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.

One may consult introductory textbooks in mathematics to see how we can prove undefinability from incompleteness or prove incompleteness from undefinability.

The definition of 'incomplete' is simple:

A theory T is incomplete if and only if there is a sentence S in the language for T such that neither S nor its negation are a theorem of T.

It is trivial to prove that there are incomplete theories, and not trivial, though pretty easy, is proving the soundness theorem that then trivially proves the incompleteness of certain theories. What is interesting about Godel-Rosser is that there are incomplete theories of a particular kind (consistent, recursively axiomatizable and arithmetically sufficient).

If we define 'true' as 'provable', then of course all bets are off regarding these theorems as they are stated. And if in baseball we define 'hit' as 'home run', then we would throw away all the baseball statistics books. Yeah, we know all that.

But to accommodate someone who insists that 'true' means 'provable', then we could simply say that wherever we have written the word 'true', it is to be replaced by 'gorue'. Then read all the proofs and discussions about them with that change. It matters not toward understanding the substance of them.

There is no proof of G in F.

That's the point.

Too miss that point is to utterly not know what the theorem is about.

"Why" is not a technical term, more a heuristic matter, and could mean different things to different people. In the most bare sense, "Why is T a theorem?" is answered by showing the proof of T . But, heuristically, there is a massive amount of discussion in the literature giving insight into the theorem and its proof; and insight is given in Godel's own paper.

With identity theory, '=' is primitive and not defined, and the axiom of extensionality merely provides a sufficient basis for equality that is not in identity theory. Without identity theory, for a definition of '=' we need not just the axiom of extensionality but also the 'xez <-> yez' clause.

As to manipulation of symbols, the incompleteness theorem can be be done in mere primitive recursive arithmetic, so the assumptions and means of reasoning are well within the scope of the methods of finite arithmetical calculations.

Regarding Tarski's undefinablity theorem, Tarski proved that in certain systems, there does not even exist such a sentence. Not only did Tarski not use such sentences as a basis, he actually proved that such sentences don't even exist in the relevant systems. To not understand that is to not understand what the theorem is even about.

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I know the context in which interrogatory sentences were mentioned lately. But the matter of interrogatories has been brought into other posts in this forum as part of incorrect attempts to refute the theorems.

Again, as has been explained several times in this forum:

G asserts that G is not provable in system P.

But P does not prove G, and P does not prove that it does not prove G.

/

Proofs don't "hide" things. From fully declared axioms and rules of inference, we may prove Godel-Rosser. We may prove versions that do not mention semantics. And we may prove versions that mention both syntax and semantics. This is all famous and understood by reading an introductory textbook in mathematical logic.

The incompleteness theorem requires no notion or terminology 'True(L, x)' where L is a set of axioms or system.

Rather, using the above style of notation, we have:

True(M x) where M is a model and x is a sentence. Read as "x is true in M".

and

Theorem(L x) where L is a set of axioms and x is a sentence. Read as "x is a theorem from L".

And we prove about certain systems:

Theorem(L x) implies that for every model M, if True(M y) for every y in L, then True(M x). (This is the soundness theorem).

And we prove about certain axiom sets:

If True(M x) in every model M, then Theorem(L x).

And incompleteness proves that there are M and L such that:

(1) For every y such that Theorem(M y), we have True(M y); and (2) True(M x); but (3) it is not the case that Theorem(L x).

"Did you lie?" doesn't have a truth value, because it is not a declarative sentence. Indeed, interrogatory sentences do not appear as lines in proofs.

Contrary to a claim made in this thread (and made by the same poster several other times in this forum), it is not the case the Godel sentence requires that there is a sequence of inference steps that prove that they don't exist (as has been explained several other times in this forum).

More generally, Godel's and Tarski's proofs do not have the defects claimed in this thread (and claimed by the same poster several other times in this forum). That can be verified by reading an introductory textbook on mathematical logic in which the groundwork and proofs of Godel-Rosser incompleteness and Tarski undefinability are provided.

I don't prefer Wikipedia as a reference on such matters, but it was asked where in the Wikipedia article on the 'Axiom of extensionality' is it said that 'equals' means 'the same'.

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.

In ordinary contexts in mathematics, including mathematical logic, including set theory, 'equals' means 'is the same as', which means the same as 'is identical with'. This is formalized by identity theory, which extends first order logic without identity, and adds a primitive binary predicate symbol '=' with axioms and a semantics.

More specifically:

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Identity theory is first order logic plus:

Axiom: Ax x=x

Axiom schema:
For all formulas P,
Axy((x=y & P(x)) -> P(y))

Semantics:

For every model M, for all terms T and S,
T = S
is true if and only if M assigns T and S to the same member of the universe.

/

Set theory can be developed in at least two ways:

(1) First adopt identity theory. This gives us:

Theorem: Axy(x=y -> Az((xez <-> yez) & (zex <-> zey)))

Then add the axiom of extensionality:

Axiom: Axy(Az(zex <-> zey) -> x=y)

This gives us:

Theorem: Axy(x=y <-> Az(zex -> zey))

Thus, with identity theory and the axiom of extensionality, every model of
Az(zeT <-> zeS)
is a model that assigns T and S to the same member of the universe.

(2) Don't adopt identity theory. Instead:

Definition: Axy(x=y <-> Az((xez <-> yez) & (zex <-> zey)))

That gives us as theorems all the axioms of identity theory.

However, if we don't also stipulate the semantics of identity theory, the axioms of identity theory along with the axiom of extensionality do not provide that every model in which S=T is true is a model that assigns S and T to same member of the universe.

I don't think they're stupid. Rather, I find that there is complacency and sloppiness in the writing of certain articles, sometimes to the extent that there are plain falsehoods in them. But in the case of the article being discussed, I'm not pointing out falsehoods, but rather the confusions the article opens.

Before the reply to my post, I deleted "To see that, you just need to read the article that you yourself say is "clear and accurate"", as I thought it would be better not to invite referencing that article again.

But to address that article, here are the proofs without ""liar"" (scare quotes in original), "ask", "truth bearer" or anything else extraneous to the mathematical proofs:

In this context, 'formula' and 'sentence' mean 'formula in the language of first order arithmetic' and 'sentence in the language of first order arithmetic'.

In this context, 'true' and 'false' mean 'true in the standard model for the language of first order arithmetic' and 'false in in the standard model for the language of first order arithmetic'.

For every formula M, let g(M) be the numeral for the Godel number of M.

/

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.)

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.

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.

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.

It's not a question of what was relevant to your point. I cited faults in the article, whether or not those faults bear on your point.

Tarski's proof doesn't work the way you describe it.

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For any sentence T, set of axioms S and set of rules R, we may ask the question "Is T derivable from S with R?" That fact doesn't entail the counterfactual that there are questions in proofs. Tarski's proof does not have questions in it.

Yet I showed exactly what is amiss in the Wikipedia article recently cited.

"Is there a proof of T?" is a question.

But a proof of T does not have questions in it.

One can couch things as questions. But the proofs themselves do not have questions in them.