You don't see a point in them, but that doesn't stop you from posting insults.

And, again, it is very important to distinguish between an ad hominem ARGUMENT and, on the other hand, stating an non-ad hominem argument but in addition remarking that a poster is confused, ignorant and dishonest, especially when detailed explanation is given the poster as to what his ignorance, confusion and dishonesty are. — TonesInDeepFreeze

I wasn't in the direct discussions on the topic of identity you have been discussing. But when I connected to the thread, the first thing I saw in your post was you throwing out the sentence saying "ignorance and confusion" to the other interlocutor. I immediately recalled what you have been saying to me in the similar way previously, and it gave a strong impression, that you have been insulting not just me, but the others who don't agree with your opinions.

Bottom line is, that you should try to avoid doing that if possible. It can happen during the discussions in the heat of the moment unintentionally. But if you keep doing that constantly, and especially at the start of your posts, then your posts will look as if they are intentionally meant for insulting others.

What you can "prove from the axioms" is irrelevant, when it is the acceptability of the axioms which is being questioned. — Metaphysician Undercover

You're making claims about the axiom vis-a-vis identity. So it is very relevant what the axiom proves regarding identity.

And as I said: There is a difference between what mathematics says and what one thinks mathematics should say. So anyone is welcome to say how they think mathematics should be formulated, and better yet, to provide an actual formulation. And anyone is welcome to say why they think the ordinary axioms are not acceptable. But to do that, one should at least understand what those axioms are and how such mathematics is formulated. And not to continually strawman about them. Also it always helps to understand context, which here includes why mathematics adopts axioms, what the axioms of classical mathematics are intended to provide, and the perspective of the use of classical mathematics for the sciences and computing.

Again, for the hundredth time, I don't remark on the ignorance, confusion and dishonesty of posters merely because they disagree with me.

And, of course, you read the first line, but not the context that justifies that first line.

Meanwhile, look at your own posts.

As said, I am not interested in keeping talking with you on who has done what. We have done that enough. If you haven't kept on the walls of the negative postings here, I wouldn't have replied to you at all. The same goes with Banno.

As said, I am not interested in keeping talking with you on who has done what. — Corvus

And as I said, nobody's stopping you from not talking about it.

negative postings — Corvus

While you are free to not post in an insulting manner.

I wouldn't have replied to you at all. — Corvus

You were replying to me in an insulting manner (criticizing my arguments while not even addressing their key points and taking me for a fool with ignorant and false red herrings that study of mathematics is just a bunch of regurgitating what is in book, or however you actually phrased it) well before I said even a word about you personally. And as I observed you replying insultingly to another poster in this thread in even the earliest part of this thread.

Prediction: Corvus will reply yet again that he no longer wants to discuss the personal aspects of the postings, while he yet again renews his claims about the personal aspects of the postings.

Now where were we?

Oh, yes, the main crank's insistence that set theory handle order in the way he thinks it should be handled, even though he is ignorant of how set theory does handle order.

So an hourglass changes its identity as each sand grain drops. — Banno

No, the law states "a thing is the same as itself". Nothing here says that the thing cannot change as time passes. But all those changes are necessarily a part of the thing's identity. That's one of the important features of the law of identity, it allows for a true understanding of the temporal continuity of things, and the reality of change itself, by allowing that a thing maintains its identity despite changing.

Have you no familiarity with the law of identity? It seems to me that you've only been exposed to misrepresentations, proposed by logicians who want to reformulate it to support their own proposals. I do not argue that it is without problems, like the one presented by The Ship of Theseus example. And as I said earlier, some philosophers propose that we reject the law of identity altogether. But that would give us no principles for understanding the reality of temporal continuity. You see, there is an incompatibility between eternal unchanging Platonic "Ideas", and the temporal continuity of objects which are constantly changing. The law of identity refers to the latter, and the identity which a set is said to have refers to the former.

You're making claims about the axiom vis-a-vis identity. So it is very relevant what the axiom proves regarding identity. — TonesInDeepFreeze

I don't think so. A proposition (or axiom) needs to be judged by the principle it states, not based on what can be proven through the use of it. If you accept an axiom because it can prove what you want it to prove, that is just begging the question.

the main crank — TonesInDeepFreeze

I'm starting to like that handle, it makes me feel powerful like the driving part of a magnificent machine. Do you think it would be suitable for me to change my name?

An axiom is a formula. It has a meaning upon interpretation of the language. But also, it has our ordinary reading of it in a natural language. To understand that ordinary reading of it, requires understanding the context, which includes what the axiom proves.

But, of course, you will resort to any argument you can to evade actually learning anything about the subject on which you dogmatically declare.

Let's refresh just recent matters alone:

You claimed that axiom of extensionality is inconsistent with identity theory. I proved it is not. You evade that, because you know virtually nothing about identity theory, the axiom of extensionality or consistency.

Most basically, you haven't a clue what the axiomatic method is about.

I explained for you how set theory does provide for the identity of indiscernibles and the indiscernibility of identicals.

You claim that a set and an ordering on the set determine the set. You were saying that years ago in this forum and it was debunked then. You still don't get it. Still don't get what even a young child can understand.

/

I don't care what you call yourself, but if you do change your name, at least I'll know that the denotation of 'Main Crank' is the denotation of 'Metaphysical Underground', which is the same foolish, arrogant, ignorant, illogical, irrational, dogmatic, confused, intellectually blocked, dishonest, lying poster he always was.

You claimed that axiom of extensionality is inconsistent with identity theory. I proved it is not. You evade that, because you know virtually nothing about identity theory, the axiom of extensionality or consistency. — TonesInDeepFreeze

That's a good example of a crackpot reply. You are avoiding the issue, by switching to "identity theory" rather than the law of identity. The problem I brought up is that "identity" in set theory is not consistent with "identity" in the law of identity. Whether or not "identity" in set theory is consistent with "identity" in "identity theory" has no bearing on the problem I've exposed.

The law of identity is:

Ax x=x

That is one of the axioms of identity theory.

I posted that earlier today, but of course you SKIPPED it.

Since the axiom of extensionality is consistent with identity theory, perforce it is consistent with the law of identity.

So I am not at all avoiding the issue or switching the issue.

/

You keep using the word 'consistent' while you show no inconsistency; we only learn from your bungled arguments that the axiom of extensionality does not accord with your confused and dogmatic views about mathematics. As I've said an uncountably infinite number of times if I've said it a countably infinite number of times, no one disallows you from positing your own framework of understanding, but the mere fact that mathematics does not adopt your framework (which is, meanwhile, confused) does not make mathematics incorrect. And you keep evading many of the other arguments and considerations about set theory, such as the application of classical mathematics to the sciences and computability, including the existence of the computer you're typing on right now. It is funny how not only is it the case that cranks can never answer that point, but they will never even recognize it.

Moreover, when you resort to dogmatically insisting that a set is determined by a particular ordering, you resort to silliness that was debunked years ago in posts to you in this forum. We went over it and over it back then, and you still don't get it. I think you don't get it because you have a mental block about such things. So I suggest that not only should you make an appointment with an ophthalmologist to find out why you can't see things right in front of your eyes, but you should see a cognitive psychologist to find out about the mental blocks you have that disallow you form understanding even such basic ideas that can be understood by a young child.

To the Lounge with this rubbish.

So, at least in a mathematical context, "Infinity is unknowable" doesn't have an apparent meaning to me. — TonesInDeepFreeze

It seems that "infinity" as an object is more a problem of natural language than mathematics and set theory.

As you also say:

In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'. — TonesInDeepFreeze

The law of identity is:

Ax x=x

That is one of the axioms of identity theory. — TonesInDeepFreeze

As I indicated earlier, the issue is with the way that x=x is interpreted. Unless the interpretation employed by "identity theory" is consistent with the way that the law of identity is stated in its original formulation, "a thing is the same as itself", then the meaning of "x=x" which is employed by "identity theory", is not consistent with the law of identity.

What is required now, is that you state the interpretation of "x=x" which is employed by "identity theory", and more specifically "set theory", such that we can judge it for consistency with the law of identity, "a thing is the same as itself".

To the Lounge with this rubbish. — Jamal

That's better, it's more relaxed in the lounge, and may serve to lower the tension by a few foot-pounds or something like that. I hope there's no drinks available here though, or things might go the opposite way.

As you also say:

In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'.
— TonesInDeepFreeze — RussellA

I wonder what "nicknamed" would imply in supposed rigorous logic.

I am not a Mathematician, and have limited knowledge about set theory.

As a very broad generalization, I think of at least these two categories: (1) Matters of fact. (2) Matters of frameworks for facts. — TonesInDeepFreeze

:up: For me, the statement "Monet's Water-lilies is an example of beauty" is a fact and is true. However, I am speaking within the framework of a European Modernist. Within a different framework, say that of a Californian Post-Modernist, the statement, may be neither a fact nor true.
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That ordinary mathematics says "1+1 is 2" is matter of fact. But whether ordinary mathematics should say that 1+1 is 2 is a matter of framework. — TonesInDeepFreeze

:up: Within a different framework, say that of binary numeral system, 1 + 1 = 10
===============================================================================

But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects. — TonesInDeepFreeze

Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists?

If I said "I am going to buy an apple", I am not referring to "an apple" as a particular concrete thing or object, but rather referring to "an apple" "as if" it were a particular concrete thing or object.

Whilst the definite article refers to a particular concrete thing "a house" "a mountain" or "a cat", the indefinite article, "a house", "a mountain" or "a cat", doesn't refer to a particular concrete thing, but rather refers to a particular concrete thing that possibly exists.

In language also, we can refer to things that exist, "I want this cat", and refer to things that possibly exist, "as if" they exist, such as "I want a unicorn".
===============================================================================

The 'it' there must refer to something — TonesInDeepFreeze

What does "it, the knight on a chess board, refer to?

"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world.

The game of chess is played between two people, and as neither player can look into the other's mind, the rules must exist in the world in order to be accessible to both players. For example, "the knight either moves up or down one square vertically and over two squares horizontally, or up or down two squares vertically and over one square horizontally". However, as rules cannot refer to themselves, in that rules cannot be self-referential, they must refer to something external to the rules, in this case, a physical object.

IE, if there were no rules there would be no game of chess, and if there there were no physical objects the rules would have nothing to refer to.

There are therefore two aspects to "it". The extension, the physical object of a knight, and the intension, the rules that the knight must follow.

Such an approach to understanding "it" is supported by Wittgenstein's Finitism. Wittgenstein was careful to distinguish between the intensional (the rules) and the extensional (the answer). Mathematics is the process of using rules contained within an intension to generate propositions displayed within the extension. For example, the intension of 5 + 7 is the rule as to how 5 and 7 are combined, and 12 is the extension. (Victor Rodych - Wittgenstein's Anti-Modal Finitism - Logique et Analyse)

Such as approach to understanding "it" also follows from natural language. The intension of the word "beauty" is a rule that determines what is beautiful and the extension of the word "beauty" are concrete instantiations, such as Monet's Water-Lilies or a red rose in a garden.

There may be many possible rules for what is beautiful. Francis Hutcheson asserted that “Uniformity in variety always makes an object beautiful.”. Augustine concluded that beautiful things delight us. Hegel wrote that “The sensuous and the spiritual which struggle as opposites in the common understanding are revealed as reconciled in the truth as expressed by art” .

It is impossible for a finite mind to have a list of all beautiful things in the world, yet can recognise when they see something is beautiful. The human mind has the concept of beauty prior to seeing a beautiful thing. Such a rule is probably innate, the consequence of millions of years of evolution existing in synergy with the outside world.

Such a rule is the intension of the word "beauty" and physical examples, such as a Monet Water-Lily are the extensions of the word "beauty".

IE, "it" refers to the intension and extension of the word "knight". The intension being the rule the "knight" follows and the extension being the physical object ,whether made of wood or plastic.
===============================================================================

And the number 1 in mathematics is an abstract mathematical object that we speak of in a similar way to the way we speak of concretes, but that does not imply that the number 1 is a concrete object. — TonesInDeepFreeze

However, if there were no concrete objects in the world, there would be no concept of the number "1".
===============================================================================

And as 'experience' and 'occurring' are the notions I start with, I must take them as primitive. — TonesInDeepFreeze

:up: Yes, there are some concepts, such as "beauty", that we cannot learn the meaning of by description from the dictionary, but are probably innate within us. Innatism is the view that the mind is born with already-formed ideas, knowledge, and beliefs.
===============================================================================

Notice that I didn't say 'experiences' plural — TonesInDeepFreeze

In my terms, thinking about the concept "beauty", which is probably innate, and therefore primitive within us, there only needs to be one intrinsic rule able to generate numerous extrinsic examples.
===============================================================================

But then I do refer to 'I' — TonesInDeepFreeze

In Kant's terms, we have a unity of apperception. The mystery is why.
===============================================================================

As I go on, I find that certain other notions such as 'is', 'exists', 'thing' or 'object', 'same' 'multiple'. etc. are such that I don't see a way to define them strictly from the primitives I've allowed myself. — TonesInDeepFreeze

Certain words such as "house" can be defined as "a building for human habitation, especially one that consists of a ground floor and one or more upper storeys". We can learn these concepts from the dictionary using definitions. But sooner or later, we come across other words, such as "is", "exists" and "thing" that are primitive terms, cannot be defined, but only learnt from acquaintance.
===============================================================================

But the very determinations of fact, let alone the conceptual organization of facts, are vis-a-vis frameworks, and it is not disallowed that one may use different frameworks for different purposes. — TonesInDeepFreeze

:up: For me, a Modernist, the statement "Monet's Water-lilies is a beautiful painting" is true, but for others, the Post-Modernists, the same statement is false.
===============================================================================

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts. — TonesInDeepFreeze

But how can there be wisdom in the absence of facts. How can we understand the wisdom of Kant without first knowing those facts he applied his wisdom to?
===============================================================================

Meanwhile, I would not contest that formation of concepts relies on first approaching an understanding of words ostensively. — TonesInDeepFreeze

:up:

They haven't responded at all, but that seems to be their way; they are in the unusual position of having less comments (8) than Discussions (11)... — Banno

A very rare case on the internet of someone who wants to listen more than they want to talk.

Setting aside whether it's good to move a thread from the main table of contents, the moderator in his role as moderator would have been better not to so subjectively, curtly and sweepingly over-reduce a thread.

It was commented "I wonder what "nicknamed" would imply in supposed rigorous logic."

The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas.

Moreover, the nicknaming (my word) I mentioned is not so much about the logic but rather about defined symbols in a theory such as set theory.

Set theory, as formalized, uses only formal symbols, not natural language words. A formal proof is not allowed to use connotations, associations or any of the suggested notions that natural language words have. However, for everyday communication of proofs among mathematicians and students it is unwieldy to recite exactly each formal symbol in the formulas that are sequenced for a proof. Moreover, it aids picturing the content of the theory to informally use words. I call that 'nicknaming'. For example, in set theory in all formality, there is no constant term 'the empty set'; instead there is a 1-place operation symbol, a pure symbol, with a purely formal definition. Moreover, as I've mentioned, the adjective 'is a set' or even a formal predicate for 'is a set' are not even required as in formal theories such as ZFC. So, as concerns the formal theory, it doesn't matter whether or not one's personal notion of sets allows that there is one special set that has no members that is called 'the empty set' but rather, the theory has a formal theorem, such as:

E!xAy ~yex
thus a definition
0 = x <-> Ay ~yex

So, in that particular regard, we could just as well use the nickname "zee zempty set". It would not change the "structure" of the mathematics, which is the relationships of the definitions and theorems.

Note that my remarks about this are not necessarily a commitment to extreme formalism expressed as "mathematics is just a formal game of symbols". Rather, in this context, we may note that, no matter what framework or philosophy one has for understanding mathematics, at least we have the formalization, even just the fact of that formalization, as a component in our understanding - whether a fully self-contained and isolated component (i.e. extreme formalism) or as merely a point of reference and a rigorous constraint against handwaving.

I agree. There was developing an interesting discussion on the law of identity and (non-ordered) sets. Or so it seems, I just glanced at it.

So an hourglass changes its identity as each sand grain drops.

A moment of clarity.

Within a different framework, say that of binary numeral system, 1 + 1 = 10 — RussellA

That's not an example of what I was talking about. I'm talking about general frameworks such as hold one's intuitions, perspective or philosophy, not matters such as variations in base numbering systems.

But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects.
— TonesInDeepFreeze

Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists? — RussellA

No.

What does "it, the knight on a chess board, refer to? — RussellA

When I wrote 'it', I was not referring to a particular piece of wood or particular array of pixels on a screen. I'm referring to the idea, the thing that players who are not even in each other's presence - thus not moving the same piece of wood or even seeing the same array of pixels - can still refer to as "the knight".

"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world. — RussellA

But it doesn't. Again, one may argue that leading up to the formation of the concept, there were particular pieces of wood or ivory or whatever that were carved to resemble a knight and that were moved around on a two-colored board. But soon enough, we have the abstraction that can be referred to. Indeed, 'chess' itself can be defined mathematically without even mentioning particular characters such as 'knight', but rather only a purely mathematical construct. We could say there are four objects, called 'WKL', 'WKR', 'BKL', 'BKR' ("white" and "black" "knights" on "left" and "right") and the other "pieces", then define a C-sequence (sequence of "chess moves") to be a certain sequence of matrices with those objects associated with cells in a matrix and successive matrices having a property that the "pieces" are in different cells only according to certain allowed ordered pairs of cells ("moves"), etc. So you see that when I say 'it', I'm talking about an abstract object, even though attaining that abstraction required previous concrete or ostensive understanding.

Innatism — RussellA

I am not opining whether or not the basic concepts 'is', 'exists', 'same', etc. are innate. Indeed, I have no ready argument that they are not first understood only ostensively. I'm only reporting that I don't know how I could arrive at successively more involved frameworks without them.

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts.
— TonesInDeepFreeze

But how can there be wisdom in the absence of facts. — RussellA

I don't say that there can be.

There was developing an interesting discussion on the law of identity and (non-ordered) sets. — jgill

More a painfully needed, though unsuccessful, intervention than a discussion.

The points are simple:

* In mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!"

{'The Maltese Falcon', 'Light In August', 'The Stranger'} = {'Light In August', 'The Stranger', 'The Maltese Falcon'}

{8, 5, 9} = {5, 9, 8}

And '=' reads fine whether 'equals', 'is identical with' or 'is'.

No law of identity is violated there.

/

Boss: Jake, tell me what is the set of items on our shipping clerk's desk?

Jake: It's the set whose members are a pen, a ruler, and a stapler.

Maria: But he also has another set on his desk! It's the set whose members are a ruler, a stapler, and a pen.

Boss and Jake: Wha?

Boss: Maria, take the rest of the day off. You're not quite with it lately.

/

{pen, ruler, stapler} = {ruler, stapler, pen}

Nobody says that the set of items on a desk is different depending on the order you list them.

On the other hand, mathematics does have ordered pairs and triples. For example:

<b c d> = <x y z> if and only if b=x, c=y and d=z.

With ordered tuples, yes, order does matter.

The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas. — TonesInDeepFreeze

Since soundness requires true premises, and the logic you are talking about proceeds from axioms, which are not truth-apt, instead of from true propositions, how do you propose that it could be "proven that the logic is sound"?

mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc. — TonesInDeepFreeze

The point I made, is that the sense of "identity" you use here, is not consistent with the sense of "identity" used in the law of identity. So it doesn't really matter that you insist that "=" stands for "identity". Anyone can make up one's own personal sense of "identity" and have a symbol for it, state the axiom, and persuade others to use the axiom, and even create a whole "identity theory", but that doesn't make that sense of "identity" consistent with the law of identity.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!" — TonesInDeepFreeze

This is an excellent example of why your sense of "identity" is not consistent with the law of identity. By the law of identity, if the identified thing is "the books on your desk", then everything about that thing, including the order of the parts, must be precisely as the books on your desk, to satisfy the criteria of "identity". Stating an order other than what the books on your desk actually have, would not qualify as an identity statement, because that specific aspect, the order of the parts, would not be consistent with the thing's true identity.

No law of identity is violated there. — TonesInDeepFreeze

I have become fully aware that you are not at all familiar with the law of identity. Therefore your statements about the law of identity, and whether it is violated under specific conditions, I simply take as off-the-cuff remarks of a crackpot.

Nobody says that the set of items on a desk is different depending on the order you list them. — TonesInDeepFreeze

Anyone with any degree of common sense recognizes that the identity of the specified thing, "the items on a desk" includes the ordering of the mentioned items. If describing those items as a "set" means that the ordering of the mentioned items is no longer relevant, then you are obviously not talking about the "identity" of the specified thing, which is "the items on the desk". You are talking about something other than "identity" as defined by the law of identity.

It's so easy for practitioners of the subject:

$2(x+5)\equiv 2x+10$ Identity

$2(x+5)=3$ Conditional

Here's how I would put it:

2(x+5)= 2x+10
is understood to be implicitly universally quantified:
Ax 2(x+5) = 2x+10
and that is true

Then, by universal instantiation, we have:
2(x+5) = 2x+10
and that is true for every assignment of a value to the variable 'x'

2(x+5) = 3
is understood to be implicity existentially quantified:
Ex 2(x+5) = 3
and that is true

In high school algebra, we are asked to state the members of the "solution set", which is to say:
{x | 2(x+5) = 3} = {-7/2}
and that is true

First order predicate logic may be formalized in two ways:

(1) With logical axioms and rules of inference. (Known as 'Hilbert style'.)
or
(2) With only rules of inference. (E.g., natural deduction.)

In either case, we have the soundness theorem, since the logical axioms are true in every model and the rules of inference are truth preserving.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

The law of identity is that a thing is identical with itself. Using '=' to stand for 'identical with' is not inconsistent with the law of identity. Indeed, the law of identity is stated:

Ax x=x

That is, for all x, x is identical with x. That is, for all x, x is x.

Not only is '=' as used in mathematics consistent with the law of identity, but the law of identity is itself an axiom of identity theory that is taken as part of the logic used for mathematics.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

"the ordering of the set" is meaningful only when there is only one ordering of the set (in this context, by 'ordering' we mean a strict linear ordering). Any set with at least two members has more than one ordering.

The set whose elements are all and only the members of the Beatles in 1965 has 4 members, and there are 24 orderings of that set.

In other words, there is no "the" ordering of the 4 membered set {x | x was a member of the Beatles in 1965} since that set has 24 orderings.

/

"crackpot" is said by the crank calling the pot cracked.

Incidentally, I argued extensively with fishfry, that to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation. — Metaphysician Undercover

After much back and forth, you finally revealed to me that you do not understand the basic material implication $P \implies Q$ of propositional logic; and that you are incapable of understanding the statement of the axiom of extensionality on its Wiki page. I'm afraid I can't dialog with you further till you remedy these very basic misunderstandings.

Ha, ha, very funny.