Of course there are many kinds of paradoxes ranging formal to informal. But you referred to Russell's barber paradox, which is an informal illustration standing for concerns in logic and mathematics, so I wrote in that context.

In any case, hopefully now you see that your presentations of Russell's barber paradox were incorrect and not paradoxes since they have easy non-perplexing answers.

The video about the liar paradox and incompleteness is atrocious, ignorant, lying disinformation.

(1) The video, in its juvenile way, dishonestly mocks people who have studied the subject. It says about the incompleteness theorem, "What exactly does [incompleteness] mean is often something ["incompleteness experts"] have difficulty expressing" while showing a picture of a man in front of a blackboard thinking, "Um..."

First, what exactly is "exactly" supposed to mean there? What form of exactitude is being asked for? Incompleteness is a mathematical theorem that is provable in finitistic arithmetic. It is utterly exact in that way. And the theorem itself is exact, even put in English mathematical terminology: There is no consistent, formal system that proves all the true statements about arithmetic. If one knows what 'consistent', 'formal', 'prove', 'true statement' and 'arithmetic' mean, then the meaning of the theorem is easily understood.

The main question that incompleteness answers is natural and not at all nebulous or "esoteric": If the logic we use obeys the principal of non-contradiction, then are there formal, consistent mathematical axioms that prove all the true statements of arithmetic? The incompleteness theorem tells us that the answer is "no". And that implies there is no algorithm to decide of statements of arithmetic whether they are true. Again, that answers a natural question that may occur to anyone who has a modicum of intellectual curiosity, even to a high school student who might wonder whether there is an algorithm that would answer all of his or her math homework questions.

Moreover there are numerous books and articles that present the proof and discuss the mathematical import of incompleteness.

And the use of scare quotes with "expert" is sophomoric. There are people who are experts on the subject.

Then he says, "And usually you'll get a regurgitation of the aforementioned statement, that is that there are additional statements in the system that are not provable".

First, there is the gratuitously pejorative "regurgitation". Second, it is utterly reasonable to say again exactly what the theorem states, which is straightforward: There is no consistent, formal system that proves all the true statements of arithmetic. To ask that that itself be explained, is to ask the meanings of 'consistent', 'formal system', 'true statement' and 'arithmetic. And anyone familiar with the subject can supply those definitions too. Rather than mathematicians being stuck to explain the subject, mathematicians are prepared to define and explain the terminology used to explicate the the theorem and its proof; there is nothing the least bit esoteric or ineffable about it, notwithstanding that the author of the video (in his egregious ignorance and will to mock what he hasn't bothered to study) postures that there is.

(2) Then the video mentions that Godel mentions that the theorem has a close relation to the liar paradox. It needs to be stressed that the liar paradox is not actually part of the theorem or its proof, but rather that the idea for the theorem and proof are suggested, by analogy only, by the liar paradox. It does not discredit the theorem or its proof one iota that an idea outside of the theorem and proof provided an insight on how to use that idea in its main outline but in a variation that, unlike the paradox, is utterly formally correct and not mathematically nor philosophically problematic. Or one can just as easily mock the invention of the airplane on the basis that it was inspired by observing the flight of birds while everyone knows that humans can't fly in the way of birds; just as anyone who has studied the subject knows that in the relevant languages, the liar sentence is not formalizable but the unprovability sentence is.

* Then, "So when Godel says that his theorem and the liar paradox are closely related, he's indicating that they are literally the same thing but with the word 'untrue' swapped for the word 'unprovable".

This sentence is false.
This sentence is unprovable.

Different by one word. The first is not formalizable in the interpreted language in question and the second is.

The fact that they are different in only one word doesn't diminish that they are vastly different in meaning. Godel never said that they are the same thing except using a different word.

This sentence has only six words.
This sentence has only six syllables.

Different by only one word. The first is true and the second is false.

The author of the video is a liar.
The author of the video is a truth-teller.

Different by only one word. The first is true and the second is false.

(3) Then this brazen flat out lie: "[...] as Godel writes in the closing lines of his introduction, it's the liar statement itself and the contradiction that it furnishes that allows him to reach his famous conclusion regarding the existence of undecidable statements in mathematics."

Godel wrote no such thing.

And while the video says that he did, it shows a supposed translation on screen of those supposed remarks; but the translation shown on screen says no such thing.

The intent must be that people not informed about the subject and not thinking critically about the video and not looking carefully at the quick jump cut text snippets on screen will nod along thinking that Godel must have wrote it because the video says he did and shows it on screen too.

It's a lie that Godel wrote it. And it's a lie that Godel's proof relies on any contradiction derived from the liar sentence, and it's additionally dishonest to purport to support that he did say it by flashing snippets that do not at all support that he said it.

For the benefit of the truth:

* The idea of the proof is heuristically suggested by the liar paradox. A heuristic notion is very different from an actual mathematical argument. And Godel' proof itself invokes only mathematical argument and stands mathematically, independent of the heuristic notions.

* The liar sentence does not appear in any step of the proof.

* The proof shows that the system can formalize "this sentence is not provable". That is very different from "This sentence is not true", and we prove that the the predicate 'is true' (not formalizable in the system itself) and the predicate 'is provable' (formalizable in the system itself) are not coextensive since there are true sentences that are not provable.

(4) Then, "If common sense tells us that the liar paradox isn't anything significant in everyday language, why the heck does it suddenly become so significant when it's translated into a mathematical, computable, or logical language?"

The answer is simple: As mentioned above, the incompleteness proof does not translate the liar paradox. The incompleteness proof adduces a mathematical formula that happens to be true if and only if it is not provable in the system. As mentioned above, that "I am not provable" is similar with "I am not true" doesn't entail that they are the same nor that the implications of formalizing them are the same. The formalization of "I am not provable" is significant even at the most basic level that it leads to answering the question even a middle school student could ask, "Isn't there surefire, step by step method I could follow that would answer every math homework question I have to answer?"

But the author says that one approach is to "assert a significant distinction between the everyday notion of truth versus the mathematical notion of provability."

* We provide a mathematical definition of truth and a mathematical definition of provability. We don't merely assert that they are not coextensive, but rather we prove that they are not.

* The mathematical definition of truth is not claimed to adhere to all the everyday senses of truth, but it does capture the important aspects in context of mathematics.

Then, "To us, such a distinction is superficial and doesn't hold up to deeper analysis."

* Who, pray tell, is "us"?

* What, pray tell, "deeper analysis" is claimed to have been performed?

* The ignoramus author of this video should get a book on symbolic logic then one on mathematical logic to learn how the definitions are formulated and the ensuing theorems.

Then, "One can easily recast the liar paradox in terms that don't invoke the concept of true and false. [...] Indicating that we can't attribute the significance of incompleteness solely to the distinction between true and provable."

* That's a strawman, red herring and non sequitur all in one:

Strawman: No one said that the significance of incompleteness is solely in the distinction between true and provable. Indeed, incompleteness can be stated and proved even without mentioning truth; we can prove the purely syntactical version of the theorem that involves only provability, without having to add the result that the Godel sentence is true.

Red herring: Yes, there are paradoxes similar to the liar paradox but not involving the notion of truth, but in the video that fact is mentioned only as a distraction from the fact that the incompleteness theorem is rigorously proven and that it is significant even at a glance.

Non sequitur: That there are paradoxes similar to the liar paradox but not involving the notion of truth doesn't entail that the incompleteness theorem lacks significance.

Then, "we recognize that Godel's incompleteness theorem was a reaction to the Hilbert program of the early 20th century. A program that basically treated math like a religion and assumed that all mathematical truths could be constructed out of a few simple ones".

* Hilbert's program was the opposite of religion. The program was to regard finitistic calculation as safe and then to prove the consistency of infinitistic mathematics by use of only finitistic calculation. Also, unless someone can adduce otherwise, Hilbert did not regard the axioms as given by a kind of divine revelation. And he did not assume that all the mathematical truths could be derived from a few simple axioms, but rather he proposed seeking a way to do it.

* Moreover, formal axiomatics is the complete opposite of subjective belief. Nothing could be more public and objective than the machine-checkability of formal proofs.

* Godel did not first set out to take down Hilbert's program. The opposite. Godel started by trying to prove the consistency of analysis by finitistic means, but his efforts gave clues that it could not be done, so he switched to proving that it could not be done, which led to the incompleteness work.

Then, "By showing that an absurd self-referential statement could be constructed in wholly mathematical terms, one could argue that Godel was cleverly demonstrating that mathematics was merely a language like any other and therefore subject to all its usual foibles."

* The Godel sentence is not absurd. It's a true arithmetical sentence. A true arithmetical statement cannot be an absurdity. But it also happens that the sentence is true if and only if it is not provable in the system. That still does not make it absurd.

* Mathematics is not merely a language. Various mathematical systems are put in various mathematical languages, but they also have formation rules, inference rules, axioms, proofs and theorems.

* Mathematical languages are not just like other languages. Mathematical languages are formalized while ordinary languages are not.

* Mathematical languages are not subject to all the foibles of ordinary languages. Most saliently, for example, the language of first order arithmetic doesn't have the ambiguities of a natural language such as English.

* Incompleteness is not about languages but rather about systems.

Then, "in order to conclude that mathematics is incomplete [...]"

* The incompleteness theorem doesn't say "mathematics is incomplete". Rather, it says that certain kinds of systems are incomplete. For that matter, for any interpreted language, there is the complete and consistent theory of all the true sentences per that interpreted language. It's complete and consistent. It's just that its not axiomatizable in a way that we can mechanically decide of a given sentence whether it is an axiom.

Continuing, "we have to first believe that the sentence 'this statement is unprovable' is truly paradoxical and not simply a gibberish nonsensical statement disguised as a well formed one".

Wow, that guy is such a self-confused ignoramus.

* He has it exactly backwards. We definitely do not believe that the sentence is paradoxical. Again, back to the earlier central point: The liar sentence is paradoxical, but the Godel sentence is not paradoxical.

* That the Godel sentence is well formed is proven by Godel in the proof itself. And we can mechanically confirm that it is well formed.

* And the sentence is not gibberish as it is well formed in the language of arithmetic.

Then, "Tarski [concluded that] the liar's paradox was something extremely significant".

* It is significant in Tarski's work as it leads to showing the need for our mathematical contexts to not allow such paradoxes. That led to Tarski proving a truth predicate is not formalizable in certain kinds of interpreted languages since such a predicate would yield a liar sentence and its contradiction.

And, "[Tarski's] undefinability theorem hinges on the liar paradox"

* Again, Internet ignoramuses have it backwards. Tarski does not claim that the liar sentence is formalized in the languages, but rather Tarski shows that if a truth predicate is formalizable then the liar sentence would be formalizable, which would yield a contradiction, thus a truth predicate is not formalizable.

Then, the video author shows a picture of Tarski with a text bubble, "I AM the logic". The video author is juvenile as well as a liar and an ignoramus.

At the end of the day, the best we can do is paraphrase the sad resignation in the movie, "Forget it, Jake. It's the Internet."

(1) Higher order logics are usually 2-valued, especially the most famous and most studied ones. One may devise other valuations, but ordinary higher order logics are 2-valued. Being 2-valued is not unique to 0-order and first order logics.

(2) The distinction between 'true vs false' and '1 vs 0' is not essential in the context of ordinary mathematical logic. They both can be understood as the two Boolean values and similar in the substantive senses in which they are used that way.

Such utterly incidental questions as to the meaning of 'barber' can't seriously be considered part of the subject of paradox.

Anyway, looking in several dictionaries, I find that merely shaving someone does not constitute being a barber.

Again, if you add the premise that there is only one man in Seville who does any shaving (viz. the barber), and that the barber does not shave himself, but that every man in Seville is shaved is by some man in Seville, then of course that is a contradiction. But so what?

On the other hand, if, as you now suggest, you're interested in looking at the subject of tonsorial practices in various societies, then you don't need Russell's paradox for that.

/

Yes, there is a logical way to look at the paradox. The most obvious is to observe the theorem of logic:

~ExAy(Rxy <-> ~Ryy)

which in set theory yields:

~ExAy(xey <-> ~yey)

"there is no set of all the sets that are not members of themselves"

which in set theory yields:

~ExAy yex

"there is no set of all sets"

(1) The proof on pages 275-276 of the Tarski paper is not a proof of undefinability, but rather it is a proof of undecidability, specifically that there is a true sentence such that neither it nor its negation are provable in the system.

(2) The proof on pages 275-276 of the Tarski paper does not rely on the liar sentence in any step. Rather, the liar sentence is the reductio ad absurdum in different proofs - those of the undefinability - such that the liar sentence is not assumed but rather from the assumption of definability we derive the liar sentence thus refuting definability. So the situation is the exact opposite of what the poster claims. Again, we don't assume that we can formulate the liar paradox, but rather, toward a contradiction, we assume definability from which we derive that we can formulate a liar sentence, from which we would have a contradiction, thus we refute the assumption of definability.

Saying it another way, since this point continues to be misrepresented by the poster: Tarski shows that the existence of a liar sentence in certain interpreted languages would imply a contradiction, thus in those languages, a liar sentence does not exist.

(3) The passage that begins "In accordance" explicitly is not a statement of a liar sentence but rather it explicitly is a statement about provability. Tarski explicitly says that we don't use a truth predicate there but rather a provability predicate. It is the very point that in the languages under consideration, we cannot form an "I am not true in the interpretation" sentence but that we can form an "I am not provable in the system" sentence.

The poster's gravamen is a brazen straw man based on brazenly reversing what Tarski and mathematicians say.

To reiterate, the paradox doesn't even need any mention of maleness, barbers, towns or even humans. It is best seen in its starkest form:

Something shaves all and only those that do not shave themselves. Contradiction.

For set theory (replacing the 2-place relation 'shaves' with the 2-place relation 'is a member of'):

There is an x that such that for all y, y is a member of x if and only if y is not a member of y. Contradiction.

And most generally, in logic alone, for any 2-place relation R:

There is an x such that for all y, x bears the relation R to y if and only y does not bear the relation R to y. Contradiction.

In symbols:

ExAy(Rxy <-> ~Ryy). Contradiction.

'the barber of Seville' is a definite description. There are different ways of handling definite descriptions, including, at least, both the Fregean and Russellian. To bring that complication into the example would require being exact in how we do it.

In any case, one can shave a person without being a barber. So, still, there is no paradox.

On the other hand, you could add a premise: There is one and only one man in Seville who does any shaving.

E!x(Sx & Ey Hxy)

But that premise is not in your statements.

Why don't you just look up the barber paradox to see that it involves:

one who shaves all and only those who do not shave themselves

You have to have both - all of those and only those.

If you say, "there is someone who shaves all and only those who do not shave themselves" then you do have a contradiction.

(1) If one insists on the premise that it makes no sense to speak of mathematical objects, then one may hold that it makes no sense to speak of the law of identity applying to mathematical objects. But it is not required to accept that premise. Moreover, different ways have been mentioned in which we may still refer to the mathematics of sets, numbers, etc. without recourse to calling them 'objects'. Whatever they are, we still have formal mathematical languages in which to posit axioms that are easily understood as expressing conditions for things like sameness.

(2) Indeed, one should not conflate symbols and occurrences of symbols. In the formula:

1 = 1

the only symbols that occur are '1' and '=', but there are two different occurrences of '1'. Indeed, the first occurrence of '1' is not the same as the second occurrence of '1', but still the symbol '1' is the symbol '1'.

The law of identity is: "a thing is itself"

Or: "a thing is identical with itself"

Or: for all x, x is identical with x

Or: for all x, x is x

Symbolized: Ax x=x

Indeed we do not thereby claim that the three occurrences of 'x' in that formula are the same occurrence. To argue that we do is a strawman.

And the point stands that we well understand that '2+2' and '4' refer to the same number. There are not two different numbers, one named '2+2' and one named '4'. There are two different expressions: '2+2' and '4', but they name the same number.

And, for example, the procedure of adding 2 to 2 is different from the procedure of subtracting 2 from 6, but the result of those procedures is the same. And '2'+2' and '6-2' do not stand for procedures but rather for the result of the procedures. So '2+2' and '6-2' stand for the same number. So mathematics writes: 2+2 = 6-2.

/

We still have not heard a reply to the challenge to state "the" ordering of The Beatles, pertinent to the claim that every set has only one ordering. After several requests to even address the challenge to state "the" ordering, I take it that the poster has no answer and no willingness even to address the challenge. And so it stands that the poster has no viable claim that sets have only one ordering.

We could add that men of Seville are shaved only by men of Seville and that every man of Seville is shaved, and still we would not have a paradox in either case (1a) or (2a):

(1a) The barber is a man of Seville, and the barber shaves all the men of Seville who are not the barber, and the barber does not shave himself, and every man of Seville is shaved by a man of Seville.

Sb &
Ax((Sx & ~x=b) -> Hbx) &
~Hbb &
Ax(Sx -> Ey(Sy & Hyx))

It is consistent that the barber is shaved by a man of Seville:

Ey(Sy & Hyb)

(2a) The barber is a man of Seville, and the barber shaves only the men of Seville who do not shave themselves, and the barber does not shave himself, and every man of Seville is shaved by a man of Seville.

Sb &
Hbx -> (Sx & ~Hxx) &
~Hbb &
Ax(Sx -> Ey(Sy & Hyx))

It is consistent that the barber is shaved by a man of Seville:

Ey(Sy & Hyb)

Consistency proof for both (1a) and (2a)

Let the universe be {barber, Charlie}

Let the men of Seville be {barber, Charlie}

Let the barber shave Charlie and Charlie shave the barber.

Symbolized:

U = {b c}
S = {b c}
H = {<b c> <c b>}

You could study the subject more to learn how to express your idea in an understandable way. But, of course, that would be a matter of how you spend your own time.

In this context, to claim that there is a paradox is to show how a contradiction is drawn. But there is no contradiction drawn from "The Barber of Seville shaves only the men of Seville who don't shave themselves". Three possibilities are each separately consistent with "The Barber of Seville shaves only the men of Seville who don't shave themselves": It is consistent that the barber is not shaven, and it is also consistent that the barber is shaved by someone who is not a man of Seville, and it is also consistent that the barber is shaved by someone who is a man of Seville.

(1) "The Barber of Seville shaves all the men of Seville apart from himself." [original post]

(2) "The Barber of Seville shaves only the men of Seville who don't shave themselves." [latest post]

Those are two different propositions.

Let 'Sx' stand for 'x is a man of Seville'.

Let 'Hxy' stand for 'x shaves y'.

(1) would be taken to mean "the barber is a man of Seville and if x is a man of Seville who is not the barber, then the barber shaves x", and probably tacit is "the barber does not shave himself".

Sb & Ax((Sx & ~x=b) -> Hbx) & ~Hbb

(2) would be taken to mean "the barber is a man of Seville and if the barber shaves x, then x is a man of Seville who does not shave himself."

Sb & (Hbx -> (Sx & ~Hxx))

But neither (1) nor (2) are paradoxical.

With (1) it is consistent that the barber is not shaven: ~Ex Hxb, and it is also consistent with (1) that the barber is shaved by someone who is not a man of Seville: Ex(~Sx & Hxb), and it is also consistent with (1) that the barber is shaved by someone who is a man of Seville: Ex(Sx & Hxb).

With (2) it is consistent that the barber is not shaven: ~Ex Hxb, and it is also consistent with (2) that the barber is shaved by someone who is not a man of Seville: Ex(~Sx & Hxb), and it is also consistent with (2) that the barber is shaved by someone who is a man of Seville: Ex(Sx & Hxb).

/

The actual contradictory formulation is (and we don't even need to mention maleness, a location such as Seville, or being a barber):

There is someone who shaves all and only those who do not shave themselves:

EbAx(Hbx <-> ~Hxx)

Put another way:

There is someone b such that for all x, if b shaves x then x does not shave x and if x does not shave x then b shaves x:

EbAx((Hbx -> ~Hxx) & (~Hxx -> Hbx))

I understood from your first post that you are not well versed in logic. So I have offered information that would help you, have pointed out where your locutions are not understandable, have asked questions whose answers might allow an understanding, and gave suggestions for proceeding methodically with clearly defined terminology, all in a sincere attempt to find out what you have in mind in this subject and thus to possibly guide you to insights on the subject, indeed as you invited questions to relieve the vagueness.

Now your latest post is even more undefined usage. If you're not interested in proceeding step by step with the concepts, then at least it is appropriate for me to remark that I cannot make heads nor tails of such things as "Godel numbering, mainly in utilizing the same logic found in linear algebra, specifically operators with matrices. My assumption being that by doing so, you could concise the amount of information required for the readout of the logic needed to ensure the lowest amount of information needed through the logical operators in matrixes for decoding RGB encoding on the matrix logical operator" even though I do know what Godel numbering, linear algebra and operations with matrices are.

As I said, an isomorphism is a 1-1 structure preserving function.

What is the domain and range of the function you wish to adduce. What is the relation on the domain you wish to preserve?

And you use yet another term, "domain of linear algebra", for which you provide no definition. Is that a term you have read somewhere, or just something you've made up on the fly here?

And you didn't address my question: What do you mean by "Godel numbering in terms of an isomorphism?" I mentioned what a Godel numbering is, and I provided the definition of 'isomorphism'. But what you mean by "a Godel numbering in terms of an isomorphism" is unknown to anyone but yourself.

Perhaps there are other people who understand what you're saying or asking, but I don't.

It would help if you specified in already understood mathematical terms exactly what you mean, step by step.

Starting with your posts today, what do you mean by "Godel numbering [...] in terms of an isomorphism such as linear algebra"?

I know what Godel numbering is. I know what an isomorphism is. And I know what the subject of linear algebra is basically about. But it would be for you to exactly, in already understood mathematical terms, say what you mean by "Godel numbering in terms of an isomorphism such as linear algebra".

Start with "Godel numbering in terms of an isomorphism".

Symbolize:

Sy <-> y lives in Seville

My <-> y is a man

Hxy <-> x shaves y

Premises:

Sb & Mb

Ay((Sy & My & ~y=b) -> Hby)

~Hbb

Consistent with those premises:

~Ey Hyb

Ey Hyb

There is no paradox in any of this.

As mentioned, Russell's paradox is not "There is a b that shaves all who do not shave themselves except himself." Rather, Russell's paradox is "There is a b who shaves all and only those who do not shave themselves." Symbolized, Russell's paradox is:

Ay(Hby <-> ~Hyy)

The basic subjects of the original post deserve to be stated clearly:

(1) Godel-Rosser is a conditional.

The antecedent is: T is a formal, consistent theory for a certain amount of arithmetic. (The 'certain amount of arithmetic' is best understood in context of each particular theory.) (And every formal theory has only countably many symbols in its language.)

The consequent is: T is incomplete.

So it's about theories in languages. I don't know what is meant by "discrete ordered systems" [emphasis added].

(2) Godel numbering assigns unique numbers to the symbols, formulas and finite sequence of formulas and in such a way that we can mechanically recover the symbol, formula or finite sequences of formulas from its Godel number. It's quite easily understandable, though it depends most saliently on the fundamental theorem of arithmetic.

And yes, if we add countably many symbols, we can devise another Godel numbering.

(3) An isomorphism is a 1-1 homomorphism. (Informally, a homomorphism is a structure preserving function.)

Starting here, the original post is quite unclear to me:

(4) "[...] add Gödel numbering to account for new [...] rules"

I don't know what that means.

Then the rest of the post is even more unclear.

(1) Formalism comes in variations, many of which are not the view that mathematics is only a symbol game. Indeed, Hilbert himself stressed that mathematics has a contentual part.

(2) Formalism is compatible with the method of models and its correspondence evaluation of truth. Indeed, consider Tarski himself. Moreover, consider mathematicians such as Abraham Robinson and Paul Cohen whose work is steeped in the method of models.

(3) Philosophy of mathematics includes a panoply of approaches and is not best characterized as a choice between between formalism and realism.

(4) Perhaps Hilbert denied that mathematical objects exist independently, but if he did, I would be interested to see a cite. Of course, Hilbert emphasized the difference between finitistic/contentual mathematics and infinitistic/ideal mathematics, but that does not, at least in and of itself, preclude that mathematical objects exist independently.

(5) This was addressed, but to stress: Using the method of models is not inconsistent with the symbol game view of mathematics, as models themselves may be taken per formalizations in the meta theories. Along with this, it is not required that the set of natural numbers be taken as existing independently of a formal theory that proves the theorems that there exist natural numbers and the set of them.

(6) This was addressed, but to stress: It is a salient result of model theory that PA does not define - not even within isomorphism, not even within equinumerousness - the system of natural numbers.

On a philosophy forum, one of its most prolific posters cannot fathom the use-mention distinction.

"To the Lounge with this rubbish" indeed!

The law of identity is a philosophical principle.

It is adopted in mathematics.

Ax x=x
is math.

/

Using '=', 'equals', and 'is identical with' interchangeably does not violate the law of identity.

Suppose I owe a creditor a certain amount of money, and ask them, "I have record of my balance as being 582 dollars plus 37 dollars. Do you have the same number?" They say, "Yes, your balance is 619 dollars and 0 cents." It would be ridiculous for me to say, "No! 582 plus 37 is not the same number as 619.00!"

582+37 is the same number as 619.00.

582+37 is identical with 619.00.

582+37 is equal to 619.00.

582+37 = 619.00.

That is not vitiated by the fact that:

'582+37' is not the same expression as '619.00'

Even a child can understand that

2+2 = 4 means that '2+2' and '4' name the same number.

/

In set theory, there are no two different sets with the same elements and different orderings.

However, for any set with at least two elements, there are different orderings on that set. To express a set S with an ordering R on S and a different ordering Y on S, we may simply say:

R is an ordering on S & Y is an ordering on S & ~R=Y

To talk about a set S and a particular ordering R on S we may mention;

<S R>

Millions of people who have studied mathematics understand that. Including those who have built the digital computers we are using at this moment.

/

There are 24 orderings of the set whose members are the bandmates in the Beatles. That doesn't entail that there is more than one set whose members are the bandmates in the Beatles. There is only one such set. It is the set whose members are the bandmates in the Beatles no matter how you order them.

So, I'm still curious what "the" ordering of the Beatles is supposed to be.

If one cannot answer that, then one ought not claim that for every set there is "the" order of that set.

And, in this particular case, by a lack of response to the question, I take it that the poster who makes that claim has no answer.

There was discussion about whether incompleteness pertains to systems with infinitely many types.

It does. Indeed Godel's original proof was about such a system.

As long as the system is recursively axiomatizable and with recursive inference rules, consistent and arithmetically adequate, it is incomplete.

Regarding placement of threads: Some of the moderation of this forum is quite irrational.

If '=' in set theory is to mean 'is the same as', it is not the case that the treatment of identity in set theory can dispense semantics.

Again, usually set theory presupposes identity theory, in which case it is by semantics that the interpretation of '=' is stipulated, and in which case '=' means 'is the same as'. And if set theory does not presuppose identity theory, then the axiom of extensionality is not enough syntactically, as we need the axiom of extensionality with an added clause. And still that is not enough to have that '=' means 'is the same as'. The details were given here:

https://thephilosophyforum.com/discussion/comment/897006

This is telling:

The poster challenged by asking where in a certain Wikipedia article it says that 'equals' means 'the same'. I pointed out: The article states that the axiom of extensionality uses '=' with regard to predicate logic, with a link to an article on 'First-order logic'. And that article correctly states that the most common convention is that 'equals' means 'the same'. Moreover, the article on 'Equality (mathematics)' defines equality as sameness, and the article on the equals sign refers to equality, and the article on 'Identity (mathematics)' refers to equality.

I don't usually reference Wikipedia, but at least it is abundantly clear that the poster's challenge regarding what Wikipedia does happen to say is answered, when he could have found out it out for himself. It's typical of the poster to stick with his method of railroading full speed ahead with his own claims and challenges while hardly ever granting that they have been answered.

Meanwhile, I'm still interested in hearing what one would claim to be "the" order of the set of all and only the bandmates in The Beatles.

That is just one of myriad examples. Without an answer, the notion that every set has its "the" ordering is dead in the water.

I didn't say anything about 'constitutive'.

And it is exactly my point that use of terminologies in different fields are often not compatible with one another, and, as I have said many times in this forum, and again in this thread, mathematics makes no claim that '=', 'equals' and 'is identical with' are used in mathematics in the same senses as in all those in everyday life and in other fields of study.

And I don't have a personal sense of 'identity theory'. I am merely referring to a publicly studied formal theory.

And I don't claim to "support" identity theory. I am merely saying what it is, what some of its theorems are, something about the semantics that goes with it, and how it relates in certain ways to set theory.

There is nothing "off the deep end" about anything I've said here. Barely clever putdowns by means of renaming posters might be at least minimally apropos if they were based on at least something.

Here are some of the details:

Theorem: There is no formula T(x) such that for every sentence S, T(g(S)) is true if and only if S is true.

Proof:

Toward a contradiction, suppose there is such a T(x).

So, there is a formula D(x) such that for every numeral m, D(m) is true if and only if m is the numeral for the Godel number of a formula P(x) such that P(m) is false. (The steps in obtaining this line from the previous line are not included in the article on which this summary is based.)

D(g(D(x))) is true
if and only if
g(D(x)) is the numeral for the Godel number of a formula P(x) such that P(g(D(x))) is false.

Toward a contradiction, suppose D(g(D(x))) is true.
So g(D(x)) is the numeral for the Godel number of a formula P(x) such that P(g(D(x))) is false.
g(D(x)) is g(P(x)), so D(x) is P(x), so D(g(S(x))) is P(g(S(x))), so D(g(S(x))) is false. Contradiction.

Toward a contradiction, suppose D(g(D(x))) is false.
So it is not the case that g(D(x)) is the numeral for the Godel number of a formula P(x) such that P(g(D(x))) is false.
So D(g(D(x))) is true. Contradiction.

So there is no formula T(x) such that for every sentence S, T(g(S)) is true if and only if S is true.

/

Theorem: There is no formula T(x) such that for every sentence S, S is true if and only if T(g(S)) is true.

Proof:

Lemma: For every formula P(x) there is a sentence D such that D <-> P(g(D)) is true. (The proof of this lemma is not included here.)

Toward a contradiction, suppose there is a formula T(x) such that for every sentence S, S is true if and only if T(g(S)) is true.

So, for every sentence S, S <-> T(g(S)) is true.

But, by the lemma, there is a sentence D such that D <-> ~T(g(D)) is true. But also, D <-> T(g(D)) is true. Contradiction.

So there is no formula T(x) such that for every sentence S, S is true if and only if T(g(S)) is true.

I've posted explanation previously in this forum. But it seems it needs to be resaid:

Tarski's undefinability theorem is that, in the relevant contexts, there is no formula T(n) that is satisfied by all and only those n that are Godel numbers of true sentences of arithmetic. That is proven by showing that if there were such a T(n) then there would be a sentence H such that H is true if and only if H is false, but since there is no such H, there is no such T(n).

Again, he is not claiming there is such an H, let alone that he is not claiming that there is such an H that is true or that is false. Rather, toward a contradiction, we suppose there is a T(n) as described above, then we derive the absurdity that there is an H that is true if and only if it is false, so we conclude, courtesy argument by contradiction, that there is no such T(n).

Again, Tarski was not trying to figure out how to deal with the liar paradox. Rather, he used the fact that there is no sentence that is true if and only if it is false to prove that there is no formula in the language of arithmetic that defines the set of true sentences of arithmetic.

When we are studying formal languages, formal semantics and formal theories, we would need to know how "This sentence is not true" would be formalized, or even if it can be formalized. Tarski is not merely addressing an informal paradox, but rather he is using that informal paradox to figure out how to prove a certain formal theorem. The figuring out how to prove is not itself formal, and the formal proof does not use a liar sentence but rather the proof is that, in the relevant contexts, there is no formalization of the liar sentence.

Again, Tarski did not "include" such a sentence, especially an informal one.

Again, in context of Tarski's undefinability, it's not a matter of whether the liar sentence is or is not a truth bearer, rather the matter is that, in the relevant contexts, there is no formalization of such a sentence.

To say that Tarski's proof is wrong because he uses a liar sentence as if it is a truth bearer is to get it all backwards. Tarski doesn't at all say that there is a formal sentence in the manner of the liar sentence that is a truth bearer. He says even more than the contrary: that, in the relevant contexts, there does not even exist such a formal sentence.

All of that can be understood in detail and with all the groundwork by studying an introductory textbook in mathematical logic.

I didn't say that it is not the case that undecidability is fully met by self-contradictory expressions. I didn't say that because I don't know what "undecidability is fully met by self-contradictory expressions" is supposed to mean.

Self-contradictions are false in all models.

For a given model M, every sentence in the formal language is either true in M or false in M, and not both.

I don't preclude anyone from posting a proof of anything they want to proof. I have no such power.

The posts have come full circle, at least three times today. If any new points arise, I'll consider addressing them.

I cannot provide for progress in a conversation by repeating that I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted.

No important point has been ignored [by me]. It's the other way around.

I pointed out that the footnote pertains to informal heuristic analogy and is not part of the formal proof itself. That is not changing the subject. And a proper analysis of the proof is not advanced by taking a footnote that is part of the informal remarks about the proof out of context. A proper analysis is to address the actual formal proof. Moreover, since Godel's original paper, the theorem has been strengthened to Godel-Rosser, and the context has been sharpened by the subjects of recursion and model theory, and the notation has been modernized, and the proofs have been streamlined, and the whole subject has been given greater elucidation and presentation. And all of that is provided by many introductory textbooks in mathematical logic. Thus, a proper analysis of incompleteness begins with study of an introductory textbook in mathematical logic. It is sure that flitting among snippets on the Internet and mere cursory readings of even original sources, while skimming all of that not toward step by step mathematical understanding and verification of inferences, to the point of seizing upon footnotes out of context and not understood, is not the way to an understanding or analysis of the subject.

By the way, the only authorized and authoritative translation is the one in van Heijenoort's anthology.

I cannot provide for progress in a conversation by repeating refutations and explanations that are ignored while what has been refuted is simply reasserted.

Again, whatever "the axiom of extensionality indicates identity means":

(1) If we use identity theory at the base of set theory, then the axiom of extensionality merely adds a sufficient condition for '='. And the semantics of identity theory provide that '=' means 'the same as' or 'is identical with'.

(2) If we do not use identity theory at the base of set theory, then then we may use the axiom of extensionality but augmented with an additional clause to define '='. However, without the semantics of identity theory, it is not the case that such an axiom alone proves that '=' means 'the same as' or 'is identical with'.

Again, as has been mentioned very many times on this forum, the use of the symbol '=' and the words 'equal' and 'identical' in mathematics are by stipulation. By use of such stipulations we do not claim that the words are used exactly as they are used in all the very many other different contexts and senses in everyday language and in philosophy. This kind of thing should not have to be pointed out so very many times in a philosophy forum.

As to sets and order, as has been demonstrated very many times on this forum, sets with at least two members have different orderings, so there is not "the" ordering of a set.

A while ago, I gave this example: The set whose members are all and only the bandmates in The Beatles is a set. But there is not "the" ordering of that 4 member set. Indeed there are 24 orderings of that set:

https://thephilosophyforum.com/discussion/comment/884421

Or put it this way, if every set has an order that is "the" order of the set, then the set whose members are all and only the bandmates in The Beatles has an order that is "the" order. If one will venture to state which of the 24 orders of that set is "the" order, then I can ensure that we could find at least 23 Beatles fans who would disagree with that being "the" order.

A definition of 'identity' was requested and the poster said he will look at it. In identity theory in mathematics, '=' is not primitive. But the semantics require that S=T is true if and only if 'S' and 'T' name the same thing. To look at this in more detail and with all the groundwork for it provided, one may look one of many introductory textbooks in mathematical logic.

Mathematical logic does not assign "fault". Fault though would be vital to assign if one were a judge in a traffic accident case.

The Godel sentence is not a contradiction and it is not nonsense. It is a statement of arithmetic. And G is true, and G is true if and only if G is not provable in a theory such as PA. That doesn't make G a contradiction nor nonsense. And the theory itself does not prove G (first incompleteness) and the theory itself does not prove that it does not prove G (second incompleteness).

This all can be understood by simply reading an introductory textbook in mathematical logic.

I didn't quote.

The proof itself does not mention 'epistemological antinomy'. Godel's footnote pertains to analogies of the proof, the proof itself does not invoke a notion of 'epistemological antimony'. Godel is talking about heuristic insight there, which is an analogy (not an identification) between certain informal antinomies and his mathematical proof. It is quite an error to grasp onto a footnote out of context while ignoring the actual hard mathematical proof.

Again, however one characterizes the Godel sentence, it is not a contradiction. Indeed it is a true sentence of arithmetic.

Godel never said any such nonsense that if a system proves a contradiction then the system is incomplete. Indeed, if a system proves a contradiction then the system is complete.

Moreover the Godel sentence is not a self-contradiction.

Again, 'incomplete' in this context is given a stipulative technical definition pertaining to mathematical logic. The use of 'incomplete' in mathematics is not claimed to pertain to all the other everyday meanings or other technical meanings in other fields of study. The nature of stipulative technical definitions is not even something that one should have to point out in a philosophy forum.

No self-contradiction is provable in a consistent theory, irrespective of incompleteness.

These are stipulative definitions. Anyone may use different definitions. To accommodate someone who insists that we don't use a technically defined term the way we have defined it, we could say 'gincomplete' instead. It would still be about the same point: If a theory T is consistent, recursively axiomatizable and arithmetically sufficient, then there is a sentence S such that neither it nor its negation is provable in T.