written an expression for F in terms of (some of) a, b, c, d, and e, because there is not enough information to do so. For example, are Bob and Cassidy friends? I have not given enough information to tell. However, I might want to express what I know about friendship. Suppose G is the relation that holds between x and y when I know x to be friends with y. That is, "xGy" expresses "Eliah knows x has y as a friend."

Suppose further that my knowledge, while it may be incomplete, is not erroneous. Note that G ⊆ F, and, taking the above information as the extent of my knowledge on the matter:

G = {(a, b), (a, c), (b, a), (c, a)}

We of course have ∀x (d, x) ∉ G. Now consider Florence, a person other than Alice, Bob, Cassidy, and Derek. Take "f" to mean "Florence". Then ∀x (f, x) ∉ G too. However, it may well be that Florence has friends, even among Alice, Bob, and Cassidy. (But not Derek, unfortunately.)

It might be handy for the relation G to be able to distinguish between d, which G does not pair with anything due to friendlessness, and f, which G does not pair with anything because it is outside the scope of G's purpose to do so since G is about my (Eliah's) knowledge. But G cannot do so, because it is constructed as its graph.

More formally, what I mean is that I cannot express, in terms of G, d, and f alone, the way d and f differ under G... because in fact they do not differ under G at all, but instead under further information we carry around with us as we think and write about G.

As another example, consider ℤ, the set of integers, and ℝ, the set of reals. We want ℤ ⊆ ℝ (since all integers are real numbers). Let's suppose that we have constructed them appropriately.

(Recall that we can consider any objects that behave like integers, with respect to some way of defining the usual operations on integers, as the integers, so long as we then use them with those operations. The same goes for real numbers. That's why I'm bringing this up. When "ℤ" and "ℝ" appear in the same text, it is nearly always assumed that the sets they name are constructed appropriately so that ℤ ⊆ ℝ.)

Now consider < as defined above. Recall that, as used above, this is the less-than relation on ℤ. Then, with that meaning of <, 2 < 3, but ¬ 2 < 2.5, because 2.5 ∉ ℤ. This is not inherently a problem -- if it is a problem, that suggests that we're doing something that makes it a mistake to use "<" to mean the less-than relation on ℤ!

However, I think this does illustrate how sometimes it is valuable to talk about "the set of things < is intended to carry information about." Since we construct < as its graph, we cannot do that. This, too, is not inherently a problem. We can always talk about some other mathematical object that contains that extra information, such as (ℤ, <). However, it is good to keep this limitation in mind. [And it will come up again, for functions, in regard to codomains and surjectivity.]

(I presented the friendship example first as I consider it to a better example, even if less compelling. There reason for this is that, armed with an understanding of some of the important properties of the integers and real numbers, we can actual discern that "<" is not the less-than relation on the real numbers, just by noticing that, given any real numbers x and y, one of them is less than the other or they are the same. All reals are related to some other real under real-number <.)

In constructing relations as their graphs, we are not merely picking a particular set-theoretic construction for a notion of a relation, but we are also deciding what notion of a relation we wish to use.

@EliahKagan there's no such demand made in the UK

@EliahKagan So for a binary relation R, there are some A and B -- in fact, infinitely many choices of them, though some may be more conceptually illuminating to talk about than others -- so that R is a relation between A and B, i.e., so that R ⊆ A × B. Whether or not R can reasonably be thought of as geometric depends on whether or not A × B can reasonably be given a geometric interpretation. Often, there will be no choice of A and B for which this is so.

The strong examples of this are with sets that are so big that their elements cannot be regarded as points in the kinds of geometry we are usually interested in. However, it is easy to come up with examples where there is no obviously useful geometric interpretation.

@EliahKagan I see XD

For example, one way to represent a game that is deterministic (except with respect to the players' choices) is with a set of possible states S and a relation, I'll call it ⮩. For states x and y, we have x ⮩ y iff it is a legal move to go from x to y in the game. (Information like whose turn it is, any any hysteresis, can be packed into the states themselves.)

It does not seem like, in general, there is any useful way to represent ⮩ geometrically.

I do not mean it cannot be represented pictorially. Sometimes it can. I just mean that this is quite dissimilar to the unit circle example.

The best way to represent ⮩ would usually be with, um, a graph-theory graph. :)

(I mean, graphs in graph theory are abstract, and they are not pictorial in nature, but you can imaging drawing dots for the states of the game and drawing arrows between the dots, and those dots are then the usual pictorial representations of vertices and the arrows are the usual pictorial representations of edges, like in the pictures graphviz draws.)

@Zanna I think that is, if you will pardon a small mathematical pun, a plus.

I don't think anyone has ever suggested to me that axes have to have arrows

I have enough trouble getting people to draw graphs with meaningful scales

@EliahKagan :)

@EliahKagan yes, right... in fact it seems like such pictures would explain a lot of games more clearly than purely verbal instructions, at least for me

@EliahKagan *can imagine drawing

@Zanna Then you may be pleased to know that a representation of games that is similar to (and contains the same information) as that one will come up again. :)

Even if we don't talk about it, which I expect we will, it's in the Smullyan book.

great :D

So does that all make sense?

I think I have somewhat followed it

Are there particular parts that do not fully make sense, or that make less sense than other parts?

@EliahKagan I didn't really get this part

Did this subpart itself make sense?

> C is a relation between ℝ and ℝ (i.e., between ℝ and itself)

but maybe I was distracted by the mention of pictures of goats

@EliahKagan no

@Zanna :)

I mean I'm sure it does but I don't see what it means

@Zanna Do you know what I mean, when I say of some binary relation R, that is a relation between the sets A and B?

I think I got that

This relates (pun!) to what I was saying there.

So, I did not say this explicitly, but should have, and actually meant to. To say that R is a relation between the sets A and B is to say that it answers questions of the form "Is x related to y?" where x ∈ A ∧ y ∈ B.

It answers yes when xRy and no otherwise.

So, given any x and y for which xRy, surely x ∈ A and y ∈ B.

Thus R ⊆ A × B.

(But unless everything in A is related to everything in B under R, R ≠ A × B.)

@EliahKagan oooh

So, where by "<" I mean the less-than relation on ℤ (the set of integers), that relates integers to integers, i.e., given any x and y for which x < y, x ∈ ℤ ∧ y ∈ ℤ. Thus < ⊆ ℤ × ℤ. Similarly, as given above, where C is the the origin-centered unit circle relation on the 2-dimensional Cartesian coordinate plane, C ⊆ ℝ × ℝ. After all, the whole 2-dimensional Cartesian coordinate plane is ℝ × ℝ.

(Note that "ℝ" denotes the set of real numbers and should not be confused with plain-written "R".)

@Zanna :)

Sorry I did not mention that earlier. It is quite important. It is also very much connected to how n-ary relations are objects that do the work of n-ary predicates, in the same sense that sets are objects that do the work of unary predicates, and n-ary functions (which I've often referred to here but not yet actually introduced) are objects that do the work of n-ary function symbols.

I claim that this is the same notion of a relation as you were already familiar with, except:

(a) that here it is in the context of set theory, where relations are constructed as sets -- for compatibility with set theories were the universe of discourse is sets, as is most commonly done, ZFC being the most popular set theory, and because it is not necessary to rely on the existence anything but sets to construct relations, even if one has no objection to doing so, and

(b) the oft-unconsidered question as to whether or not a relation knows what it reigns over (i.e., what kinds of things it relates).

@EliahKagan right, I see

So, for example, you have probably drawn a graph (i.e., a plot) of < on the reals, some interval of the reals. That is, shaded the upper-left side of the line y = x. Or some similar exercise. This is a direct pictoral representation of the graph of < (i.e., the set of pairs (a, b) where a < b). And we regard < to be that graph.

But what is conceptually important about relations is not that they are constructed as their graphs, but that n-ary relations answer questions of the form "Are (these n things, given in this order) related?

Do you see what I mean when I say that n-ary relations relations are objects that do the work of n-ary predicates, in the same way that sets are objects that do the work of unary predicates?

@Zanna Also, does the rest of that message make more sense now, in light of that? (Please feel free to say no.)

@EliahKagan yes I think so

@EliahKagan The reason I said "I claim that...", rather than just saying it is so, is that it is conceptually significant that relations are objects, that you can say things about them, that they can be elements of sets, that you can have relations that hold between relations, and that all these things (even though "higher-order" relations) are objects, and that a first-order theory is sufficient.

I remember the day, years ago, when I learned that symbols like "<" and "+" are names for objects. It changed my life. :)

@EliahKagan yes. In the rest of the message, we have some goats and doorknobs which are purely there for entertainment

Yes, they are not related to anything under C.

@EliahKagan oh yes?

@EliahKagan I am assuming no goat or doorknob is a real number. Or at least not a real number in the interval [-1, 1]. I should have actually mentioned this explicitly, since we could construct goats or doorknobs as real numbers. I'm not saying that's likely to be fruitful, only that we could do it. (Or we could end up constructing them the way way we happened to construct some real numbers.)

I'm not insisting that we must construct goats or doorknobs set-theoretically, only seeking to avoid neglecting the possibility that we might.

hahaha

@Zanna It is a long and, at least to me, very interesting story that I hope to tell you sometime. I was 16 and interested in the meaning of the math used in physics. A physics professor answered my extensive questions about this topic, and along the way taught me a large portion of all the math I have ever learned (thus far).

@Zanna Are you uncertain? And if so, is your uncertainty about relations, or about the connection between sets themselves and unary predicates?

@EliahKagan just feeling that I need more, sort of, practice with these things. Not so much uncertain

@EliahKagan sounds like a great story!

@Zanna It occurs to me that there is also some information I've alluded to a couple of times but not presented, which is very important to the relationship between sets and unary predicates. That is, even though that's something I presented, I have not justified the notion that a formal theory of sets actually achieves this goal in a useful way. (It does; I just haven't yet justified it.)

As mentioned far above, one solution to the problem posed by Russell's paradox is type theory, and such a system was developed by Alfred North Whitehead and Bertrand Russell in Principia Mathematica, but this is not the system that caught on. (Type theory is used today--though not usually as a general foundation for mathematics, though that can be done--but not usually the system developed in PM.)

As I alluded to but did not cover, even though this is obviously far more relevant to your interest in set theory, set theory (built on first-order logic) was the other major approach. Set theory is considered to have been initiated by Georg Cantor, but it was formalized by Ernst Zermelo, whose system was then improved with the aid of Abraham Fraenkel.

In using Wikipedia to look up some people's first names and their spellings, I note the claim that Thoralf Skolem contributed in a manner on par with Fraenkel, which accords with some things I recall reading, but I don't remember the details. (I am familiar with some significant contributions by Skolem, just not which parts of ZF and ZFC he contributed to.)

but his name got missed out

Well, it's in the name of various theorems. :)

Recall that, in Frege's system, given any formula with one free variable, there was a set of all and only those objects that satisfied the formula. The problem with this was the Russell set r = {x | x ∉ x}, since r ∈ r ↔ r ∉ r, which is Russell's paradox.

PM's solution to Russell's paradox was to make it so there was no well-formed way to even say x ∈ x or x ∉ x, by making it so that the type of terms that appear on the left of ∈ must be of lower order than the type of terms that appear on the right of ∈ in the same formula. In that system there are classes (in that setting I'd rather call them classes, as they're not quite what we think of as sets today) of all orders, but a class can never contain anything of the same order as itself.

Zermelo's solution--the modern set theory solution--to Russell's paradox is to weaken comprehension. That is, instead of having axioms that give you, for each formula with one free variable, the set of all and only those objects that satisfy the formula, have axioms that give you, for each formula with one free variable and an existing set S, the set of all and only those objects that satisfy the formula and are members of S.

It turned out that this was not quite enough, and it was later strengthened in a way I shall eventually describe (which was a major change from Zermelo set theory to ZF). However, the key point remains that, in modern set theory, the unrestricted comprehension of Frege's system has been replaced with axioms that, given a set, let you prove there are various sets that are smaller or the same size as it. It does not give you an sets initially! For that, other axioms are required.

The goal in selecting the axioms that allow one to prove that some sets of the sizes you need exist is to pick ones that are simple and reasonable. And I don't mean to say that all the axioms of ZFC are for this purpose. Those that are for this purpose are the axioms of pairing, union, infinity, and power set. BRB, but you may want to look at some of those axioms.

@Zanna So the key, to how the relationship between sets and unary predicates is retained, is that once I have a set S, given a unary predicate F there is a set T of all and only those objects x in S for which Fx. (Actually, it is broader than that; it can use any formula with one free variable, it doesn't have to be an atomic sentence.)

I expect this may make more sense when I actually present the axiom schema of restricted comprehension (a.k.a. the axiom schema of selection, a.k.a. the axiom schema of specification) formally, but what this means is that some (though not all) forms of set-builder notation have an associated mechanical procedure for generating existence proofs.

Given the axioms of a modern set theory like ZFC, we know there is such a thing as {x ∈ S | Fx}, given a set S and a predicate "F". To the right of the | (which some people write as a :), we can write whatever formula we like in place of Fx so long as it has exactly one free variable, then make sure that's the variable we put in front of ∈ S on the left of the |, and that's itself a sketch of the proof of the set it refers to.

Note that you can also replace S with a term of arbitrary complexity, but if it's a definite description that fails (or the analogous situation with function symbols), then the existence of the set that one has written the set-builder expression for is not substantiated.

That applies for ternary relations, quaternary relations, etc. (though showing so requires the additional construction of ordered triples, ordered quadruples, etc., which can be built atop one's construction of ordered pairs).

@EliahKagan By this I mean it wasn't enough to do all the math that one wants to do, not that it wasn't enough to escape Russell's paradox.

This also shows that there is something quite natural, in the context of set theory, of regarding an n-ary relations as not knowing or, this is perhaps a better way to put it, not having an opinion about which n it should be regarded to relate.

After all, the T you define as T = {x ∈ S | Fx} doesn't care which set S was, even though there may be contexts where we would like to carry that information around (in which case we can still do so, e.g., by using such an oejct as (S, T) instead of just T).

By this I mean that any object a for which ¬ Fa can be added to S. If S' is produced from S by doing that arbitrarily many times, then {x ∈ S | Fx} = {x ∈ S' | Fx}. Sets are extensional; two sets whose answers to "do you contain x?" always agree, no matter what x you pick, are the same set. Similarly, we define relations extensionally, so two binary relations whose answers to "do you relate x to y?" always agree, no matter what x and y you pick, are the same relation.

(And so forth for n-ary relations of higher n than 2. Note that a set could be regarded as unary relation.)

So I don't want to disagree with the idea that practice is important, since it is. But I think there was also some missing information, which I think that should clarify.

Please let me know what you think, and if the connection between relations and n-ary predicates (as well as why, in set theory anyway, we regard relations as not caring about what specific sets they relate, and construct them as their graphs) is clearer based on it.

I will after sleeping:)

There is no hurry.

@Zanna It may be that this is no longer demanded in the US. I don't know. I actually do not know for sure that it was ever demanded all throughout the US. It was one of the things that points were officially required to be deduced for on some of the math Regents exams in the state of New York. So they taught us always to draw the arrows.