## What is the difference between actual infinity and potential infinity?

• 130
Explain to me then, how this set '2+2', is the same thing as this set, '4'. They look very different to me, and also have a completely different meaning. By what principle do you say that they are the same?

They are the same according to the game of identity called as "equality theory". There is a confusion here between expressions and what they denote, "The Sun" , "The nearest star to Earth" are two DIFFERENT (i.e. not identical) expressions, yes, but they denote the same object! so when we say for example "The Sun = The nearest star to Earth", what we mean is that the object denoted by the expression "The Sun" is Identical to the object denoted by the expression "The nearest star to Earth", that's very clear, it is identity of the denoted and not of the denoting expression, of course the denoting expressions are different. That an object can be denoted by different expressions is well known, and it poses no problem whatsoever. Along this understanding the expression "2+2" is meant to denote some object x, and the expression "4" is also meant to denote some object x, however both expression (though different) denote the SAME object exactly.
• 7.2k
They are the same according to the game of identity called as "equality theory".

We've been through this already. "Equal" does not mean "the same", or "identical". Identity is defined by the law of identity, equality is defined by mathematical principles. If you think that there is a principle of identity which makes equality into identity, then please produce this new law of identity, what you call "equality theory". I've asked fishfry for this principle of identity, to no avail.

There is a confusion here between expressions and what they denote, "The Sun" , "The nearest star to Earth" are two DIFFERENT (i.e. not identical) expressions, yes, but they denote the same object! so when we say for example "The Sun = The nearest star to Earth", what we mean is that the object denoted by the expression "The Sun" is Identical to the object denoted by the expression "The nearest star to Earth",

You are using '=' here in a way other than how it is used in mathematics. Argument by equivocation is useless to me. Sure you can use '=' to mean identical if you want, but we are talking about the way it is used in mathematics, 2+2=4 for example, and it is not used here to mean identical.

Along this understanding the expression "2+2" is meant to denote some object x, and the expression "4" is also meant to denote some object x, however both expression (though different) denote the SAME object exactly.

This is obviously not true. There are three symbols on the left side '2', '+', and '2'. There is only one symbol on the right side '4'. That is the first indication that the right side does not denote the same object as the left side. there is a plurality of symbols on the left, so there is likely a plurality of objects symbolized on the left. Furthermore, if the three symbols on the left side are meant to signify one entity, it is not the same entity as is signified on the right side, or else it would be signified in the same way as the right side. The symbols on the left are not meant to signify the same thing as on the right, or else the same symbol would be used. If they happen to symbolize the same thing this would be by mistake, but there is no mistake here. If two distinct symbols are meant to symbolize the very same thing, this would only be intended to create the illusion of different things being represented, and there would be no reason for this except to deceive. Therefore we must conclude that they are meant to symbolize something different. If the two sides were meant to symbolize the very same thing, the equation would read '4=4', and this would be a useless equation. And if the very same thing was meant to be symbolized by different symbols this would be an act of deception. Since '2+2=4' is not a useless equation, nor an expression of deception, we must conclude that what is signified on the left side is not the same object as what is signified on the right side.
• 130
then please produce this new law of identity, what you call "equality theory". I've asked fishfry for this principle of identity, to no avail.

There is no new law at all. It is a schema of statements, in first order logic it would be expressed as: that x = x, i.e. everything is equal (identical) to itself, and that if phi(x) is an expression in which x occur and if phi(y) is obtained from phi(x) by merely substituting all occurrences of x in phi(x) by the symbol y, then the law is:

x=y implies [phi(x) iff phi(y)]

in a more informal manner, x is equal (identical) to y if every expression true of x is also true of y and vise verse, what we mean by true of is the truth of the denotation of that expression about objects and not the truth of its grammatical structure.

Actually equality is nothing but identity. In first order logic it boils down to substitutivity, as mentioned above.

But you need always to discriminate between what an expression is denoting and what an expression is. I already gave a simple example "The Sun" and "The nearest Star to Earth", in physics those two expressions are referring to exactly the same object but they are indeed two distinct expression! The former has two words the latter had five! So truly they are distinct expressions but they are denoting exactly the same object. In an exactly similar manner in mathematics the expression "2+2" is nothing but a functional term, it denotes an object that is exactly the same (identical) object that the constant term "4" is denoting. This is no deception, much as the different statements in the first example are no deception. The idea is about what can be called a "consequential truth" here. In the game of arithmetic the expression "2+2" is identical to "4", in the sense that they both denote the same object. i.e. this is a consequence of the axioms and rules of inference of that game, this is a theorem, and consequential fact, we need to determine exactly when two different expressions in the language of arithmetic denote the same object, just because they are different it doesn't mean that they can't denote the same object. We have a formal game here, and we want to know which of those expressions denote the same object and what are not, this is no deception, it is not even trivial, that's what we want to know.

All the rest of your account on trying to a kind of prove that "2+2" must be an expression denoting something that is different from the expression "4", is NOT correct, neither conceptually nor formally.

By the way the objection you stated that if they are indicating the same object then 2+2=4 would be equal to 4=4 and therefore would be vacant, this objection is already a property that Kant had spoken about when he defined analytic truths, i.e. its just repetition of what has been already said. Which is correct.

Consequential results are not that easy to figure out, they turn to be very tricky, that even if at the very conceptual root they are repetitions of statements, yet the recognition about which statements boils to be repetitions of which other statements is not that easy to determine and sometimes its even impossible to know relevant to a fixed set of axioms in arithmetic.

However the reality of 2+2=4 is not only linked to the above formal consequential game reasoning. One might say that all of that consequential game is just vacant, and that mathematics is not vacant as analytic reasoning is, so there must be a kind of truth to 2+2=4, something more akin to synthetic truth Kant was speaking about. The answer is that the truth of 2+2=4 is inherited from the truth of the axiomatic system in which 2+2=4 is a theorem of. The axioms themselves are not analytically derived from prior sentences, if they are consistent, then they are true of some model, and the truth of the axioms whatever it might be is inherited down to all of theorems derived in the system axiomatized by those axioms. On can say that all theorems are just repetitions of what's in the axioms, so the truth of 2+2=4 is related to the background of the axioms of the axiomatic system in which it is proven, which is CORRECT! That doesn't prevent 2+2=4 being the same as 4=4 at all, it doesn't make it trivial because it is only an aspect exposing the truth of what's in the axioms.

Again to sum it up, although "2+2" and "4" are two distinct expressions, yet they both denote the same object, much as how expressions "the Sun" and "The nearest star to Earth" are distinct and yet denote the same object.
• 1.5k
I've asked fishfry for this principle of identity, to no avail.

I was going to reply to you later but just ran across this, which could not be more false.

I have repeatedly explained to you that the axiom of extensionality is directly derived from the logical law of identity. I thought that had already been mentioned by someone even before I joined this thread. So if you don't understand what it means, or want to see more detail, just ask.

If you are claiming that equality isn't identity in natural language, you might have a philosophical point.

But if you are making a mathematical claim, you're just factually wrong. Mathematical equality is identity of sets. A mathematical equality states that the sets on either side of the equation are the same set.
• 7.2k
in a more informal manner, x is equal (identical) to y if every expression true of x is also true of y and vise verse, what we mean by true of is the truth of the denotation of that expression about objects and not the truth of its grammatical structure.

OK, I've said true things about '2+2' which are not true about '4'. Therefore the two are not identical. It's what I've been doing for last number of posts, explaining how '2+2' signifies something different from '4".

Actually equality is nothing but identity. In first order logic it boils down to substitutivity, as mentioned above.

Clearly equality is not identity, because different things are true concerning what is expressed on the right side of an equation than are true concerning what is expressed on the left side. This is by your own definition of "identical", above. How can you deny this?

But you need always to discriminate between what an expression is denoting and what an expression is. I already gave a simple example "The Sun" and "The nearest Star to Earth", in physics those two expressions are referring to exactly the same object but they are indeed two distinct expression!

I don't dispute that distinct symbols can refer to the same object. What I dispute is that '2+2' refers to the same object as '4'. So your example proves nothing.

In the game of arithmetic the expression "2+2" is identical to "4", in the sense that they both denote the same object..

If this is true, then show me the object which both '2+2' and '4' refer to. If that object is a concept, then explain to me this concept which they both refer to. I've already explained how they each refer to a different concept, but you refuse to listen. So now it's your turn to describe to me this one concept which you believe both '2+2' and '4' refer to.

This appears to be the extent of your argument, a simple assertion that '2+2', and '4' both refer to the same object. Now see if you can justify this assertion by showing me the object which they both refer to. It will be difficult for you, because I already see that '2+2' refers to something completely different from '4', as I've explained. So you need to dispel this false belief that I have, demonstrate how the differences I described are not real, and show me how they really refer to the same object. Assertion does not justify.

I have repeatedly explained to you that the axiom of extensionality is directly derived from the logical law of identity.

You provided no such explanation, only an assertion. The web pages you've referred me to do not support your claim. They speak of "equality", not identity. This is from the Wikipedia page on the axiom of extensionality which you referred.
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B. — Wikipedia
Where's the reference to the "logical law of identity" which you are asserting?

A mathematical equality states that the sets on either side of the equation are the same set.

That is not true, you are inverting what is stated in the axiom of extensionality. It is stated that if two sets have the same members, the sets are equal. It does not state that the sets are the same, it states that if the members are the same, then the sets are equal. Therefore the sets remain distinct, as two equal sets, not one and the same set.

But if you are making a mathematical claim, you're just factually wrong. Mathematical equality is identity of sets. A mathematical equality states that the sets on either side of the equation are the same set.

This is exactly the point of contention. Let's say that two things are said to be mathematically equal. By what law of identity do two equal things become the same thing?
• 130
OK, I've said true things about '2+2' which are not true about '4'. Therefore the two are not identical. It's what I've been doing for last number of posts, explaining how '2+2' signifies something different from '4

The two expressions are of course not identical, they are indeed distinct expressions, I already said that, that's clear because the expression 2+2 contains three symbols in it, while the expression 4 contains only one symbol, of course they are not identical. But that doesn't by itself entail that what they are denoting is not identical! There is a difference between "identity of expressions" and "identity of what expressions are denoting". The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object. Now the identity symbol "=" between any two expressions phi, pi , i.e. the expression phi = pi , means phi and pi are denoting the same object, it doesn't mean that phi and pi are identical expressions. You are confusing identity of expressions and identity of what they denote.

If this is true, then show me the object which both '2+2' and '4' refer to

Your wish is my command! First one must note that expressions 2+2, 4 , 2+2=4, all of those doesn't have any innate meaning by themselves, we need to assign meaning to those symbols, otherwise they are just blind string of characters. For example the string of symbols "0 + 0 + 0 = 10" is true in Arabic language, since 0 denote number 5 in English, and 10 denote number 15 in English, while obviously it is false in English. So symbols by themselves are blind, they only acquire meaning by conventional definitions. So 2+2=4 is only true relevant to the context that assigns meaning to its symbols, for example in the system of arithmetic. Now lets take some arithmetical system, for example PA (peano arithmetic) as our background system which assigns meaning to symbols 0, 2, + , =, 4. Now in Peano arithmetic 0 is a constant symbol, it means it is an expression denoting a single object of the domain of discourse of PA. Now the expression S(x) is a functional expression it means its a term of the language of PA that denotes only one object for each particular substitution of x, similarly the expression x + y for any particular substitution of x and y, denotes a single object because + is a two place function symbol. The meaning of "phi = pi" when phi and pi are functional expressions in the language of PA means " phi denotes the same object pi denotes".

Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:

phi denotes || phi ||.

so according to that 2 is denoting the object || S(S(0)) ||.

Also 4 is denoting the object || S(S(S(S(0)))) ||

Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:

2+2=4

The proof of that is present in PA.

What you had in mind is an example of theoretic x meta-theoretic confusion.Which is something that almost everyone passes through!

However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though.
• 7.2k
But that doesn't by itself entail that what they are denoting is not identical!

Right, and I've explained how what is denoted by '2+2' is different from what is denoted by '4'. Principally, '2+2' denotes two units of two whereas '4' denotes one unit of four. So, there are things which we can say that are true about 2+2 which are not true about 4, and vise versa.

The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object.

Sure, but '2+2' denotes two objects whereas '4' denotes one object. And, even if you construe '2+2' as one object, that object is divided in a vey specific way, in half. No such division is specified by '4'.

Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:

phi denotes || phi ||.

so according to that 2 is denoting the object || S(S(0)) ||.

Also 4 is denoting the object || S(S(S(S(0)))) ||

Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:

2+2=4

The proof of that is present in PA.

You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))). Sorry to have to inform you of this, but you haven't provided the premise required to draw your conclusion. Consequently, you have no proof.

However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though.

You really don't seem to understand the difference between equal and identical. Here's some principles which might help.
1. Two distinct things may be equal. For example, distinct human beings are said to be equal.
2. Two distinct things cannot be identical, "the same". "Same" refers to one and only one thing, (Leibniz principle for example, if x is the same as y then x is y, there is only one thing).
3. We do say sometimes, that a thing is equal to itself, as well as being the same as itself.
• 1.5k
You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'.

Let me remedy that omission.

Before I start I hope we're agreed that there are two levels to this discussion:

1) The philosophical point that 2 + 2 is not identical to 4 because the former conveys the information that a thing, namely 2, that is manifestly different than 4, is being combined with itself to produce something entirely different, namely 4. This I take to be your viewpoint.

I might argue that point with you, but I would not be on firm footing. There are subtle philosophical issues that I'm ignorant of; but that at the very least I can see I'm ignorant of them. So I'm not entirely conceding your point; but I must depart the field. I haven't the capacity to defend my side.

2) But on the mathematical side, I claim that 2 + 2 and 4 designate identical numbers and identical sets and of that I have not got the slightest doubt. I regard this as simply a technical matter that I'm educated about and that you are about to be educated about. You may disagree but at least you know where I'm coming from.

So: I say that when in math we write $x = y$ we are asserting that x and y are identically the same. I shall now state my case. (* See note at the end).

1.1 We have the law of identity that says that for each natural number, it is equal to itself.

1) We assume we have the natural numbers as given to us by the Peano axioms. These are denoted by the symbols 0, 1, 2, 3, ... I don't know if you regard this as an objectionable premise. We have to start somewhere.

PA says:

(1) There is an undefined symbol $0$, which we call a "number."

(2) There is a function $S$, called the successor function, that inputs a number and outputs a number.

(3) If $n$, $Sn$ is a number.

There are some other axioms to make sure numbers are suitably well behaved.

https://en.wikipedia.org/wiki/Peano_axioms

With these three axioms we have an endless sequence of numbers: $0, S0, SS0, SSS0, SSSS0, \dots$. As a matter of convention we introduce the following names: $S0 = 1, S2 = 2, S2 = 3, S3 = 4, \dots$. I hope these are not unfamiliar.

Now we need to define the arithmetic operations. We define $+$ inductively as follows:

(*) $n + 0 = 0$

(*) $n + Sm = S(n + m)$

With these definitions in hand we may now evaluate $2 + 2$.

$2 + 2 = 2 + SS0 = S(2 + S0) = SS(2 + 0) = SS2 = SSSS0 = 4$

This puts the matter to rest. The expressions $2 + 2$ and $4$ refer to the same number. It's practically a definition, following so easily from the Peano axioms and the definitions of the symbols $2$, $+$, and $4$.

If you think it means something else, you are mistaken. You may have some intuitions that $+$ means "combining two things to make some other thing," but nothing in the math supports that point of view. I can't help what they told you in first grade.

These are strings of symbols manipulated by formal rules. A computer could implement the rules. The symbols are devoid of meaning except for what we bring to them with our intuitions. And our intuitions are part of our philosophy. They are not in the math itself.

As I say that's the mathematical story. I will concede that you may have a point if you overload the symbols with your intuitions about what they mean. That's the philosophical question. If you think $2 + 2 = 4$ "means" something that the math doesn't say, then you are making a metaphysical point, not a mathematical one. If you got your intuitions in first grade, I'd ask you to update them in the light of how the math actually works.

On the math there is no question. $2 + 2 = 4$ is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of $+$. As I say it's practically a definition.

For completeness this is only half the story. The Peano axioms aren't strong enough to develop a theory of the real numbers, for example. For that we need set theory, including the axiom of infinity. With the appropriate assignment of sets to numbers, and a definition of the successor function, the universe of sets contains a model of the Peano axioms; and the proof I gave can be lifted directly to a proof that the sets designated by $2 + 2$ and $4$ are the same set.

(*) Note -- I did not prove the starred claim that every mathematical equality is a statement of set identity. That would be part of the extension of the discussion to set theory, and I did not want to add those details to this post.
• 62
Imagine what the regular expression accepts, are expressions like this:

{
{1.2323,343.3333}
,{344.2,0,34343.444,6454.6444}
,{2323.11,834.33}
,{}
,{5 12.1,99.343433}
}

So, it only accepts sets, the members of which must be sets themselves, and these member sets must only contain real numbers.

So, it only accepts elements from the power set of real numbers. (Correct?)

What occurs to me is that you only have rational numbers in your sets. Moreover all the subsets of R (actually of Q in this case) are finite, albeit arbitrarily large. How will you represent irrational numbers with a finite number of symbols, especially those that aren't computable? And how will you represent infinite subsets of R?

Another argument: the strings of finite length over a finite alphabet are countably infinite. It's not hard to write a program that spits them out one at a time. So any subset of those strings is at most countably infinite.
• 1.3k
What occurs to me is that you only have rational numbers in your sets.

Agreed.

The positional notation always expresses rationals, and not even all of them, such as 1/3 or so. Furthermore, if the digit stream for a number is infinitely long, then it will never proceed to matching the next number. This problem looks insurmountable.

How will you represent irrational numbers with a finite number of symbols, especially those that aren't computable?

Point taken. Can't be done, indeed.

So any subset of those strings is at most countably infinite.

Yes, I came to realize that now.
• 130
You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))).

Yes the reason is because I'm holding PA, and it shows you the rules about +, so I didn't want to go to all of that technical side. So I just mentioned that the proof is present in PA, and I didn't want to go to this technical detail. But if you follow the axioms of PA you will begin with S(S(0)) + S(S(0)) which denotes || S(S(0)) + S(S(0)) || (you won't see this explicitly written in references about PA, but that's what PA is actually saying, I'm just clarifying it), to just end up with:

S(S(0)) + S(S(0)) denoting || S(S(S(S(0)))) || .

So just go to PA to fill in the missing part, you'll see that for yourself.

Two distinct things may be equal. For example, distinct human beings are said to be equal.

Thanks for this account and the two points after it. But in mathematics when we are speaking about equality we don't mean this really. Equality in mathematics which occurs between expressions, especially when it occurs between functional expressions then it meant to be identity of denotation by those expressions.

You think 2 + 2 is denoting a process that involves a combination of two units to form another unit, which is wrong. 2+2 is "describing" such a process, but NOT denoting such a process, it is denoting what results from that process, that's your error. 2+2 is a functional term in mathematics, it denotes ONE and just ONE particular object, the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair, so x +y = z is meant to be an assignment that sends single substitution of x and single substitution of y to a SINGLE substitution of z. so it sends the ordered pair (x,y) to a single z for each particular substitution of x,y. so 2+2 is meant to be the object that + sends the pair (2,2) to. When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all.

I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG.

We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity.

There is a difference between the details involved in an expression and what that expression is denoting. Denotation of expressions is determined by definition of the rules of the language in which that expression is meaningful. So I agree that there is a lot of say information going on in the expression 2+2, much more than the simple reference involved in the expression 4, that's right, but that doesn't affect their denotation, because their denotation is set by the rules of arithmetic and not by these aspects, by the rules of arithmetic 2+2 is a term of the language and it can ONLY be substituted by a SINGLE object of the universe of discourse, which is as single as 2 is and as single as 4 is, it is as single as any number is (which are the singular objects of the universe of discourse of PA), this is a rule of the language of PA, that + is a function. This is a stipulation, consider it an axiom. And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4. Much as physics say that the single object denoted by "The Sun" is exactly identical to the single object denoted by "The nearest star to Earth and Jupiter", even though there are particular differences in details of those expressions including differences in syntax (particular wordings, number of them, grammatical differences etc..) and difference in semantics (information involved in these sentences), still both sentences are "denoting" the same object. So definitely different expression can convey different set of information to just denote the same object, that's obvious, and 2+2=4 is just once case of that situation.

In nutshell in mathematics the denotation of 2+2 is already stipulated to be a single object that is as single as 2 is and as single as any natural number is, and the denotation of 4 is of course a single object since its a number, and that that single object denoted by 2+2 is exactly the same( is identical to) the single object denote by 4, and that's what 2 + 2 = 4 exactly means.
• 62

Good point about the rationals that would be left out. And thanks for absorbing my feedback in a good spirit. (I like that you know programming. I'm working on becoming a better programmer. )
• 130

I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it.

Axioms:

Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of

for all x,y (x=y -> [phi(x) <-> phi(y|x)])

are axioms

x + y =/= 1
x + y =/= x
x + y = y + x
(x + y) + z = x + (y + z)

Define: x=2 iff x=1+1
Define: x=3 iff x=2+1
Define: x=4 iff x=3+1

Theorem: 2 + 2 = 4

Proof:
By definition of 4 we have: 3+1=4
By definition of 3 we have 2+1=3, use identity axioms to replace this and get:
(2+1)+1=4
By associative law we have (2+1)+1 = 2+(1+1), use identity axioms and replace to get:
2+(1+1) = 4
then by definition of 2 we have 2= 1+1, so by identity axioms replace to get:
2 + 2 = 4
QED

Actually what is used in the above proof is only the definitions of 2,3,4, and the identity and associative laws.

I think theory of addition is complete as far as I know.
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1.1 We have the law of identity that says that for each natural number, it is equal to itself.

This is our point of disagreement. The law of identity does not say this, you are claiming this. If there is a law in mathematics which states that each natural number is equal to itself, it is not the law of identity. it is a law of equality. So what I am asking is, on what grounds do you say that this law of equality is a law of identity, and supplant the real law of identity with this one?

This puts the matter to rest. The expressions 2+2 2+22 + 2 and 4 44 refer to the same number.

It shows me that they are equal, it doesn't show me that they are the same. That's the disputed point, that being equal means being the same.

These are strings of symbols manipulated by formal rules.

Your mistake is in your citation of the 'rules'. The law of identity does not say that a number is equal to itself. I suggest you revisit the difference between identity and equality. "Identical" means the same whereas "equal" means having the same value. Do you see a difference between these two? When two things have the same value they are not necessarily the same thing. Being the same thing is what is necessary to fulfill the conditions of the law of identity.

On the math there is no question. 2+2=4 2+2=42 + 2 = 4 is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of + ++. As I say it's practically a definition.

You keep saying this, that it is "derived directly from the law of identity", and you refer me to websites which discuss equality. Nowhere have I found the law of identity mentioned in this discussion of equality. So I really think that it is just you (and perhaps many others) who mistakenly believe that equality is derived from identity, and I am trying to point this out to you. Perhaps I am the one who is wrong, and equality is really derived from identity, but if so, where is the evidence of this?

So just go to PA to fill in the missing part, you'll see that for yourself.

Well fishfry seems to have done this already, but it doesn''t show identity, it shows equality. That is the point being discussed, you and fishfry seem to think that in PA equal things are the same thing.

the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair

Right, this is what I said,. By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'.

When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all.

This is not true, '=' means equal, it does not mean identical. You are arbitrarily replacing what '=' really denotes, with "identical" and this produces a false statement. When you arbitrarily change the meaning of symbols in your interpretation, you create false statements.

I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG.

This is totally irrelevant. What is WRONG, is to arbitrarily claim that two equal (having the same value) things are identical (the same).

And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4.

This is what fishfry claimed to show above. But the demonstration does not show that the two are exactly identical, it shows that they are equal. Then fishfry states a misrepresentation of the law of identity, claiming that the law of identity states "that for each natural number, it is equal to itself". Where is your understanding of the law of identity?

We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity.

What we need to agree on is definitions of "equal", and "same".

Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of

This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules!
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This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules!

No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology:

Glossary of First-Order Logic

Just use the find function on your browser, and search for "identity" and read all of what it says about it in that site.

The "=" symbol is used to symbolize identity, so x=y actually means that x and y are exactly the same object, i.e. they are identical, and not that they are having the same value and remain discriminate at the same time.

To be more precise, due to shortages of formal languages, it is better to call identity as indiscernibility, because under that theory in question we say that x and y are identical if the theory in question cannot have an expression phi(x) (written in the language of the theory and in which x occur) and an expression phi(y|x) [which is the expression obtained by merely replacing all occurrences of x by y in formula phi(x)] such that phi(x) is true of x and phi(y|x) is not true of y. So we say that x and y are indiscernible under the language of that theory. Of course that doesn't necessarily mean that they are in reality identical, it just means that the theory in question cannot discriminate between them and so it see them as "identical", i.e. it says that they are identical.

The indiscernibility of identicals is a famous law, and in first order logic it is the law that I wrote (and that ironically you said it is not about identity??] see:

Leibniz's law

About the first law of identity which is reflexivity law, i.e. that every thing is identical to itself, this is just a trivially true statement about identity, there is no dispute about that.

So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions??

However, as I said you can "technically" speaking have some theories that see some objects as identical, but other theories can discern between them, yes this can happen, much as we human can see a star and think its one while in fact it is two or more stars.

On the other hand if we are to understand Equality in YOUR sense as assignment to a common object, like in having a fixed function F over a certain domain D, so we'll say that all elements of D are equal under F, to just mean they are assigned the same value (image) under F. Note here that D can have many members. This use of 'equality' is not perfect, it is mentioned in common languages like that, yes, but it is imprecise, it hides a lot of details, and certainly it is NOT what is meant by equality which is symbolized by "=" in mathematics. In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think.

Equality as used in PA and in ZFC, and generally in first order logic with equality, that is symbolized by "=", in those contexts it exactly means identity or sameness of objects, more precisely speaking that the theory in question cannot discriminate between x and y if it proves that x=y.
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By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'.

You say '2+2' denotes an object divided in half. Well I'd say: OK no problem.

You continue saying whereas '4' denotes no such division.

Yes the correct wording is that '4' doesn't denote such a division, this is clearer. However it doesn't deny it? You seem to be confusing : Not denoting phi , for denoting not phi. So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error. Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam.

So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half.

Absence of denotation doesn't mean denotation of absence.

Absence of denotation just signal incompleteness of information.

We are not claiming that expressions supply FULL information about what they are denoting.

2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object. I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting.
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No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology:

Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic. That the symbol "=" means identical in first order logic does not demonstrate that the symbol "=" means identical in mathematics. Here's a quote from the section of the referred site on "identity" : " Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."

In all the mathematical sites which fishfry referred me to, none of them spoke of equality as identity. So you are just continuing with your hollow assertions.

So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions??

So you and fishfry just decided to change the name of the theory from "equality" to "identity", for some arbitrary reason. Or was that done for the purpose of deception?

In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think.

Now I'm convinced, you've changed the name from "equality" to "identity" for the purpose of deception, and you are flat out lying here. Why do you seek to deceive?

So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error.

Please reread this, and I hope you can see your mistake. Is '4' denoting an object? Yes. Is it denoting that the object is divided? No. Therefore '4' denotes an object not divided. Look, '4' does not denote that the object denoted is divided in half. That is clear. Therefore what is denoted is an object not divided in half. That the object, like any object, may potentially be divided in half, divided some other way, manipulated in any other way, or be converted into an infinite number of other objects, is irrelevant to what is denoted. What is denoted is an object not divided, and you are lying when you say that to interpret in this way is an error, because I can tell from the contorted way that you've written the passage that you are trying to disguise the truth. Why lie?

Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam.

We are not talking about claims, we are talking about denotations. If '4' denotes an object, it is impossible that the object denoted by '4' is divided in half or else it would not be an object denoted, but two other objects, the halves.

So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half.

What is denoted is what "is" denoted. If what is denoted is an object then that object is not divided in half, or else it would be two objects, regardless of whether the object is divisible. I hope you understand that it is contradictory to talk about one object which is two objects.

Absence of denotation doesn't mean denotation of absence.

Absence of denotation just signal incompleteness of information.

This is utter nonsense. What is denoted is what is denoted. It's completely nonsensical and illogical to say that the denotation could possibly include an infinity of other things which are not actually included in the denotation. Can't you see that you're just blabbering nonsense in an attempt to cover up your lies? The denotation is of something specific and what is not included in the denotation is not denoted. It's nonsense to argue that something else could have been denoted, therefore we should allow that what could have been denoted is part of what was actually was denoted

2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object.

But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration?

I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting.

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It does not state that the sets are the same, it states that if the members are the same, then the sets are equal. Therefore the sets remain distinct, as two equal sets, not one and the same set.

I can't respond to this. You're factually wrong. There's only one set {0,1,2, pi}. There isn't "another" set that happens to have the same elements and is therefore equal. Any set with the same elements is identical to this set.

If you don't get that or you don't want to get that or you think I'm completely wrong, that's your privilege. You're making mathematical claims that are false. It's ok. A lot of people do that and I can't fix them all. I've said my piece here. All the best.
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I just wanted to add, that we can actually have a very simple system in which 2 + 2 = 4, that of first order logic and add to it primitives of identity (equality) symbolized as "=" which is a binary relation symbol, and of "+" denoting addition which is a two place function symbol, and of "1" denoting what we customarily know as one, which is a constant symbol. I'll try to coin a system in which 1 is the first number, i.e. doesn't have zero in it.

Yes thanks for making that point. In fact Russell and Whitehead famously took 400 or whatever pages to prove that 1 + 1 = 2 directly from logic; and presumably they could do 2 + 2 = 4. The only reason I didn't mention it is that I'm not familiar with the development of numbers as in R&W. My knowledge is mostly in the math domain which means I need to start with the Peano axioms. But Russell and Whitehead is probably the right answer to how you show that 2 + 2 = 4 is a logical identity.
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1.1 We have the law of identity that says that for each natural number, it is equal to itself.
— fishfry

This is our point of disagreement. The law of identity does not say this, you are claiming this.

Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself. Why wouldn't this apply to numbers? A rock is identical to itself, the number 3 is identical to itself. You are claiming the former and denying the latter? Perhaps this is a clue to why we disagree. How can a number not be identical with itself by virtue of the law of identity?
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But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration?

Here is your error, you think that '2 + 2' denotes TWO objects. This is wrong. You are not understanding the operator "+", this is a two place FUNCTION symbol, you need to read some logic related to mathematics, i.e. foundational work on mathematics. "+" is a two place FUNCTION, it means that it is a ternary relation that sends a pair of objects to ONE object for that particular pair, so suppose you are summing 9 and 8 here the addition "+" function would send the pair {9,8} to ONE number that is 17, in other words view addition as some process that at each time it has TWO INPUTS and ONE output such that whenever you input the same values again you get the same output again. Now it is important to understand what the expression "2 + 2" means, it means the OUTPUT of summing 2 with 2. In other words the expression "2+2" denotes the object that the operator + would send the pair {2,2} to. I hope this is clear. So '2 + 2' by definition of functionality of "+" cannot denote two objects. The appearance of two symbols in it, i.e. the symbol "2" appearing twice, doesn't mean that "2 + 2" is denoting two objects at all, "2 + 2" is the VALUE of the function + for the pair {2,2}, and it is ONE object. You are not discriminating between "denotation" and 'information "predication" accompanied with that denotation', '2+2' denotes ONE object and only ONE object which is the value of the + operator on the pair {2,2}, but '2 + 2' carries information [this is not denotation] related to that denotation, that the single object denoted by '2+2' can be split in half, i.e. it is the value of a pair having identical projections, that doesn't mean that it is denoting two objects at all. On the other side the expression '4' is a constant symbol, it also denotes a single object, but a constant symbol is a zero function symbol, so it does NOT carry with it any additional information about what it denotes, but at the same time it doesn't denote absence of any kind of information about what it denotes, so it doesn't denote an object that is not divisible in half, i.e. cannot be the value of + function from a pair with identical projections, it cannot assert that negative information about what its denoting because it is a ZERO place function symbol.

Your main error is that you think that "2 + 2" is denoting two objects.
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possibly I don't agree with R&W on that. + is not a logical operator, it is a mathematical operator, but as you know we can speak logically about extra-logical concepts, we can add them to any logical system, but of course the result is not a purely logical system, but a logically compatible system you may say, usually refereed to as logically extended system. The trivial complete system that I've depicted is not a pure logical system, it is a logico-mathematical system. I think it's complete. i.e. not subject to Godel's incompleteness theorems, but I'm not sure really.
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Your main error is that you think that "2 + 2" is denoting two objects.
One object but two digits.

You can erase one digit and be left with an object here, but not with 4. Clearly there's a subtle difference, just as between soup and its components.
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Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic

Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking. Even older mathematical systems like ordinary math, all those can be recaptured more effectively as systems extending first order logic with identity. I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics. In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity.

Now the question is what about older systems that are not formalized as extensions of first order logic with identity, can we understand the = in them as something other than identity, the answer is yes of course, it can be taken to be an equivalence relation, and I agree you'd better name them as equality, since it is not necessarily the identity relation. But formal recapturing of them as extensions of first order logic with identity with the = sign taken to represent identity, is by far a much sharper and more well defined an rigorous approach.

But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION.
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One object but two digits

Yes! On the informal level I would agree, but formally NO. It only symbolizes a number that the operator + is sending the pair (2,2) to. It doesn't speak of anything of that number having two or more digits. In reality the best representation of four is as four strokes, but this is besides the formal system of arithmetic actually.
• 856
Four strokes as in four ones.
But four ones is not one four, due to the obvious notion that it's a single stroke.

I think the point trying to be made here is that a shared value does not denote that two things are identical, as obviously they have at least one differing quality.

Take a stick.
Snap it in half.
Is it the same?
• 7.2k
Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking.

Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them. Now, the question is where does ZFC derive its meaning of "=", from traditional mathematics, or from the law of identity. As we seem to agree, though you are intent on making an impossible reduction, the two meanings of "=" are distinct. Since the axioms of ZFC do not mention the law of identity, but mention a theory of equality, I think it is quite obvious that ZFC derives its meaning of "=" from traditional mathematics, and not the law of identity.

I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics.

It is not a terminology preference. In traditional mathematics "equal" means having the same value (and I'm sure you are fully aware of this), implying that two distinct things may have the same value. In the law of identity "same" means one and only one thing.

It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity.

In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity.

There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose?

Now the question is what about older systems that are not formalized as extensions of first order logic with identity...

Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive.

But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION.

Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me.
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Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive.

I didn't ask about those which CANNOT, I asked about those which are not. I mean are not presented in a formal manner as an extension of first order logic with identity. Of course they can be formalized as an extension of first order logic with identity, but I'm asking about when we don't do that and leave it un-formalized. Then in this case how are we to understand "=" symbol in them. I'd say that it is not necessarily the identity symbol. Yes that is correct of course.

Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity.

if ZFC is presented as an extension of first order logic with identity, then of course "=" would stand for identity. IF we don't do that, then of course it would not necessarily stand for identity.

Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them

Not only ZFC, you have PA (peano arithmetic) nowadays presented as an extension of first order logic with identity. And there are many other systems also so presented, you can read about reverse mathematics. Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity.

It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity.

As I said above, this depends on how you formalize ZFC, if you formalize it as extension of first order logic with identity then the = symbol in it would stand for identity. If not then it can stand for some other equivalence relation.

There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose?

I don't know why you keep assuming that I'm lying? Anyhow. The fact that nearly all of traditional mathematics can be formalized as extension of first order logic with identity is well known, you can see reverse mathematics for that. And you can serf the web for Harvey Friedman's grand conjecture, etc..

Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me.

The ordered pair of the two objects, here in your example (2,2) is not what is denoted by "2 + 2", I'm trying to tell you that but you keep refusing to listen, the expression "2 + 2" is the object that the + operator send the ordered pair (2,2) to. To clarify this: the + operator is sending the pair (2,2) to some object call this object k, to represent that for you by an informal sketch:

(2,2) ---+---> k

Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it.

You may say No. not necessarily, 2 + 2 is denoting an object k, and 4 is denoting an object L, where L is not identical to k, but L is equal to K, i.e. L is possessing some relation R to k where R is some equivalence relation that can occur between distinct (non-identical) objects. So according to this 2 + 2 is denoting an object that have the relation R to the object denoted by 4 where R is some equivalence relation that is not necessarily the identity relation, of course the intention is that the = sign stand for that equivalence relation R. OK this is a possible case of course, but this is more complicated! It is much easier to stipulate that R is the identity relation itself.

Anyhow as I said before if you present arithmetic or any mathematical theory that contain the symbol = as an extension of first order logic with identity, then = would be taken to symbolize identity itself. if not then it can stand for some other equivalence relation.

I hope that settles matters.
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Of course they can be formalized as an extension of first order logic with identity,

Obviously this cannot be done validly, because "=" does not always mean "same", something you refuse to acknowledge, for some strange reason.

Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity.

Sure, you can interpret "=" as identical, but that would only produce a false presentation based in the fallacy of equivocation, as the website which you referred me to clearly states: "Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."

I don't know why you keep assuming that I'm lying?

The evidence shows that you know what you are asserting to be false. This is lying.

(2,2) ---+---> k

Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it.

There is no "k" though. What is symbolized is "2+2", two objects and an operator, not one object "k". So this object represented by "k" is not represented by "2+2", it has been wrongly created by you mind, false imagination, nothing here represents it.

That this is the case is evident from the fact that you proceed to say "that object, i.e., k is exactly the natural number denoted by the symbol 4". There is no symbol "4" in the expression "2+2". The object "k" is only on the right side, where the operator sends the ordered pair. So I can ask you, what does "k" really represent? Does it represent what "4" represents, as you say here, or does it represent what "2+2" represents, as you say above? You are only contradicting yourself.

Or perhaps that little arrow represents the same thing as "=", and all you are doing is stating "2+2=4".

However you look at it, there is no operator signified by "4", so it is very clear that "2+2" does not represent the same thing as "4". You've just stated my case for me. It is your repeated demonstration that you clearly understand that "2+2" signifies something different from "4", though you assert the opposite, which makes me say you are lying.

Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself.

The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing.
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So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing.

OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extension of first order logic with identity where the symbol "=" is taken to mean "identity" i.e. "being the same as". Actually this is a well known result, actually most of that kind of mathematics can be formalized in second order arithmetic, you can read about it in reverse mathematics which also can be re-formalized as an extension of first order logic with identity. Actually ZFC itself can be formalized as an extension of first order logic with identity, and ZFC is way stronger than almost all of mathematics before the era of set theory. This is a very well known result.

You say that it is fundamental to mathematics that the two sides of 2 + 2 = 4 must represent distinct objects.

I say that if = stands for identity, then it would mean that 2 + 2 denotes (represents) exactly the same object that 4 represents (denotes). Obviously you object to that, you say that there is something fundamental against this.

what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. Notice that identity relation is a kind of equality relation, but the converse is not true, you can have an equality relation that is not identity. OK. But why = as used in mathematics, for example in arithmetic, why it is not reducible to identity in your understanding?

Notice that Peano arithmetic which is a very famous theory of arithmetic, is indeed formalized nowadays as an extension of first order logic with identity, of course with the understanding that "=" is taken to represented identity relation and not any other kind of equality.

If there is something fundamental to mathematics against the use of = symbol in it to represent identity, then how PA is formalized as such??? How ZFC is formalized as such and it is generally regarded by many as the official foundation of mathematics? Both are indeed formalized with = in them understood as identity.

What you are saying is that the current foundational systems of mathematics are committing a fundamental error? (notice that most of those are coined as extensions of first order logic with identity) According to your account they must instead represent the = as an equivalence relation that can hold between distinct objects, and that the object denoted by 2 + 2 must be considered as a distinct object from that denoted by 4. This is strange? why?
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