I don't speak for Banno, but I have said that there is no set named with the noun 'infinity', but rather there is the adjective 'is infinite' defined:

x is infinite iff x is not finite — TonesInDeepFreeze

You do. But of course you won't admit it.
The concept of infinity is for description of motions, actions and operations.
The use of infinity in the set theory is ambiguity.

You speak for Banno — Corvus

I explicitly said I do not speak for Banno.

You say that in mathematics 'infinite' means 'finite', but 'infinite' means 'not finite'. Then I say that I do not speak for Banno and you say that I do. I think the problem might be that you don't know what the word 'not' means.

and now trying to speak for me — Corvus

I haven't presumed to speak for you.

Meanwhile, you've put words in my mouth, and failed to recognize that when I caught you doing it.

It seems obvious your whole purpose of coming into the forum is forcing people to admit errors when the error is on your side. — Corvus

That's a stupid thing to say.

You do. But of course you won't admit it. — Corvus

I haven't presumed to speak for Banno.

You're lying again.

I explicitly said I do not speak for Banno. — TonesInDeepFreeze

Your sayings and actions are totally different. You don't even know what you have been saying, but denying it. That is truly incorrigible.

You're lying again. I committed no action that constitutes speaking for Banno.

Right, A=B means that the value of A is equal to the value of B. This does not mean that A is identical to B, so the "=" signifies a relationship of equality, it does not signify a relationship of identity. — Metaphysician Undercover

The value represented by the symbol "A" is identical to the value represented by the symbol "B".

Two dollar bills are non-identical, but equal value. — Metaphysician Undercover

They are of identical value.

But this creates a procedural problem in practice. Let's take the example "1+1=2". The value represented by "1+1" would be exactly the same, identical, to the value represented by "2". The problem is that "1+1"contains the representation of an operation, and "2" does not. And in order that an operation can fulfill what is intended by the operator, the operation must have a very special type of value. Because it is necessary to recognize this special type of value, that signified by the operator, it is impossible that "1+1" signifies the exact same value as "2", because there is no operation represented by "2". In other words the value represented by "1+1" consists of an operation, and the value represented by "2" does not, therefore they are not representations of the exact same value. — Metaphysician Undercover

Given that 1 + 1 = 3 - 1, the value given by the procedure "add 1 to 1" is identical to the value given by the procedure "subtract 1 from 3" – that value being 2.

It's not the case that there are two equal but non-identical values of 2.

I haven't presumed to speak for Banno.

You're lying again. — TonesInDeepFreeze

It sounds like you are a little string controlled doll in Banno's pocket.

You're lying again. I committed no action that constitutes speaking for Banno. — TonesInDeepFreeze

Stop distorting the facts, and be your own man and honest to yourself.

It sounds like you are a little string controlled doll in Banno's pocket. — Corvus

First you say I speak for Banno, then you say that Banno controls me. But if Banno controls me, and I speak for him, then I speak for him at his control, so then it should be just fine for me to speak for him. (Though I don't speak for him and he doesn't control me.)

and now trying to speak for me
— Corvus

I haven't presumed to speak for you. — TonesInDeepFreeze

I am only replying to your posts, the way they are. But you two Laurel and Hardy are not worth the time. All the best.

Stop distorting the facts — Corvus

You've not shown that I've distorted any fact. Meanwhile, you've been distorting all over the place, as I have shown.

I am only replying to your posts, the way they are. — Corvus

Whatever that might mean in your own mind.

Laurel and Hardy — Corvus

As long as I can be Laurel. Stan Laurel is a great hero of mine. Right up there with Buster Keaton.

You've not shown that I've distorted any fact. Meanwhile, you've been distorting all over the place, as I have shown. — TonesInDeepFreeze

Most of your own posts are filled with distortions. See that's what I meant. You don't recall you have been writing in your own posts.

Most of your own posts are filled with distortions. — Corvus

You argue by mere assertion.

Anyway, you said this in not worth your time and signed off with "All the best", yet you're still going at it.

You argue by mere assertion. — TonesInDeepFreeze

Ok whatever. Have a good day. cheers.

The value represented by A is identical to the value represented by B. — Michael

A = B

A is B.

The value named by 'A' is the value named by 'B'.

A is equal to B.

The value named by 'A' is equal to the value named by 'B'.

A is identical to B.

The value named by 'A' is identical to the value named by 'B'.

Seven ways of saying the same thing.

But you will never bring the crank to understand that.

Given that 1 + 1 = 3 - 1, the value given by the procedure "add 1 to 1" is identical to the value given by the procedure "subtract 1 from 3" – that value being 2. — Michael

No that is clearly not the case, because these two procedures are completely different. They are said to result in the same value, 2, but the operations represented do not have the same value, nor are they identical.

Look at the two operations claimed to have an equal value. One is to take two distinct individuals and unite them producing a group of two. The other is to take a group of three and remove one individual, producing a group of two. Surely you cannot believe that these two procedures could have the same value. For example, if you had one dollar and someone gave you a dollar, that would be a far more valuable operation than if you had three dollars and someone took one dollar from you, even though they both result in you having two dollars.

And it is not the case that I equivocate with "value" here, because as I explained in the last post, the reality is that operators signify a different type of value from numerals. And, we must account for this if we are to assert that the value represented on the left side of the equation is identical to the value represented on the right side.

What we can see is that the conclusion of these two different operations results in the same value, 2. But it is clear that we do not have that "same value" unless we come to the correct conclusions in carrying out the procedures. So we have two very different operations each concluding with the same value as one another. The value, which is the same for both, is assigned to the conclusion, not the operation itself. But the operations are what is signified on the right and left sides.

If we assert that the two operations "1+1", and "3-1", each themselves have the same value, we neglect the very important fact that having the same value is really dependent on correctly carrying out the operations which are signified. Therefore "the same value" is attributed to the two conclusions, not to the two operations, themselves.

I propose that what you present here is a very sloppy analysis of what an equation actually is. The operation presented on the right side does not inherently have the same value as the operation presented on the left side, as you propose. What is really the case is that correctly carrying out the two operations, to their respective conclusions, produces the same value. I say it is very sloppy because it neglects the essential aspect of applied mathematics, which is to produce conclusions.

This sloppiness appears to be endemic to the philosophy of mathematics, and is very relevant to the issue of "infinite". The very meaning of "infinite" implies that there can be no conclusion to the operation. But the tendency in the philosophy of mathematics is to ignore the need for the human task of carrying out the operation (the consequence of Platonism which removes the requirement of human conception, I would argue), as you demonstrate with your example. So we find this mistake commonly with examples such as what @ssu suggested a bijection between the natural numbers. Obviously, by the conception of "the natural numbers", that they are infinite, it is impossible to conclude such an operation. Therefore it is impossible that there is such a bijection, or that it could produce a quantitative value.

No that is clearly not the case, because these two procedures are completely different. They are said to result in the same value, 2, but the operations represented do not have the same value, nor are they identical. — Metaphysician Undercover

Operations don't have a value. Operations return a value. The value returned by the operation of adding 1 to 1 is identical to the value returned by the operation of subtracting 1 from 3.

We can go with that position if you want. It is irrelevant to the rest of the post, which demonstrates that "the value" of the right side, and of the left side is only produced by carrying out the procedure to its correct conclusion.

It is irrelevant to the rest of the post, which demonstrates that "the value" of the right side, and of the left side is only produced by carrying out the procedure to its correct conclusion. — Metaphysician Undercover

Yes, and the values returned by both sides are identical.

The values returned are the same. What is represent by the right and left sides is not the value itself, but the operation. Therefore the "=" signifies an equality between two operations, it does not signify "the same".

The values returned are the same. What is represent by the right and left sides is not the value itself, but the operation. Therefore the "=" signifies an equality between two operations, it does not signify "the same". — Metaphysician Undercover

You're conflating an extensional and intensional reading. To hopefully make the distinction clear, consider the below:

1. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (1) is false because being the President of the United States isn't identical to being the husband of Jill Biden.

Under an extensional reading (1) is true because the person referred to by the term "the President of the United States" is the person referred to by the term "the husband of Jill Biden".

The intensional reading of "1 + 1" is the operation, the extensional reading is the value returned by that operation. Under that extensional reading, 1 + 1 = 3 - 1 where the "=" symbol is used to mean "is identical to".

You're conflating an extensional and intensional reading. To hopefully make the distinction clear, consider the below:

1. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (1) is false because "X is the President of the United States if and only if X is the husband of Jill Biden" is false.

Under an extensional reading (1) is true because the person referred to by the term "the President of the United States" is the person referred to by the term "the husband of Jill Biden". — Michael

Sorry Michael, I cannot follow you. You've strayed from mathematics, just like Tones did with the example of Twain=Clemens. Your example, like Tones' appears to be completely irrelevant. To me, you've changed the subject and I cannot follow the terms of the change. If you want to continue this course, please demonstrate how it is relevant to mathematics. However, in the meantime I ask that you consider the following

es, and the values returned by both sides are identical. — Michael

Because of the issue with Platonism, It is not even proper to designate these values, the one produced by the right side, and the one produced by the left side, as "identical". Identity is what is assigned to an object, by the law of identity, "a thing is the same as itself". Notice it is a thing which is the same as itself, "identical".

When we recognize that the value produced by carrying out the procedure on the right side is "equal" to the value produced by carrying out the procedure on the left side, we implicitly acknowledge with the use of "value", that this is something within the mind, dependent on that mental activity of carrying out the procedure. If we use use "identical", instead of "equal" it is implied that what is really a value (something mind dependent) is an object with an identity. This is why Platonism is implied when we replace "equal value" with "identical value". It is implied that the value is an object with an identity.

Sorry Michael, I cannot follow you. You've strayed from mathematics, just like Tones did with the example of Twain=Clemens. Your example, like Tones' appears to be completely irrelevant. To me, you've changed the subject and I cannot follow the terms of the change. If you want to continue this course, please demonstrate how it is relevant to mathematics. However, in the meantime I ask that you consider the following — Metaphysician Undercover

Well, I can't explaining the mistake you're making in any simpler terms, so if you don't understand that then I can't help you further.

When we recognize that the value produced by carrying out the procedure on the right side is "equal" to the value produced by carrying out the procedure on the left side, we implicitly acknowledge with the use of "value", that this is something within the mind, dependent on that mental activity of carrying out the procedure. If we use use "identical", instead of "equal" it is implied that what is really a value (something mind dependent) is an object with an identity. This is why Platonism is implied when we replace "equal value" with "identical value". It is implied that the value is an object with an identity. — Metaphysician Undercover

You really read too much into words. There's just no substantial metaphysical implications in saying that the value returned by one operation is identical to the value returned by some other operation. It's just language and just maths. We don't need to believe in the mind-independent existence of abstract entities.

Well, I can't explaining the mistake you're making in any simpler terms, so if you don't understand that then I can't help you further. — Michael

Like Tones' you refuse to stick to mathematics, committing the folly @Banno pointed to, a pretense of mathematics. Until you define and demonstrate how the distinction between extensional and intensional is relevant to a discussion of mathematical values, your reference to physical objects is completely irrelevant.

It's just language and just maths. — Michael

It's not maths, as both you and Tones have clearly demonstrated, by needing to refer to physical objects rather than mathematical values to support your claims of "identical".

I gave the Mark Twain / Samuel Clemens example as an illustration, not an argument, of the distinction between sense and denotation. And I mentioned the distinction between sense and denotation not as an argument for the point that, in mathematics, '=' stands for identity, but rather to refer to another aspect of the matter. And this point is not confined to physical objects.

Like Tones' you refuse to stick to mathematics, committing the folly Banno pointed to, a pretense of mathematics. Until you define and demonstrate how the distinction between extensional and intensional is relevant to a discussion of mathematical values, your reference to physical objects is completely irrelevant. — Metaphysician Undercover

It's not maths, as both you and Tones have clearly demonstrated, by needing to refer to physical objects rather than mathematical values to support your claims of "identical". — Metaphysician Undercover

It's an analogy to explain to you the mistake you're making.

a. 1 + 1 is identical to 3 - 1.

Under an intensional reading (a) is false because adding one to one isn't identical to subtracting 1 from 3.

Under an extensional reading (a) is true because the value returned by adding one to one is identical to the value returned by subtracting 1 from 3.

Compare with:

b. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (b) is false because being the President of the United States isn't identical to being the husband of Jill Biden.

Under an extensional reading (b) is true because the person who is the President of the United States is identical to the person who is the husband of Jill Biden.

You may try for, literally, years and he will not understand.

So we find this mistake commonly with examples such as what ssu suggested a bijection between the natural numbers. — Metaphysician Undercover

No, you are making a mistake.

I suggest you to read an elementary school book on set theory. There indeed are infinite sets and there can be a bijection between these sets. It's not just "mistake" like you think.

From "cuemath" describes this perfectly well:

A finite set is a set with a finite number of elements and is countable. An infinite set, on the other hand, has an infinite number of elements, and an infinite set may be countable or uncountable. Yes, finite and infinite sets don't mean that countable and uncountable. There is a difference. For example, sets like N (natural numbers) and Z (integers) are countable though they are infinite because it is possible to list them. In other words, we can have a one-to-one correspondence (bijection) from each of these sets to the set of natural numbers N, and hence they are countable. On the other hand, the set of all real numbers R is uncountable as we cannot list its elements and hence there can't be a bijection from R to N.

And furthermore, just how important is a bijection in the definition of cardinality:

Cardinality of Countable Sets
To be precise a set A is called countable if one of the following conditions is satisfied.

A is a finite set.
If there can be a one-to-one correspondence from A → N. i.e., n(A) = n(N).
(This point is used to determine whether an infinite set is countable.)
If a set is countable and infinite then it is called a "countably infinite set". Some examples of such sets are N, Z, and Q (rational numbers). So, the cardinality of a finite countable set is the number of elements in the set. On the other hand, if it is an infinite countable set, then its cardinality is equal to the cardinality of the set of natural numbers.

See Cuemath: cardinality

Perhaps you should start here:
Lecture on infinity and countability

Or here, just what is an infinite set:
Mathworld Wolfram: Infinite set

Or simply the axiom of infinity in ZF-logic:
Axiom of Infinity

Or if you think that there is no set of the natural numbers N, I think your contribution to any set theoretic discussion or to the subject of infinity is quite limited, to say at least. Otherwise I do value your opinions and remarks on various other subjects.

If one rejects the view that abstract objects exist (and obviously, as abstractions, they don't exist physically), then, of course, the left term and the right term in an identity statement cannot refer to abstract objects. But that is a different objection than objecting to taking '=' as standing for the identity relation.

And if one objects to calling whatever mathematics refers to as 'objects', then we note that the word 'object' is a convenience but not necessary, as we could say 'thing' instead, or 'value of the term', or 'denotation of the term', or even none of that, and just say 'members of the domain of discourse' so that 'T = S' is interpreted as, for any model M for the language, M(T) is M(S).

It is not required to have any particular ontological view of what mathematical terms refer to just to understand that '=' stands for the identity relation. That is, whatever the terms T and S refer to (no matter what one regards mathematical terms as referring to), we understand that 'T = S' stands for the statement that whatever 'T' stands for is the same as what 'S' stands for.

Moreover, there is a difference between what is meant in mathematics by '=' and what one thinks mathematics should mean by '='. Whatever one thinks mathematics should mean by '=' doesn't change the fact that in mathematics '=' stands for identity.