Ok. For things in the real world, they are already in some order, even if it's a complete state of disorder. Even a completely disordered collection of gas molecules in a container, at every instant each molecule is wherever it is. And that set of coordinates, locating every molecule in space, is the order.

I get that. But by the same token, there is no preferred order. Suppose for example that I got my schoolkids from the playground to line up single-file in order of height. And now YOU come along and say, "Ah, that is the inherent order, and all other orders are disorders of that."

But of course your observation was a complete accident. I could have lined them up alphabetically by last name.

So even among physical objects, if we allow that they are always in some order, even if it's disorderly; but nevertheless, there is no preferred or inherent order.

I believe you are saying there's an inherent order, have I got that right? — fishfry

Well now that you mention it, no. 1, 2, 3, ... is NOT the inherent order of the set N

, believe it or not. On the other hand it sort of is, in a sneaky way. Von Neumann defined the symbols 1, 2, 3, ... in such a way that n∈m

∈

if it happens to be the case that we want n < m to be true. — fishfry

I know this is hard for normal humans to accept, since it's pretty obvious that 1 < 2 < 3 and so on. But mathematicians insist on being picky about how numbers and other things are defined. In the set-theoretic view of modern math, the numbers 1, 2, 3, ... are defined as particular sets, with no inherent order; and then we impose their order by leveraging the ∈
∈
operator. — fishfry

Have I got any of that right? — fishfry

Anyway, back to the question. How do we know that 2 and 3 are not the same set?

Well 2∈3
2
∈
3
, but 2∉2
2
∉
2
.

Therefore by extensionality, 2≠3
2
≠
3
, because they don't have exactly the same elements.

Perhaps you can begin to see the virtues of working a the set level separately from its order properties. We can see the mechanics of how to use the axiom of extensionality. No order properties are needed to determine that 2 and 3 are different sets. It's just a matter of ignoring hypotheses that you don't need for a particular argument.

Nobody is saying that a given set doesn't have an order, as well as a lot of other stuff. A topology, some algebraic operations, a manifold structure perhaps. But we can learn a lot just from restricting our attention to the membership relation and seeing what we can learn just about that. — fishfry

Then you have been proven wrong. I don't need to mention or consider or use any of the order properties of 2 and 3 to determine that they're different numbers. — fishfry

Entirely without rational basis. This para is a wild generalization of your complaint about 2 and 3, but I already showed how we can distinguish 2 and 3 using only their membership properties and not their order properties. — fishfry

You are thrashing away at a strawman you've created out of your imagination, and under the mistaken belief that we can't tell 2 from 3 without their order properties. But we can. — fishfry

No, you are consistently wrong about this. If A and B are sets and I can prove that A = B, then A and B are the same set. They are in fact the identical set, of which there is only one instance in the entire universe. They are NOT "two copies" or two distinct entities that we are calling the same by changing the meaning of the word "same." — fishfry

DUH that is what it MEANS to be the same set. That is the ONLY thing it means to be the same set. — fishfry

Yes that is what it MEANS for two sets to be the same. That they have the same members. That's ALL it means and EVERYTHING it means.

You simply can't accept that and I don't know why. — fishfry

For set theory, an example is Hindman's 'Fundamentals Of Mathematical Logic' — TonesInDeepFreeze

I think we have to look at context here. What is our subject of discussion, what are we talking about here? — Metaphysician Undercover

Are we talking about things (individuals), of which there is a multitude, or are we talking about a group (set) of individuals, of which there is one? Your description above, seems to imply the former. You are talking about separate things, many schoolkids, and there is many possibilities as to the order they could have. On the other hand, if you were talking about the group as a whole, as your subject, then the parts of that group, the individuals, must have the order that they have at that time, even though it could be different in past or future times. If you were talking about the same individuals in a different order, this would require a change to that specific group, so you would be talking about that group, at a different time, because you'd be talking about the individuals, changing places. — Metaphysician Undercover

You might understand this better through what is known as internal and external properties. To each individual, as a subject, its relations to other individuals are external properties. To the group, as a unit, and the subject, the relations between the individuals is an internal property. — Metaphysician Undercover

You talk about the schoolkids as distinct individuals, where the various relations between them are the external properties of each and everyone of them. There are no internal relations here. Each schoolkid is a subject to predication, age, height, etc.. and you might produce an order according to those predications. The order is external to each schoolkid, people say it transcends, and changing the transcendent order does not change any of the schoolkids in anyway. — Metaphysician Undercover

Now, let's take the group as a whole, as an object, and produce a corresponding subject, the set, and make that our subject. Since the whole group is our object of study, any change to the order of the individuals is an internal change to that object, therefore a change to that object itself. The order of the individuals (as the parts of the whole) is an internal property of that object, and a change to that order constitutes a change to the object, which we must respect as predicable to the corresponding subject. Therefore we can say that the order of the individuals, as the parts of the whole, is an intrinsic property of the whole, which is represented as the set. — Metaphysician Undercover

Notice however, the switch from "subject" to "object", and this I believe is the key to understanding these principles. There is an implicit gap, a separation, between the meaning of "logical subject" and "physical object". When we make a predication, "the sky is blue" for example, "the sky" is the subject, and if there is an object which corresponds with that subject, the predication may be judged for truth. However, we can manufacture subjects and predications with complete disregard for any physical objects, and so long as we have consistency, we have a valid "subject", with no corresponding object. — Metaphysician Undercover

Consider the following proposition, "There is a group of schoolkids". We have a propositional subject, without a corresponding object, what some people would call "a possible world". Since there is no assumed corresponding object which would cause a need for conformity, we can predicate any possible order we want, so long as it is not contradictory. The hidden problem of formalism which I referred to lies in the naming of the group, "schoolkids". That name needs to be clearly defined and the definition will place restrictions on what can be predicated without contradiction. — Metaphysician Undercover

Perhaps, we can remove these restrictions, by making the things within the group, the elements of the set that is, completely nondescript. "There is a group of nondescript things". We still have the name "things", with implied meaning, so this name has to be defined, and this would put restrictions on what we can predicate. So we go to a simple symbol, "x" for example, and assume that the symbol on its own, has absolutely no meaning, and this would allow any individual predication whatsoever without any risk of self-contradiction. X is a subject which has absolutely no inherent properties. — Metaphysician Undercover

It might appear like we have resolved the problem in this way, we have a subject "x" which can hold absolutely any predication, so long as the predications don't contradict.. However, when we assume that the subject has no inherent properties, we disallow any predication because the predication would be a property and this would contradict the initial assumption. So this starting point allows no procedure without contradiction. — Metaphysician Undercover

Now look what happens when we say "there is a group of x's". There is actually something implied about x, which is implied simply by saying that there is a group of them. It is implied that x has a boundary, separation, etc.. We may start with the assumption that there is no intrinsic properties of X, but as soon as we start to predicate, we negate that assumption. And the symbol, x, without any predications is absolutely useless. — Metaphysician Undercover

I agree, what I meant is that this appears to be the inherent order, but it's not necessarily, that's why I went on to say that we can deny that order. — Metaphysician Undercover

I think so, but I also think, that sort of inherent order has minimal effect, and the real issue comes up with the restrictions, or limitations to order which are constructed. What I am arguing is that how the inherent order manifests, is as a limitation to the order which one can select. If there is absolutely no inherent order, then we can select any order, but if there is limitations to what can be selected, we cannot choose any order. — Metaphysician Undercover

The examples you give are, I believe, selected, therefore they're probably no true inherent order. The example I gave, is that we cannot give 2 and 3 the same place in the order, they cannot be equal, so we need to proceed toward understanding how this limitation exists. — Metaphysician Undercover

So this is where the real problem lies, in defining a symbol, such as 2 or 3, as a set. — Metaphysician Undercover

Check back to what I said about the difference between internal and external properties. The subject now is a set, say 2, and a set necessarily has internal properties. We have the elements which compose the set, 0,1, which are also sets. As the set is also related to other sets, it has external properties, represented by the ∈
operator. The external properties are not necessary, and are a matter of choice, but whatever choice is made, that choice dictates the nature of the internal properties. — Metaphysician Undercover

Now here's where I think the illusion lies. A set necessarily has internal properties, even though there may be infinite possibility as to the nature of the internal properties, making the specific nature of the internal properties dependent on choice, in this case von Neumann's choice. The illusion is that since the specific nature of the internal properties is dependent on a choice from infinite possibilities, it would therefore be possible to have a set with no internal properties. Clarification of the illusion implies that a set cannot exist prior to the choice of external properties, which dictate the internal properties. Internal properties are essential to "a set", and so a set has no existence prior to the choice of external properties, which determine the internal properties. This makes the empty set, as a set with no internal properties, impossible. The problem now, is what is zero? It can't be a number, because numbers are sets, and an empty set is impossible. — Metaphysician Undercover

I think you misunderstand. As I explain above, you refer exactly to the internal (intrinsic) properties of 2 and 3, as sets, — Metaphysician Undercover

to show that they are different numbers. What the set theory has done is denied order as an external property of those things, 2 and 3, as numbers with order relative to other numbers, and made it into an internal property of those things, as sets. An internal property is an intrinsic order. The fact that the intrinsic order is ultimately dependent on choice is irrelevant, because some order must be chosen for, or else the system would be meaningless. — Metaphysician Undercover

No, you've simply shown how external order has been switched for internal order. And now I've shown the problem which arises from this switch, the contradictory, therefore impossible "empty set", which makes the inclusion of zero an inconsistency. — Metaphysician Undercover

As I say, the idea that you've gotten rid of the order properties is just an illusion. The order inheres within each individual number, as the definition of that specific set. Rather than simply being an external property of a number, as an object, and how it relates to other numbers, order is now an internal property of the number itself, as a set.. — Metaphysician Undercover

I argue the exact opposite, that you are consistently wrong about this. It is exactly "two copies", just like the word "same" here, and the word "same" here, are two distinct copies, even though we say it's the same word. Look, we are talking the meaning of symbols here. "A=B" means that that symbol A has the same meaning as B, it does not mean that A signifies the same entity as B, without additional information. However, the additional information in this case indicates that what is signified by A and B is a set, "the same set". But a set is not a thing, it is a group of things, grouped by a categorization such as type. Therefore this is an instance of "the same meaning", signified by A and B (indicated by "type"), not an instance of the same entity signified by A and B. This is just like when we use the same word twice when the word has meaning, rather than referencing a particular object. We say that the word has the same meaning, just like we might say A and B have the same meaning, in your example. — Metaphysician Undercover

Exactly, and this is a different meaning of "same" from the meaning of "same" in the law of identity. That is the point. In the law of identity "same" means a lot more than simply having the same members (what I called a qualified "same"), it means to be the same in every possible way ("same" in an absolute, unqualified way), — Metaphysician Undercover

I totally agree with that, that's what "same" means in this context. — Metaphysician Undercover

The problem is that it does not mean what you stated above: "They are in fact the identical set, of which there is only one instance in the entire universe". The set is an imaginary thing, indicated by meaning, it is not something in the universe. So it's not even coherent to say that there is one instance of that set, it's not even a thing which has an instance of existence, it's just the meaning of a symbol. So you speak of "the same set", and claim there is only one instance of that set, but this would be taking a different meaning of "same", which refers to instantiated things, and applying it to "same set", which really means having the same meaning, and not referring to one instantiated thing. Do you see the difference between referring to one and the same thing with a name, "MU", and using a word which has meaning, like "person", without any particular thing referred to? Person refers to a type, so it has meaning, just like "set" refers to a type, so it has meaning. These do not refer to instantiated things of which we could say there is one instance of, they refer to ideas. — Metaphysician Undercover

I will agree with you that identity is implicitly in extensionality, in the sense that two sets are equal if they have the "same" elements. — fishfry

We need identity to know when two elements are the same. — fishfry

Now, since the elements of sets are other sets (barring urelements for the moment), I can see that there's a bit of a pickle. I''m not sure how this pickle is resolved. — fishfry

Perhaps this is what you're trying to explain to me.

Is it? — fishfry

Enderton's set theory text is a great book. But, as with many excellent set theory books, it doesn't mention all the technical details. — TonesInDeepFreeze

I didn't say that identity is implicitly in extensionality, whatever that might mean. — TonesInDeepFreeze

I've said that usually set theory is based on first order logic with identity. That includes the identity axioms (such as found in Enderton's logic book). Then set theory adds the axiom of extensionality that provides a sufficient condition for identity that is not in identity theory. — TonesInDeepFreeze

if A and B are both sets
    use extensionality from set theory
else 
    use identity from logic

I don't know how I could be more clear about that. Explicity:

Start with these identity axioms:

Ax x=x (a thing is identical with itself)

and (roughly stated) for all formulas P(x): — TonesInDeepFreeze

We need identity axioms to prove things we want to prove about identity, including such things as: — TonesInDeepFreeze

Suggestion: Learn the details of the axioms and rules of inference of first order logic with identity. Then start with the very first semi-formal proofs in set theory (such as a set theory textbook usually gives semi-formal proofs), and confirm how those proofs would be if actually formalized in first order logic with identity. Then you would see how the axioms and rules of inference of first order logic with identity play a crucial role in set theory. — TonesInDeepFreeze

I have no idea what pickle you see. — TonesInDeepFreeze

Perhaps this is what you're trying to explain to me.

Is it?
— fishfry

No. — TonesInDeepFreeze

If you read again the first post in this thread on this particular subject, with regard to exactly what I've said, step by step, then it may become clearer for you. But also, as mentioned, learning the axioms and inference rules of first order logic with identity would be of great benefit. My suggestion would be to start with:

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar
(but you could skip it if you feel strong enough already in doing formal proofs in symbolic logic and making simple models for proofs of consistency and of proofs invalidity by counterexample)

Then:

A Mathematical Introduction To Logic - Enderton — TonesInDeepFreeze

I found both of those books to be a special pleasure and profoundly enlightening. The Enderton book especially blew my mind, as I saw in it how mathematical logic so ingeniously, rigorously and elegantly gets to the heart of the fundamental considerations of logic while making sure that no technical loose ends are left dangling. — TonesInDeepFreeze

if A and B are both sets
use extensionality from set theory
else
use identity from logic — fishfry

I don't know how I could be more clear about that. Explicity:

— TonesInDeepFreeze

Ok at this point, I am wondering: Why are you telling me this? I don't understand what you want me to know about this. What problem are we trying to solve? — fishfry

I am certain I never said we don't need identity! — fishfry

I'm all for the law of identity. A thing is equal to itself. That's good do know. In fact it helps make equality an equivalence relation with exactly one item per equivalence class. — fishfry

Alas. — fishfry

I have no idea what pickle you see.
— TonesInDeepFreeze

Then my attempt to explain my take on the subject we're discussing failed. — fishfry

If that is the price of conversing further with you on these matters — fishfry

I don't recall the context in which I recommended Enerton's set theory book, but if it was about first order logic with identity for set theory, then I mis-recommended. — TonesInDeepFreeze

Who was the famous logician? — TonesInDeepFreeze

Shoenfield's logic textbook is rich and has lots of stuff not ordinarily in such a book. But it is difficult, and he uses some terminology inconsistent with ordinary use in the field. — TonesInDeepFreeze

As I recall, many posts ago, my initial point was that, contrary to your assertion, the axiom of extensionality, as ordinarily given, is not a definition. — TonesInDeepFreeze

Then I went on to explain how there are other ways to set up the logic and the set theory axioms so that a different version of extensionality would be a definition. — TonesInDeepFreeze

An ordinary presentation of set theory either explicitly or implicitly has set theory based upon first order logic with identity theory. — TonesInDeepFreeze

Yes, of course, set theory has non-logical axioms, so set theory is not just first order logic with identity. — TonesInDeepFreeze

if A and B are both sets
use extensionality from set theory
else
use identity from logic
— fishfry

That's not right. — TonesInDeepFreeze

In set theory, we use both the logic axioms (which include the identity axioms) and the set theory axioms (which include the axiom of extensionality). lf our focus now speaking is identity theory and the axiom of extensionality, then it suffices to say that we use both. — TonesInDeepFreeze

I don't know how I could be more clear about that. — TonesInDeepFreeze

I was trying to solve the problem that you had not been understanding me as you characterized my point again incorrectly, so I tried to state it in as simple terms as I could. — TonesInDeepFreeze

I didn't say that you did. Rather you gave your reason that we need identity. And I take it that 'identity' in that context is short for 'the axioms and semantics regarding identity', and I gave better reasons that we need them. — TonesInDeepFreeze

However, several posts ago you did indicate (as best I could tell) that you think the axiom of extensionality is all we need for proving things about identity in set theory, — TonesInDeepFreeze

which would comport with your view that the axiom of extensionality is a definition. — TonesInDeepFreeze

So, I mentioned that, for example, from the axiom of extensionality alone we cannot prove:

(x = y & y = z) -> x = z — TonesInDeepFreeze

We need the law of identity, but we also need the indiscernibility of identicals. — TonesInDeepFreeze

(But Wang has an axiomatization in a single scheme.)

Yet, interestingly, from the axiom of extensionality we can derive the law of identity:

(1) Az(z e x <-> z e x) logic

(2) x = x from (1) and the axiom of extensionality

But the law of identity does not ensure that '=' stands for an equivalence class. It only provides

x = x

It does not entail

x = y -> y = x

nor

(x =y & y = z) -> x = z

To get all the needed identity theorems, we need both the law of identity and the indiscernibility of identicals. — TonesInDeepFreeze

Moreover, we want to ensure that '=' stands not just for an equivalence relation but for, indeed, the identity relation. — TonesInDeepFreeze

And to do that we have to make the semantic stipulation that '=' is interpreted as standing for the identity relation on the universe. — TonesInDeepFreeze

Me too. There is so much I didn't learn a long time ago but should have learned. I never got past a pretty basic level. And now I am very rusty in what I did learn, and don't have very much time to re-learn, let alone go beyond where I was a long time ago. — TonesInDeepFreeze

You said that sets have sets as members and that there is a pickle about that viv-a-vis identity. — TonesInDeepFreeze

Of course it is not. — TonesInDeepFreeze

Of course it is. It's an axiom. It says what is true about all the things we call sets. Therefore we can characterize the world of things into sets and non-sets, according to whether they satisfy the axiom. So axioms serve as definitions and vice versa. They are the same thing. — fishfry

Then I went on to explain how there are other ways to set up the logic and the set theory axioms so that a different version of extensionality would be a definition.
— TonesInDeepFreeze

That's fine, but that's one of the points where you lose me. Why do you care, or why do you think your doing so will make me understand something I didn't understand before? — fishfry

I've never heard of identity theory except in the context of many of the Wiki disambiguations. And when I showed you the most likely meaning, you rejected it. So I have no idea what identity theory is. — fishfry

Yes, of course, set theory has non-logical axioms, so set theory is not just first order logic with identity.
— TonesInDeepFreeze

I don't recall even having an opinion about this, let alone expressing it in this thread. — fishfry

But if set theory adds an axiom, then clearly it is not the same thing. It's something else, a new thing. — fishfry

I said reasons and you said better reasons? Ok. Your reasons are much better than my reasons for believing things we both agree on. — fishfry

Every attempt I make to understand you is wrong. So maybe just give up because I don't get it. — fishfry

We need the law of identity, but we also need the indiscernibility of identicals.
— TonesInDeepFreeze

Not really. — fishfry

So, I mentioned that, for example, from the axiom of extensionality alone we cannot prove:

(x = y & y = z) -> x = z
— TonesInDeepFreeze

Of course we can, straight from the axiom. — fishfry

Moreover, we want to ensure that '=' stands not just for an equivalence relation but for, indeed, the identity relation.
— TonesInDeepFreeze

The identity relation is an equivalence relation. — fishfry

EVERY idea I toss out to try to relate to what you're saying, you reject. — fishfry

To get all the needed identity theorems, we need both the law of identity and the indiscernibility of identicals.
— TonesInDeepFreeze

Irrelevant to anything I can relate to, in this conversation or in general. — fishfry

And to do that we have to make the semantic stipulation that '=' is interpreted as standing for the identity relation on the universe.
— TonesInDeepFreeze

I never stipulated to it. — fishfry

Physical collections have inherent order. Sets don't. That's all I'm saying. You seem to agree. What are you concerned with then? — fishfry

Is this related to the intensional and extensional meaning of symbols as has been discussed previously? — fishfry

I have no idea what topic you are discussing at this point. I've agreed with you about the playground and I thought I'd explained to you about sets. What is left to discuss? — fishfry

I understand what you're saying, I just don't have much interest in the subject. — fishfry

I am not your philosophy professor and this is not going to get you a good grade in my class. Why are you telling me all this? Honestly. I don't get it. I'm sorry. — fishfry

I'm not referring to internal properties of anything. 2 and 3 are MODELED within set theory as sets. — fishfry

There's only one instance of each set. You seem to disagree. Don't know what to say. — fishfry

How can you disbelieve in a rule in a game? — fishfry

Nobody but you is making ontological or metaphysical claims about sets. — fishfry

If you could just clearly summarize your concerns, it would help. The internal and external stuff, I'm sure it's interesting, but I was not able to relate it to anything we've ever talked about. So just toss me a clue if you would. — fishfry

I post for at least as an end in and of itself, and also meaningful record for whomever may read it, no matter how few people or even accepting that it might be none at all. It would be good if my best efforts in explanation were understood, but I cannot ensure that they are, especially given that they are ad hoc and out of context of the required material they depend on. — TonesInDeepFreeze

We're going around full circle. — TonesInDeepFreeze

(1) I said it may be more commonly called 'first order logic with equality'.

(2) For about the fourth time, a only a few posts ago I gave the axioms. And you responded by asking why I posted it! — TonesInDeepFreeze

(3) And I gave you a reference to Enderton where he stated an axiomatization equivalent with the one I gave. And Hinman also, and moreover as he states set theory as based on first order logic (which is to say, first order logic with equality). — TonesInDeepFreeze

(4) You said yourself that you recognize that set theory is based on first order logic. Set theory is based on first order logic with equality. That is what identity theory is, as I've said before. Whether called 'identity theory' or 'first order logic with equlality', it's the same set of axioms. — TonesInDeepFreeze

Yes, because the reasons I mentioned go the heart of the motivation for the axioms. — TonesInDeepFreeze

That's up to you. But I am not errant for correcting things that are wrong. — TonesInDeepFreeze

And you studied with Shoenfield. On page 21 lines 13 and 15 of his book you will see the equality axioms that are the indiscernibility of identicals, similar to the way I formalized and that you asked why I posted it. — TonesInDeepFreeze

So what? In logic it is ordinarily stipulated. — TonesInDeepFreeze

The reason why physical collections are different from sets, in this way, is that physical objects are different from intelligible (including mathematical) objects. What I am concerned about is that the law of identity, as formulated from Aristotle, is specifically designed from a recognition of this difference, and intentionally designed to protect, and maintain the understanding and acceptance of that difference. To put it simply, an abstraction, intelligible object, is a universal, and a physical object is a particular. The law of identity refers to the identity of a particular. And, because intelligible objects are different from physical objects, as you recognize and acknowledge, they cannot be held to this law. So mathematical ideas, if they are called "objects", are objects which naturally violate the law of identity. In short, that's how we distinguish physical objects from ideas, with the law of identity. — Metaphysician Undercover

In classical sophistry physical objects are confused, mixed up, and conflated with intelligible objects. The difference between the particular and the universal, as "objects" is ignored. This allows sophists to logically prove things which are absurd. The law of identity is intended to enforce that difference, and expose the faults of the sophist. The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity. — Metaphysician Undercover

I'd have to say, no, not really. Internal/external properties is a distinction we make concerning the properties of particular physical objects, the object's internal relations, and the object's external relations. Intensional/extensional meaning is a distinction concerning the meaning of a word, how the word relates to ideas, and possibly physical objects. This is a matter of semiotics, and Charles Peirce provides some very good insight into the use of symbols. But that is a completely different matter from what I was discussing, as the internal/external properties of a physical object. — Metaphysician Undercover

The problem, is that you continually cross the boundary of separation between physical objects, and intelligible objects, in your manner of speaking, in the sophistic way, without even noticing it. — Metaphysician Undercover

That's what happened with your example of schoolkids. In order for the example to work, "schoolkids" must refer to a multitude of particular physical objects. Yet "set" must refer to an intelligible object. So in speaking the example you cross the category separation, back and forth in the way of sophistry, without even realizing it. — Metaphysician Undercover

Imagine if we were to maintain the boundary. Instead of having schoolkids in a playground, we would be talking about the idea of "schoolkid", or an imaginary schoolkid. This appears to deny the possibility of any extensional meaning. Further, if we want a number of schoolkids, then we need a principle of separation to distinguish one from the other. But that principle of separation would either create an order amongst the imaginary schoolkids, or else produce a complete separation of type, making distinct types of schoolkids. — Metaphysician Undercover

OK, you have no interest in the difference between a subject to be studied and an object to be studied. — Metaphysician Undercover

That's fine by me, but until you learn this difference you are likely to continue to speak in a way which mixes these two up, and makes your examples and arguments appear like nothing more than sophistry, and arguing by equivocation, just like Tones. — Metaphysician Undercover

This is what happens when a subject is called an object (mathematical) and the difference between the physical object and the mathematical object, (as defended by the law of identity) is ignored. — Metaphysician Undercover

That's right, you are not my philosophy professor, that would reverse credentials. I am your philosophy professor, and your lack of interest deserves a failing grade. — Metaphysician Undercover

Right, this is why a set is not an object, objects have internal properties and external properties, sets have meaning. — Metaphysician Undercover

There is no "instance" of any set. — Metaphysician Undercover

You recognize that there is a difference between physical objects an sets, why do you not see that there is no such thing as an instance of a set? — Metaphysician Undercover

Sets are not the type of thing which have an instantiation. "Instance" refers to a particular, a set is a universal. That sort of misleading statement is where the sophistry kicks in, even though I know you are not intending to be misleading.. — Metaphysician Undercover

That's a simple question with a simple answer. When a rule in a game contradicts another rule in a game, this is cause for disbelief in the whole game. That was the point of the example I gave you of waves in physics. — Metaphysician Undercover

That has become obvious to me. But in a philosophy forum, things ought to be the other way around. We ought to be discussing the ontology of sets and working through the problems which arise. — Metaphysician Undercover

There's too many concerns to summarize. But let's look at a most fundamental problem of set theory as an example. You recognize the difference between physical objects, and sets, so let's start there. — Metaphysician Undercover

Now, consider the elements of a set, these might be sets as well. — Metaphysician Undercover

The elements of a set are not physical objects, just like sets are not physical objects. — Metaphysician Undercover

The elements are ideas, universals, they are not particulars or individuals. — Metaphysician Undercover

Since they are not particulars the set cannot be measured as particulars. A set cannot have a cardinality. That's a basic problem. — Metaphysician Undercover

rantings — fishfry

Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'. — TonesInDeepFreeze

(4) You said yourself that you recognize that set theory is based on first order logic. Set theory is based on first order logic with equality. That is what identity theory is, as I've said before. Whether called 'identity theory' or 'first order logic with equlality', it's the same set of axioms.
— TonesInDeepFreeze

Ok — fishfry

You have MUCH BETTER REASONS than I do. Ok. — fishfry

And you studied with Shoenfield. On page 21 lines 13 and 15 of his book you will see the equality axioms that are the indiscernibility of identicals, similar to the way I formalized and that you asked why I posted it.
— TonesInDeepFreeze

I admitted to being a logic slacker. — fishfry

I no longer know what we were talking about. — fishfry

Making clear corrections, giving generous explanations, and posting ideas in general is not ranting. — TonesInDeepFreeze

And by starting with Enderton's logic book, which does present the axioms for '=', you would see how they work in set theory even if not explicitly stated in his set theory book. — TonesInDeepFreeze

But when you complained that it does not mention identity theory, I said that I would have been mistaken if I offered it for reference on that matter. And, now that I see the context, I grant that, since the context was general, it would not be entirely unreasonable for you to take it that at least part of the reason for my recommending the book is that it mentions identity theory, so, in that respect, my recommendation my be faulted. — TonesInDeepFreeze

But then I followed up by pointing to Enderton specifying the equality axioms in his logic book (though he doesn't mention in that book the fact that set theory is based on first order logic with equality). And that was pertinent to your complaint that you couldn't find anything on that topic. — TonesInDeepFreeze

And I cited Hinman's book that both gives the axioms for equality as part of first order logic, equivalent to the axioms I posted, and he says that set theory is based on first order logic. — TonesInDeepFreeze

And I referred you to Shoenfield's book that specifies the axioms for '=', equivalent to the axioms I posted. — TonesInDeepFreeze

And, you yourself agree that set theory is based in first order logic. — TonesInDeepFreeze

So, all that is needed is to show citations that first order logic ordinarily includes identity theory (i.e. first order logic with equality) and that was accomplished by citing Enderton's logic book, Hinman, and Shoenfield. — TonesInDeepFreeze

But I guess that, despite my sin of overlooking that a certain book doesn't supply reference to a particular point (though it still is an excellent reference for the context of this subject and on other points) it seems I am finally past needing to explain over and over and over that the identity axioms are in first order logic and set theory is based in first order logic, as you post: — TonesInDeepFreeze

Giving pinpoint corrections, copious explanations, and sharing ideas in general is not ranting — TonesInDeepFreeze

In that instance, yes, and made clear by what I wrote. — TonesInDeepFreeze

Of course, that's hardly even a foible. But it's at least odd that someone who knows nothing about the matter would categorically say that it false that the indiscernibility of identicals is not included in first order logic with '=' as primitive. — TonesInDeepFreeze

We were talking about how '=' is interpreted. — TonesInDeepFreeze

Too late to wriggle out — fishfry

Ah, you have 'fessed up after all. — fishfry

You are making more of this than I intended for you to make. — fishfry

I'm mocking you for saying that you agreed with a point I made — fishfry

I have a bad habit of tweaking and needling people — fishfry

I don't know "nothing" about the matter. I know logic as it's used in math, but did not study enough formal predicate logic. Indiscernibility of identicals I know of in other contexts, and am genuinely surprised to hear that it's incorporated into set theory. — fishfry

We were talking about how '=' is interpreted.
— TonesInDeepFreeze

It's interpreted as the axiom of extensionality in set theory. Which doesn't actually require identity, and I've asked for a specific example to prove otherwise. — fishfry

If I have two sets, and I want to know if they're equal, I apply extensionality. Not identity. — fishfry

I ask for a clear clear clear clear clear refutation or counterexample. — fishfry

The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity.
— Metaphysician Undercover

Drat those sophists. Are they in the room with us right now?

Oh I see. Tones. — fishfry

You don't know enough — TonesInDeepFreeze

The crank — TonesInDeepFreeze

The Enderton reference was to the identity axioms. See page 112 in the logic book. And also, on page 83, he specifies satisfaction regarding '=' so that it adheres to interpreting '=' as the identity relation. — TonesInDeepFreeze

And you studied with Shoenfield. On page 21 lines 13 and 15 of his book you will see the equality axioms that are the indiscernibility of identicals, similar to the way I formalized and that you asked why I posted it. — TonesInDeepFreeze

I hope you know that 'the crank' does not refer to you. If that was not clear in the context, then I should have made it clear. — TonesInDeepFreeze

I'm giving you a lot of the same information and explanation over and over, since you skip over it over and over. — TonesInDeepFreeze

I said that the indiscernibility of identicals is formalized in identity theory. I didn't say that any particular formalization mentions it with the phrase 'the indiscernibility of identicals'. — TonesInDeepFreeze

The principle was enunciated by Leibniz. But in mathematics, it's often called 'the principle of substitution of equals for equals'. — TonesInDeepFreeze

And in modern logic, it is an axiom schema in the manner I've posted, which is equivalent (though notation and details differ) to Enderton and Shoenfield, for example. — TonesInDeepFreeze

And you can look at the SEP article 'Identity' where you'll see:

Leibniz’s Law, the principle of the indiscernibility of identicals, that if x is identical with y then everything true of x is true of y. — TonesInDeepFreeze

So I saw a ref to equality on 112 of enderton that had nothing to do with set theory — fishfry

This is my point. — fishfry

can't find anything at all on page 83 — fishfry

That is where he gives the semantics for '=', as I mentioned that '=' is given a fixed interpretation. — TonesInDeepFreeze

So in set theory Ax x=x is redundant. — TonesInDeepFreeze

(x = y & y = z) -> x = z — TonesInDeepFreeze

I reply to your posts, then I see your replies back, to which I then reply back ...

I don't use any other protocols. — TonesInDeepFreeze

They did say that "Leibniz’s Law must be clearly distinguished from the substitutivity principle ..." so perhaps that's pushback to your claim. — fishfry

But there are actually two principles — fishfry

So in set theory Ax x=x is redundant.
— TonesInDeepFreeze

in which case you agree with my main point and there is nothing more to say. — fishfry

So I appreciate that you are now writing much shorter posts — fishfry

I can't keep up. — fishfry

That might be. I'm speaking in broad terms about them in that regard. If the article draws a needed distinction then I should say that they are at least akin. — TonesInDeepFreeze

The converse of the axiom of extensionality where he says that the converse "follows from the substitution property of equality." — fishfry