https://youtu.be/O4ndIDcDSGc — Amalac

Like others here, I don't know what he has in mind about division.

https://www.math.wisc.edu/~miller/old/m571-08/smith.pdf — Aryamoy Mitra

Peter Smith's 'An Introduction To Godel's Theorems' is a real good book. I recommend it. But that PDF is only a shorter warmup for the actual book published a couple years later. There were some fairly bad mistakes in the original edition that were corrected in a later edition, so the PDF might have some of those mistakes, I don't know.

What's a meta-language for F? Does [a meta-language for F] concern or describe the language of F? — Aryamoy Mitra

A meta-theory for F is either a formal theory or an informal context. The meta-theory for F formulates the language for F, the syntax for F, the semantics for F, and formulates F itself, and has meta-theorems about all that. In that context, the meta-language for F is the language of the meta-theory for F.

We say informally "the meta-theory for F" and "the meta-language for F" when it should be "a meta-theory for F" and "a meta-language for F" since F may have many meta-theories and meta-languages, depending on what the mathematician chooses. For incompleteness, a very roomy meta-theory for F in which to work easily is set theory, but even PRA ("finitistic arithmetic") could be the meta-theory, or an informal English context. Godel himself used a mix of German and mathematical and logical symbols.

I want to see something like this: AxAy(x + y = y + x) in a Godel sentence — TheMadFool

Rest assured that the Godel sentence G_F is a purely symbolic formula of arithmetic, using symbols like the ones you mentioned, though it is not a universal generalization of an equation and it is much more complicated. It is for this form:

~ExPx where P is a purely symbolic mathematical formula in the language of, say, first order PA, but too complicated to type out here.

When reading informal accounts of incompleteness it might be mistaken that G_F is not a purely symbolic mathematical formula, because in informal accounts we say that G_F says "G_F is not provable in F". Yes that is what G_F "says" in that G_F is true if and only if G_F is not provable in F, but G_F actually is a purely symbolic mathematical formula and not the English captioning of it as "G_F is not provable in F".

the key proposition in Gödel's proof is K = The proposition with Gödel number G is not provable (in T) and the "coincidence" is that K is the proposition with Gödel number G. In other words, K is not provable. — TheMadFool

Please, let's stick with one set of letter-symbols, so 'F' rather than 'T'.

The Godel-sentence G_F is a formula in the language of number theory. It can be reduced to the primitive language of PA, with only the symbols '0', 'S', '+', and '*', and a quantifier and connectives (even just one connective would work). It is a pure mathematical formula. The ordinary interpretation of the symbols is that they refer to arithmetic.

Then, we show in the meta-theory that G_F is true if and only if G_F is not provable in F. So we say that G_F "says" that G_F is not provable in F. Note that we put 'says' in scare quotes; what we mean more technically than "says" is that we show that G_F is true if and only if G_F is not provable in F.

has to be a mathematical theorem, no? — TheMadFool

I think you are conflating 'sentence' with 'theorem'.

Theoremhood is always relative to a theory. A sentence is a theorem of a theory if and only if the sentence is provable in the theory. In the case of incompleteness, we sometimes leave tacit the theory, and say 'provable' rather than the actually correct 'provable in F'. Also, every sentence is a theorem of some theory or another. The only sentences that are theorems of all theories are the logical validiites.

G_F is not a theorem of F. And the negation of G_F is not a theorem of F. That's the point of incompleteness.

If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds. — Metaphysician Undercover

My points don't depend on whether equality is or is not independent of human judgement.

Formal languages, including the language of identity theory, are more precise than natural languages. But the point I made was not so much about precision but that 'equality of human beings' in the sense of equal rights or whatever is a very different meaning of 'equality' in mathematics.
— TonesInDeepFreeze

These two senses utilize the same principle. — Metaphysician Undercover

To say that 2+1 and 3 are equal is saying that 2+1 is 3.

To say that John and Mary are equal (in the sense of equal rights) is not saying that John is Mary. Rather it is saying that the rights of John are the same as the rights of Mary.

These are very different uses of the word 'equal'.

Your argument is ridiculous.

now you admit that you do not mean that they are "necessarily" the same — Metaphysician Undercover

I don't "admit" in the sense of conceding or retracting some earlier point. I just never stated regarding necessity to begin with, and I don't state now because it would require a discussion about modality that is not needed to present the basic mathematical framework I've mentioned.

What good is such a principle? — Metaphysician Undercover

It provides a clear and straightforward framework for doing mathematics.

You judge "2+1" as referring to the same thing as "3", because they are equal, but there is no logical necessity there, which proves that they are? — Metaphysician Undercover

What do you intend the pronoun 'they' refer to there?

I don't say that '2+1' and '3' are equal. I say that 2+1 and 3 are equal.

But in a context where '2', '+', and '3' were symbols not standing for the number 2, the addition operation, and the number 3, then it may not be the case that '2+1' and '3' denote the same number in that context. And it is not necessary that symbols always denote the same. Denotation of symbols is by stipulation or convention not by necessity.

One can stipulate premises and then infer conclusions. That is not question begging. Also, we don't have to stipulate that Henry Fonda is the father of Peter Fonda, since we can arrive at that claim by empirical or historical evidence.
— TonesInDeepFreeze

Sure, in that case we can refer to empirical judgement, but in the case of numbers we cannot, because we cannot sense numbers in any way. — Metaphysician Undercover

(1) So in the empirical context, your objection was refuted.

(2) In the mathematical context, numbers are not physical objects. And over the course of this discussion I said that we arrive at mathematical conclusions by mathematical proof or by performing mathematical procedures. You are not caught up in the discussion because you ignore and skip.

Can you show me through your "method of models" — Metaphysician Undercover

I can't cram it all into a post or even several posts. You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step, including the notions: concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, et. al. And prerequisite would be understanding basic mathematical notions, including: sets, tuples, relations, functions, et. al

Over many posts, you keep telling me what I think or said, and you're wrong. You're a bane.

And you claimed that you asked me a question I didn't answer. I then asked you to link me to that, because I don't recall you asking me a question I did not answer recently. Please link me to it.

Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet.
— TonesInDeepFreeze

That itself is a judgement, that these unwritten equalities are equalities. Clearly equality remains a human judgement. See "equal" is a human concept. To say that there are equalities which humans haven't discovered, is to already judge them as equalities. — Metaphysician Undercover

What you just said, whatever its merit, doesn't vitiate anything I've said.

all you did was assert that equality in mathematics is more precise than equality in other subjects. — Metaphysician Undercover

Formal languages, including the language of identity theory, are more precise than natural languages. But the point I made was not so much about precision but that 'equality of human beings' in the sense of equal rights or whatever is a very different meaning of 'equality' in mathematics.

I have not said that numbers are special regarding denotation.
— TonesInDeepFreeze

This is exactly what you are saying. By insisting that "equal to" in the case of numbers means 'denotes the same object', you are saying that numbers have some special quality which can make two distinct but equal things into the same thing. You are claiming that numbers have a special status which makes equal things into the same thing. — Metaphysician Undercover

Wrong. I'm not saying any of that.

You don't have to use the word "necessarily", to mean it. — Metaphysician Undercover

True, but I don't mean it.

When you say that being equal implies that they are the same, you refer to a logical necessity — Metaphysician Undercover

Nope. I am not bringing the notion of logical necessity into play.

Otherwise it would be false to say 'if they are equal then they are the same'. — Metaphysician Undercover

That's a non sequitur.

I've already explained to you how you do not have the premise required to say that "2+1" denotes the same object as "3", when the two signify different things ("have different senses"). — Metaphysician Undercover

You don't explain. You assert and then argue fallaciously.

"The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". — Metaphysician Undercover

You said it both denotes and is its sense. Denotation and sense are different.

You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. — Metaphysician Undercover

One can stipulate premises and then infer conclusions. That is not question begging. Also, we don't have to stipulate that Henry Fonda is the father of Peter Fonda, since we can arrive at that claim by empirical or historical evidence.

The same thing is the case with "2+1" and "3". They signify different things (have different senses). Now, you do not have the required premise to conclude that they denote the same object. — Metaphysician Undercover

We prove that 2+1 = 3. Then we prove that '2+1' and '3' have the same denotation by the method of models. I've told you that about a half dozen times now.

Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness.
— TonesInDeepFreeze

You mean that false premise? — Metaphysician Undercover

You've not shown any false premise in the method of models.

Now, please link me to the post in which you claim you asked me a question I did not answer.

To grasp how exactly it all works and makes perfect and rigorous sense, you really would need to read a book in mathematical logic

When SEP says "true but unprovable" it understood that 'unprovable' is informally brief for 'unprovable in F'.

Therefore (1), GF cannot be false, and must be true. For this reason, the Gödel sentence is often called “true but unprovable (2)”
— SEP

The word "therefore" (1) suggests an argument i.e. there's a proof for the Godel sentence GF. However, the next line asserts that GF is ...often called "true but unprovable (2)". — TheMadFool

The word 'therefore' is not being used in the proof of G_F in the theory F. Rather, 'therefore' is being used in the argument in the meta-language that G_F is true.

the unprovable status of GF is not what matters — TheMadFool

No, that G_F is unprovable in F is at the crux of the incompleteness proof.

GF should be asserting a mathematical theorem, call it T, and asserting that T is unprovable and not that GF itself can't be proved. — TheMadFool

No, that's where you're mixed up. G_F is a statement in the language of F. G_F is a mathematical statement about natural numbers. But G_F is constructed so that in the meta-theory we show that G_F is true if and only if G_F is not provable, and in the meta-theory we show that G_F is true, i.e. that G_F is not provable.

Two different things:

(1) G_F is a statement in the language of F but not a theorem of F.

(2) 'G_F is true' is a theorem of a metatheory for F.

It's crucial to distinguish between (1) the statement G_F itself in the theory F and (2) the statement 'G_F is true' which is a meta-theoretic statement about G_F.

There are two levels of proof: (1) proofs in F and (2) proofs (such as the incompleteness theorem and the statement 'G_F is true' in the meta-theory.

You quoted Wikipedia and put my name on it as if they are my own words. Please don't do that.

This discussion is becoming unwieldy with different letter-symbolizations with your own your letters and also letters from Wikipedia and SEP. For example, you use 'A' when Wikipedia uses 'F'. I'm going to stick with Wikipedia here, since its explanation is coherent and quotable.

Godel sentence = G = There's a mathematical theorem, call it T, in a given axiomatic system A, such that T is unprovable/undecidable (which word is apt?) in A. — TheMadFool

That is all messed up.

You seem to be conflating the statement of the incompleteness theorem itself with the Godel-sentence G_F. The statement of the incompleteness theorem is not the Godel-sentence.

By 'the incompleteness theorem' I am referring to the Godel-Rosser incompleteness theorem. I'll call it 'C'. You attempted to express a part of C, but with terrible errors when you wrote:

"There's a mathematical theorem, call it T, in a given axiomatic system A, such that T is unprovable/undecidable (which word is apt?) in A."

Let's fix that (and recall that by 'true' in this context I mean 'true in the standard model of arithmetic').

I think the part of C you have in mind is:

There is a sentence G_F that is true but not provable in F.

I'll call that statement C* (it is a part of the incompleteness theorem C). Both C and C* are not stated in the language of F, but rather in a meta-language for F. And neither C nor C* are the Godel-sentence. In other words, the statement of the incompleteness theorem is different from the Godel-sentence that is used to prove the incompleteness theorem.

Each (appropriate) theory F has its own Godel-sentence that we call 'G_F'. And G_F "says" that G_F is not provable in F.

Kurt Godel's tour de force was proving G, the Godel sentence, is true. Am I right? — TheMadFool

There's a lot more in the proof of incompleteness that is remarkable other than the fact that G_F is true.

I asked you for an instance of equality which is not a human judgement. You didn't give me one. — Metaphysician Undercover

I don't recall you asking me such a question. If you did, then please link to the post where you asked it so that I can see the context. Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet. And I don't see any relevance to what I've said about equality.

Meanwhile, there are many decisive points I have raised that you have skipped.

In case you're having a hard time to understand, I see this as very clearly false. You and I are equal, as human beings, but we are not the same. — Metaphysician Undercover

The case is that you can't read. I replied about the notion of human equality many many posts ago, and you skipped recognizing my reply, and I even mentioned a little while ago again that I had made that reply and you skipped that reminder too!

You seem to think that numbers are somehow special — Metaphysician Undercover

I have not said that numbers are special regarding denotation.

necessarily the same — Metaphysician Undercover

I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic.

we can proceed to say that a thing is indiscernible from itself — Metaphysician Undercover

Yes we can, but that alone is not the principle of the indiscernibility of identicals.

2+1 is discernible from 3 — Metaphysician Undercover

'2+1' and '3' have different senses but not different denotations. No matter how many times I point out the disctinction between sense and denotation, and even after I linked you to an Internet article about it, you keep ignoring it.

how in hell are you supporting this obviously false assumption that "if 2+1 and are equal 3 then they are the same"? — Metaphysician Undercover

Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness.

The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. — Metaphysician Undercover

You've not shown any inconsistency.

signified — Metaphysician Undercover

If I recall, I have not used the word 'signify'. Again, a term has both a denotation and a sense.

mathematicians use "=" to relate two distinct expressions with distinct meanings — Metaphysician Undercover

Yes, since meaning includes both denotation and sense. I explained to you probably more than half a dozen times already that ordinary mathematics concerns itself only with denotation and that if you want to have sense handled also, then you need a more complicated framework.

I cannot understand the fundamentals because they are unsound. — Metaphysician Undercover

You've never even read page 1 in a book on the foundations of mathematics. So of course you can't understand anything about it.

All I see is "=" here. Where's the proof that "=" means the same as? — Metaphysician Undercover

Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models.

Wikipedia in general is not reliable regarding mathematics. Much better is the Stanford Encyclopedia of Philosophy. However, possibly that particular article might be okay. I looked at it only briefly just now. I am wondering what you think is in that article that is not compatible with anything I've said.

From the article:

"The first incompleteness theorem shows that the Gödel sentence G_F of an appropriate formal theory F is unprovable in F. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true. For this reason, the sentence G_F is often said to be "true but unprovable." However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF may only be arrived at via a meta-analysis from outside the system." - Wikipedia

Those are the very points I said also. Most poignantly, it does not say that the Godel sentence says that the Godel sentence is true, but rather it says that the Godel sentence does not say that it is true and that the truth of the Godel sentence is shown outside the theory.

G is unprovable in A.

And G does not say

G is true and unprovable in A — TonesInDeepFreeze

Here we go again.

Yes, it is the case that G is unprovable in A. And it is the case that G is true if and only if G is unprovable in A. And G is true in the standard model for the language of A.

But G does not say that G is true in A.

(1) 'True in A' doesn't even make sense. Sentences are not true or false in a theory. Rather, sentences are true or false in models for the language of the theory.

(2) G does not even say that G is true in a model for the language of the theory. The language for A cannot even express 'true in a model for the language of A' since then A would be inconsistent.

Put yet another way, without some of the previous simplifications:

A is a recursively axiomatizable, consistent, sufficiently arithmetically expressive theory.

L_A is the language of A.

PA (Peano Arithmetic) is a theory.

L_PA is the language of PA.

M is the standard model for L_A.

M is a model of PA.

Z is set theory.

G_A is the Godel sentence for A.

G_A "says" 'G is not provable in A'.

G_A is true if and only if G is not provable in A.

In Z we prove that G is true in M.

And, get this in your head:

G does NOT say 'G is true in A'.

'is true in a model for A' cannot even be formulated in A, since A is consistent (Tarski's theorem).

correct me if I'm wrong, given an axiomatic "theory" A, and Godel sentence G = the theorem T is true and unprovable in axiomatic theory A". — TheMadFool

I'm correcting you; you are wrong. And you're not even coherent. You've got an extra symbol 'T' that makes no sense.

You're not even reading what I wrote:

Again, G says "G is not provable in A" and G does NOT say "G is true and unprovable in A".

Not only does G not say "G is true in A" but A can't even form the predicate 'is true in A' or A would be inconsistent (this is Tarski's theorem that is a kind of semantic variant of incompleteness).

G is claiming that T is true in A — TheMadFool

I said it about six different ways in the previous posts that that is not correct. I explained in detail why it is not correct.

T isn't true in some other "theory" like you seem to suggesting when you say "Rather what we mean unprovable in whatever theory is in question" but in A. — TheMadFool

You are not just confused, but you are abysmally confused.

(1) Whatever sentence you mean by T, it's not relevant. The only sentence we are concerned with is G (you called it 'F1' elsewhere).

(2) I did NOT say that anything is true in any theory. About six times I said that sentences are not true or false in a theory but rather sentences are true or false in models for the language of the theory.

Why do you keep getting this wrong?

(3) When I said 'unprovable in whatever theory is in question' I was just pointing out that provability is relative to a theory. A sentence may be provable in some theories but not in others. And I am not evading that we are specifically concerned with a given system whether it be PA or Robinson arithmetic or whatever system A might be.

Again, if the theory in question is A, then G (the Godel sentence for A) says:

G is unprovable in A.

And G does not say

G is true and unprovable in A.

Get it through your head!

I'll leave it to you to connect the dots. — TheMadFool

If you have a question or a point to make, then please ask it or state it rather than dropping cryptic instructions for me to connect whatever dots I'm supposed to connect.

Perhaps you think there is an incompatibility here:

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition).
— TonesInDeepFreeze

Sentences are not true or false in a theory.
— TonesInDeepFreeze — TheMadFool

I explained why those are not incompatible here:

https://thephilosophyforum.com/discussion/comment/513465

and with detailed definitions and points about mathematical logic here:

https://thephilosophyforum.com/discussion/comment/513476

And again in my post above.

And I'll say it yet another way:

Truth and provability are different notions in mathematical logic. While they are different, mathematical logic does study relationships between them. Some of those relationships are the subject of the incompleteness theorem.

I asked you:

What is the source you read about this subject? — TonesInDeepFreeze

You are using teminology and mentioning concepts in the subject, so it seems you've read something somewhere about it. Would you please tell me the articles or books you read? Then possibly I can look them up to find the passages you've misundertsood.

you didn't answer my question. Why? Please reread your reply to my question and my response to it. — TheMadFool

Why what? And if you have particular posts you want me to look back at then please link to them so I know specifically which ones you want me to look at.

The last qustion you asked is this:

the Godel sentence becomes, "proven" (as true) AND "unprovable". Isn't this a contradiction? — TheMadFool

No, it is not. I explained here:

https://thephilosophyforum.com/discussion/comment/513465

I have no clue whether you even read that post.

And I even followed up with extraordinary specifics here:

https://thephilosophyforum.com/discussion/comment/513476

But I'll say it again:

For convenience, let's suppose the theory is PA. And by 'true' I mean 'true in the standard model for the language of PA'.

G is not provable in PA. The sentence 'G is true' is provable in, e.g. Z set theory. That is, in Z set theory we state the standard model of PA and we can prove that G is true in that model.

Conflating provability with truth is one of the most common mistakes by people who only skim an article here and there about incompleteness and don't apprise themselves of the crucial technical specifics of mathematical logic and the incompleteness theorem.

I'll say it one more time:

G is provable in a theory

and

G is true in a model for the language of the theory

are DIFFERENT notions.

And don't overlook that your claimed reductio ad absurdum was refuted:

https://thephilosophyforum.com/discussion/comment/513487

what I am looking for is an indication that 2+1 really is the same thing as 3 — Metaphysician Undercover

'S' stands for the successor operation.

def: 1 = S0

def: 2 = 1+1

def: 3 = 2+1

The proof in this case is utterly trivial, from the definition of '3'.

You are completely confused about this subject.

What is the source you read about this subject?

In all these posts about incompleteness, by 'true' I mean 'true in the standard model'.

(1) Sentences are not true or false in a theory. Rather, sentences are true or false in a model for the language of the theory.

(2) G is true but G is not provable in the theory.

(3) G says that the Godel-number for G is not the Godel-number of a theorem of the theory. In other words, G says that G is not provable in the theory.

(4) G is true if and only if G is not provable in the theory.

(5) G does NOT say anything about its own truth value. Indeed, if a theory had a predicate for truth, then the theory would be inconsistent.

Here by 'provable' and 'unprovable' we don't mean absolutely unprovable (i.e. not provable from any set of axioms) since there are no absolutely unprovable formulas. Rather we mean unprovable in whatever theory is in question (let's say it's PA for simplicity of exposition).



"1. The theorem F1 is true AND the theorem F1 unprovable (Assume for reductio ad absurdum)"

By definition, a theorem is a provable sentence. So there is no F1 that is a theorem and unprovable.

So, to be generous, let's save 1:

F1 is true but unprovable.

"2. IF the theorem F1 is unprovable THEN we don't know the truth value of theorem F1"

False premise. A formula may be unprovable but known to be true by the method of models in the meta-theory.

"3 The theorem F1 is true (1 Simp)"

Okay.

"4. IF the theorem F1 is true THEN we know the truth value of theorem F1"

False premise. There are true formulas that we don't happen yet to know the truth value.

"5. We know the truth value of theorem F1 (3, 4 Modus Ponens)"

Incorrect since 4 is false.

"6. The theorem F1 is unprovable (1 Simp)"

When 1 is corrected as I have, then okay.

"7. We don't know the truth value of theorem F1 (2, 6 Modus ponens)"

Incorrect since 2 is false.

"8. We know the truth value of theorem F1 AND We don't know the truth value of thoerem F1 (5, 7 Conj)"

Incorrect since neither 5 nor 7 have been shown.

"9. False that the theorem F1 is true AND the theorem F1 is unprovable (1 to 8 reductio ad absurdum)"

Incorrect since 8 has not been shown.

If the theorem F1 is unprovable — TheMadFool

A theorem by definition is a provable sentence.

So what you wrote as the very first line of your argument is a contradiction in terminology.

And the rest of your argument is rife with other terminological mixups and misconceptions about mathematical logic.

It is irrational to mix up your own terminological use with the way the terminology is actually used for the mathematics you are critiquing.

If you wish to critique mathematical logic, then you should learn something about it first.

Here's a simpified synopsis of the terminology:

SYNTACTICAL:

1. We have formal languages. These are sets of symbols. And there are rules for sequencing the symbols to make terms and formulas. Sentences are a certain kind of formula. It is computer-checkable whether a given sequence of symbols is or is not a formula for a given formal language. And it is computer-checkable whether a given sequence of symbols is or is not a formula of a given formal language.

2 We have rules of proof. A proof is a certain kind of sequence of formulas. It is computer-checkable whether a given sequence of formulas is or is not a proof.

3. A theory is a set of sentences closed under proof. A theorem is a member of that set of sentences.

4. An axiomatization for a theory is a set of formulas such that every member of the theory is provable from the axioms. So a theorem is a sentence that is provable from a set of axioms for the theory.

4. A theory is decidable if and only if it is computer-checkable whether any given sentence is or is not a member of the theory.

5. A theory is complete if and only if for any given sentence in the language, either the sentence or its negation is a member of the theory.

6. A theory is consistent if and only if there is no sentence such that both the sentence and its negation is a member of the theory.

7. A theory is recursively axiomatizable if and only if there there is a computer-checkable set of axioms for the theory.

8. The notion of a theory being "sufficiently arithmetically expressive" is complicated and can't be summarized here. But basically it means that the theory has the ordinary basic truths about arithmetic.

9. Godel-Rosser incompleteness is that there are recursively axiomatized, consistent, sufficiently arithmetically expressive theories that are incomplete.

P.S. Note that when we say 'unprovable' we mean 'unprovable from certain axioms'. No formula is absolutely unprovable, since any formula is provable from certain axioms (even if just from an inconsistent set of axioms).

SEMANTIC:

10. A model for a language is a non-empty set that is the universe for the model and a mapping from the symbols of the language to individuals in the set, relations on the set, and functions on the set.

11. A model then describes a "state of affairs". That is, some members of the universe and tuples of members are either in the relations or not, or in the functions or not.

12. The Tarski method is applied so that 'truth in the model' is built up in stages for simple sentences to more complicated sentences. A sentence is either true in a given model or false in that model.

13. PA, in this context, is a certain first-order theory.

14. By 'the standard model for the language of PA' we mean the model where the symbols of the language map to the natural numbers and functions on the natural numbers in the way we would expect. For example, '0' maps to the number 0, and '+' maps to the addition operation, etc.

15. (As I explained in my previous post) incompleteness implies that there are sentences true in the standard model but that are not members of the various theories. (Recall that 'member of a theory' just means 'provable in the theory'.) The only sentences that are unprovable from all consistent sets of axioms are sentences themselves that imply inconsistency.

I still don't understand how something can be "true" and "unprovable" — TheMadFool

This is explained in any textbook in mathematical logic, usually chapters 1 and 2.

Proof concerns just formulas in the language - purely syntactical objects.

Truth concerns models for the language. A sentence of the language might be true in some models and false in other models. With incompleteness, by 'true' we leave tacit that we actually mean true in one particular model, which is the standard model for the language of PA.

Proof takes place merely with regard to the theory itself. Truth is handled by a meta-theory (usually set theory) for the theory.

A theorem of a theory is a sentence that is provable from the axioms for the theory.

We say a model is a model of a given theory if and only if every theorem of the theory is true in the model.

If a sentence is provable from the axioms of a given theory, then that sentence is true in all models of the theory.

But a sentence might be true in a given model of the theory and yet not be provable from the axioms of the theory.

Incompleteness of a theory is that there are sentences such that neither the sentence nor its negation is provable from axioms for the theory. But in any given model, a sentence is either true or false. So for an incomplete theory, there are sentences that are true in certain models but not provable. In particular, for certain theories, we show that there are unprovable sentences that are true in the standard model for the language of PA.

First-order logic, in most contexts, cannot exist without an underlying propositional logic (once again, unless I'm mistaken). — Aryamoy Mitra

You are correct.

'adding' and 'subtracting' propositional conditions from one another — Aryamoy Mitra

I don't know what you mean.

a finite number of axioms — Aryamoy Mitra

Then I don't know what relevance you have in mind. G-theories can have finitely many or infinitely many axioms.

First-order logic (unless I'm mistaken) is a corollary of propositional logic — Aryamoy Mitra

No, the opposite. First-order logic subsumes propositional logic.

[First-order logic] quantifies the interrelations between its subjects - as opposed to delineating them with logical connectives. — Aryamoy Mitra

First order logic allows predicate symbols, operation symbols, and quantifers, which are not present in propositional logic. But first-order logic does have the connectives also.

With a 'logical edifice', I was referring to a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators. Won't any constraints on the latter, inclusive of Gödel incompleteness, emerge for the former (propositional ideas)? — Aryamoy Mitra

I cannot make sense of that. I don't know what you mean by "a set of ideas that stemmed from propositional conventions, which were then affixed with arithmetic operators".

I am not one to dismiss things off hand, without some understanding of the fundamental principles. — Metaphysician Undercover

That is rich from someone who dismisses approaches in ordinary mathematics while insisting on remaining ignorant of understanding their fundamental principles or even reading a single page in a book or article about the subject.

And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement.
— TonesInDeepFreeze

If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds. — Metaphysician Undercover

This is another instance of imposing your view as if it entails something I said that I did not say. You believe that equality holds based on human judgement. That doesn't entail that I said that equality or identity does. It's a strawman to represent me as saying something I did not say.

Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different.
— TonesInDeepFreeze

You said that from a judgement of equality you can infer that they are the same. I'll quote for the third time:
"Rather, we infer they share all properties from having first proved that they are equal."
You are clearly arguing that if they are equal then they are the same. — Metaphysician Undercover

You've mixed up two different issues, and got me wrong on both of them.

Issue 1: You claimed it is question begging to say that '2+1' and '3' denote the same object.

Of course if 2+1 and are equal 3 then they are the same. But what you claimed was that I was begging the question by saying that '2+1' and '3' have the same denotation. And I explained that it is not question begging since we infer that '2+1' and '3' have the same denotation from first determining that 2+1 equals 3.

Issue 2: You claimed that I said that indiscernibility implies identity.

But I did not. I said that identity implies indiscernibility.

There are three principles of identity/indiscernibility:

(1) If identical then indiscernible.

(2) If Indiscernible then identical.

And we may combine for:

(3) Identical if and only if indiscernible.

All I said is (1).

And again, not question begging.

you cannot use the indiscernibility of identicals to support your claim that they are identical. — Metaphysician Undercover

You have a cognitive problem that prevents you from discussing this without getting grievously mixed up about it. You keep reversing the direction of implication.

Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility.

But you keep saying that I say:

indiscernibility -> equality

even after I've told you that is not what I say.
— TonesInDeepFreeze

This is the only way that the principle of indiscernibility could be used to support your claim that equality means the same as. — Metaphysician Undercover

Still yet again, I am not using indiscernibility to support that 'equal' means 'identical' or that 'equal' means 'the same'. Stop mixing up what I've said and then representing your own mixed up version as if my own.

Are you and I the same because we are equal? — Metaphysician Undercover

I answered that many posts ago! Again you argue by just skipping past many key points in the replies to you.

you have no special definition for "equal". — Metaphysician Undercover

In ordinary mathematics, 'equal' is not defined but rather is a primitive. It is the sole primitive of first order identity theory. In that context 'equal' and 'identical' are two words for the same undefined primitive.

to define that sense of "equality" with "value" — Metaphysician Undercover

And I don't define any sense of 'equality' with 'value'.

In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning.
— TonesInDeepFreeze

See, you admit right here, that you only concern yourself with a part of what "2+1", and what "1" refer to. — Metaphysician Undercover

Not so much what I concern myself with personally, but rather what ordinary mathematics concerns itself with.

I mentioned the extensional vs intensional distinction many many posts ago. And again the two different terms '2+1' and '3' refer to the same object. They have the same reference. However, of course, they do not have the same sense. Again, yes, ordinary mathematics is extensional and concerns only the denotation part and not the sense part. I referred you to the Stanford philosophy encyclopedia article that discusses this. I said many posts ago that you may also consider formulations in which intensionality is considered.

Highly recommended:

'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' - Torkel Franzen

Probably the best book ever written for introducing the subject of incompleteness to everday readers.

Neither of them are terminological, in the first place. — Aryamoy Mitra

Sincerely, I would like to help you understand this topic and to provide answers, but at many points I don't know what you're trying to say because you use unrecognizable or too vague terminology.

What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems.
— TonesInDeepFreeze

That is literally, what 'self-consistency' denotes (demonstrating a self-referential consistency, from within a system). — Aryamoy Mitra

'self-consistency' is not ordinarily used in the sense of "proves its own consistency". Rather, 'self-consistency' is just a longer phrase for 'consistency'.

A theory is consistent if and only if there is not a formula such that the theory proves both the formula and the negation of the formula.

A theory T proves the consistency of a theory S if and only if T proves that that S does not prove a formula and its negation. In particular, a theory T proves the consistency of T if and only if T proves that T does not prove a formula and its negation.

A theory may be consistent and not prove its own consistency.

axiomatic finiteness — Aryamoy Mitra

Do you mean that there are only a finite number of axioms? Or do you mean that the axioms entail that there are only finite sets, or something like that?

whether [...] Gödel incompleteness extends outside Peano Arithmetic. — Aryamoy Mitra

What do you mean by 'extends outside' a theory?

Do you mean to ask whether there are theories stronger than PA that are incomplete. Yes.

Or theories with all the axioms of PA plus more axioms and that are incomplete? Yes.

What does a non-elementary axiom entail? — Aryamoy Mitra

Often, 'elementary' means first-order. Or, in different contexts, it refers to elementary arithmetic with elementary functions, which have a specific technical definition as a certain subset of the set of recursive functions.

might the Gödelian constraints on certain proof-statement correspondences in formal languages, lend itself to the underlying logical edifice? — Aryamoy Mitra

I don't know what you mean by 'proof-statement correspondences' nor what you mean by 'underlying logical edifice'.

Sincerely I say that your understanding of this subject would depend on familiarizing yourself with good books or articles on it, and with that you would have recognizable terminology in which to couch your questions about it.

So get these straight already:

(1) My explanation runs in this order:

Determine equality, then it is justified to assert that the terms denote the same.

(2) Equality implies indiscernibility. I did not opine one way or the other whether indiscernibility implies equality.

(3) Substitutivity holds in extensional contexts, and it may fail in intensional contexts.

You are ignorant of the view in which meaning has at least two components: denotation and sense.

Denotation is only part of the meaning of a term.

In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning. And, as I've pointed out a few times already, if you wish to have mathematics that concerns also sense, or the intensional aspect, then you are welcome to formulate such mathematics or to look up formulations that have been given by other mathematicians and philosophers of mathematics.

You mangle nearly everything.

(1) Claiming I've said things when I did not say them.

(2) Screwing up the direction of my explanation so that your representation of my explanation is not my explanation.

(3) Reversing the direction of conditionals.

(4) Ignore explanations and decisive points and instead keep repeating yourself past them.

(5) Ignore distinctions explicitly stated.

Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again.
— TonesInDeepFreeze

Do you know the law of identity? It states that a thing is the same as itself. It says nothing about equality or equivalence. That two things are equal is a human judgement. — Metaphysician Undercover

I know about identity vastly more than you do. And your reply merely repeats your own thesis. And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement.

Rather, we infer they share all properties from having first proved that they are equal.
— TonesInDeepFreeze

See, no strawman. You prove that they are equal (human judgement), then you infer from this, that they are the same. — Metaphysician Undercover

I said to stop claiming I've said things I did not say.

We don't judge two things are equal. — TonesInDeepFreeze

And now taking me out of context. Here is the context:

When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness.
— Metaphysician Undercover

We don't judge two things are equal. We judge that two terms refer to the same thing. And, of course, such judgements may be mistaken due to human error. — TonesInDeepFreeze

Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different.

You very clearly stated "having proved that they are equal". — Metaphysician Undercover

You're totally mixed up as to the order of the statements in my explanation. Again:

First we determine (by proof or whatever method) that 2+1 is 3. From that determination we are justified in claiming that '2+1' and '3' refer to the same number.

Don't screw up the sequence of statements in my explanation to thus mangle it.

The indiscernibility of identicals does not provide the principle required for substituting equal things. — Metaphysician Undercover

Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle.

Things are judged to be equal not on the basis that they are indiscernible. — Metaphysician Undercover

Again, you got it backwards! For the third time I've told you, I did not claim to infer equality from indiscernibility. I claimed to infer indiscernibility from equality.

What I said:

equality -> indiscernibility.

But you keep saying that I say:

indiscernibility -> equality

even after I've told you that is not what I say.

Don't reverse the direction of my conditionals.

what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. — Metaphysician Undercover

There is both denotation and sense. Substitutivity holds as rule only for denotation. I was the one who gave the early example in this thread where substitutivity fails when the context is not extensional.

Meanwhile, if you don't recognize the use of substitutivity in even just basic math, then you can't do math.

I really can't see how a relation is an object. — Metaphysician Undercover

You can't see because you ignore mathematics. A relation is set of tuples. That set is an object.

systematically finite — Aryamoy Mitra

characterized by Peano Arithmetic — Aryamoy Mitra

simultaneously unprovable — Aryamoy Mitra

entirely bereft of composite statements — Aryamoy Mitra

ceases to unequivocally demonstrate its own mathematical consistency — Aryamoy Mitra

Where do you find such terminology in discussions of incompleteness? Where did you read such things?

Meanwhile, it's better to look at Godel-Rosser incompleteness, since it is stronger than Godel incompleteness and is what most people mean by now when they refer to incompleteness. Also, we should take advantage of certain refinements in mathematical logic that were not present when Godel proved incompleteness.

Godel-Rosser is that any recursively axiomatizable and consistent theory that "expresses enough arithmetic" (such as Robinson arithmetic or Peano arithmetic) is incomplete in the sense that there are sentences in the language of theory such that neither the sentence nor its negation is a theorem of the theory.

From the above it follows that for such theories, there are true (in this context meaning true in the standard model of PA) sentences that are not theorems. Indeed, we may construct a specific such sentence.

Moreover, no such system proves its own consistency.

/

Godel-Rosser has proof that is constructive, intuitionistically (cf. philosophy of intuitionism in mathematics) acceptable, and finitistic (the proof can be carried out in primitive recursive arithmetic).

/

when accorded a finite list of axioms or postulates, do there always result either indeterminate, or unprovable truth values — Aryamoy Mitra

If a theory is recursively axiomatizable, sufficiently arithmetically expressive, and consistent (let's call these 'G-theories'), then it is incomplete, no matter whether the set of axioms is finite or infinite. Some theories that are not G- theories are complete (and some are finitely axiomatizable).

absence of self-consistency — Aryamoy Mitra

What lack of consistency? Incompleteness doesn't say that e.g. PA or ZFC are inconsistent. Rather, a proof of consistency is not available within their own systems.

finite set of elementary axioms — Aryamoy Mitra

Some theories are recursively axiomatizable with even an infinite set of axioms. For some reason you are stuck on a notion of finite axiomatization that is not relevant in this regard.

Also, 'elementary' has a technical meaning different from your use.

in order to illustrate a logical system's consistency, one must transcend the system entirely — Aryamoy Mitra

You shouldn't generalize about "logical systems" but rather you should be accurate by addressing just G-theories in this context. And the answer is yes; to prove the consistency of a G-theory we have to do that in some other theory (which itself could be another G-theory). For example, Z set theory proves the consistency of PA.

Isn't that tantamount, for instance, to asserting that all finitely synthesized constructs of reasoning, are by their existence inconsistent? — Aryamoy Mitra

Not at all. The question itself shows a confused understanding of this topic. If you are interested in understanding incompleteness, I suggest studying it from good sources.

how many new axioms one defines a set or structure with, an unprovable sentence can always be derived within it — Aryamoy Mitra

The wording of that is not good, but basically yes, you can't add axioms to a G-theory to escape incompleteness.

blank post

(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.

(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do.
— TonesInDeepFreeze

I told you already, extensionality provides a false premise. — Metaphysician Undercover

Whatever views you have about the distinction between extension and intension, and between denotation and sense, I gave you more than "vague reference" about them.

extensionality provides a false premise — Metaphysician Undercover

You may think you've shown a false premise, but you haven't.

When a human being judges two distinct things as having the same properties, and says therefore that they are equals, this does not make them into the same thing. — Metaphysician Undercover

I never said that we infer that distinct things are equal, let alone that they are equal due to having the same properties. You're resorting to strawman again.

The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing. — Metaphysician Undercover

Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again.

When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness. — Metaphysician Undercover

We don't judge two things are equal. We judge that two terms refer to the same thing. And, of course, such judgements may be mistaken due to human error.

proving that two things are equal does not imply that they are the same (share all properties) — Metaphysician Undercover

No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics.

If in reality, language use is filled with vagaries, and we want to discuss the truth about language use, then we need to account for the reality of those vagaries. To assume a context without vagaries as your prerequisite premise for proceeding toward an understanding of certain principles of language use, is simply to assume a false premise. — Metaphysician Undercover

Over and over you swing this "false premise" charge like a crudely made cudgel. It's mere assertion.

You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion.

In the case of Henry Fonda, we have observed with our senses, the very object being referred to. In the case of numbers we have not observed any such objects. — Metaphysician Undercover

It was my point that the Fonda example involves first making an empirical determination. On the other hand, mathematics is deductive and axiomatic.

You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. — Metaphysician Undercover

Of course, if numbers are not at least abstract objects, then they cannot be referred to as objects. Then '2' has no denotation to a number. Go ahead and formulate your mathematics that way if you like, but you offer no formulation or even hint of one. And, I also mentioned that if we confine our attention to mathematics at the pure formula level, then object and denotation don't even need to be mentioned.

Claiming a denotation when there is only meaning, — Metaphysician Undercover

Denotation is part of the meaning.

Do you understand the fallacy of "begging the question", assuming the conclusion? — Metaphysician Undercover

I understand it better than you. And you've not shown I am question begging. Moreover, I showed exactly how I am not question begging, and you skipped that. This is like your claim that I was inconsistent - you never showed an inconsistency.

you [altheist] and Tones are the ones confusing denotation and signification. — Metaphysician Undercover

I never used the term 'signification'.

Therefore the argument that "the father of Peter Fonda" denotes the same thing as "Henry Fonda" is a fallacious argument, by means of begging the question. The argument relies on assuming the conclusion. — Metaphysician Undercover

This is at the heart of it.

The claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person is not an argument! It is a conclusion. It is a conclusion from the premise (however it has been established) that Henry Fonda is the father of Peter Fonda. No one every suggested otherwise!

You blame others for fallacies in arguments they never made! (I.e., variation on straw man.)

There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound. — Metaphysician Undercover

That is ridiculously captious and sophomoric. It is deserves all three tropes: red herring, blowing smoke, and grasping at straws.

Of course in natural language and everyday discourse there may be vagaries that make definitive determinations difficult or impossible. So if you want to demand a context in which there are no vagaries, then of course all bets are off with natural language usage. So to proceed with understanding certain principles, of course we must assume, for sake of discussion, some context in which we are not thwarted by such vagaries as you mention.

Moreover, of course, in this context, we are assuming that there has been ample evidence that Henry Fonda is the father of Peter Fonda, and that we can agree on that for the purpose of the illustration regarding denotation of the names.

Again for about the fifth time: When I claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person, I don't claim that I have shown that Henry Fonda is the father of Peter Fonda. That's not the point, and only someone pretty obtuse would miss this. To show that Henry Fonda is the father of Peter Fonda is a matter of empirical inquiry. Of course in this context I assume that we take it for granted that we know empirically (or by whatever means of such common knowledge) that Henry Fonda is the father of Peter Fonda, and on that basis, we make the linguistic observation that the names 'Henry Fonda' and 'the father of Peter Fonda' denote the same person.

Similarly, of course I take it for granted that we already understand that 2+1 = 3, either by proof or by common mathematical knowledge. It is from that understanding that we then observe that '2+1' and '3' denote the same number.

For about the dozenth time, get it straight:

First we find out that 2+1 = 3, and then we may justifiably claim that '2+1' and '3' denote the same number.

/

The philosophy of language does take on issues with problematic, indeterminate, equivocal, conflicting, temporally complicated, and paradoxical denotation. But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number.

"equal" is assigned according to some system of judgement, so only the properties deemed significant within that system are accounted for, and this is insufficient for the conclusion of "the very same object". — Metaphysician Undercover

You have it backwards again. Mathematics does not prove that objects are equal by showing they share all properties. Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms.

Also, mathematical theories are in mathematical languages and regard models with objects and their relevant properties for particular areas of mathematical interest. In mathematics, one is free to state other languages, theories and models in which other properties are relevant besides those in some previous treatment.

I can see Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda." — fishfry

What he doesn't understand is that denotation is only one part of meaning. There is both denotation, which is extensional, and sense. 'Henry Fonda' and 'the father of Peter Fonda' denote the same thing. But indeed the they have different senses.

Ordinary mathematics deals with terms extensionally. For mathematics to deal with terms also with regard to sense requires a more complicated linguistic/logical system. As I mentioned there are proposals for such systems, but they are beyond ordinary mathematics.

The "Fonda" example was provided as an argument for the truth of it. — Metaphysician Undercover

The example was given not so much as an argument but as an illustration for you to understand a basic idea.

Ordinary mathematics regards '2+1' and '3' as having the same denotation, because we prove

2+1 = 3

In general, for any terms T and S, we infer

T = S

when we prove it and then we may say that T and S have the same denotation.

As I mentioned before, this is the case both by ordinary mathematical practice and as made rigorous in mathematical logic. That's just a fact about certain conventions in ordinary mathematics. Whether ordinary mathematics should use that convention is a separate issue, of which I have not taken a position except to point out that alternatives are complicated.

A natural language example, such as the Fonda example, doesn't prove anything about mathematics, but, as I mentions, it illustrates the general principle.