I never foresaw a page-long explanation, but it was most certainly worth it.

I'm affixing the entirety of it herein, with a few observational comments - for it definitely simplifies the overarching rigor of the subject, and necessitates a read.

'First of all, when I say "proved", what I will mean is "proved with the aid of
the whole of math". Now then: two plus two is four, as you well know. And,
of course, it can be proved that two plus two is four (proved, that is, with the
aid of the whole of math, as I said, though in the case of two plus two, of
course we do not need the whole of math to prove that it is four). And, as
may not be quite so clear, it can be proved that it can be proved that two plus
two is four, as well. And it can be proved that it can be proved that it can be
proved that two plus two is four. And so on. In fact, if a claim can be proved,
then it can be proved that the claim can be proved. And that too can be
proved.'

What this predominantly suggests, is that if an assertion can be proved, then one may also (successfully) prove that it can be proved, in a recursive manner. I don't know why this bears veracity, especially in a formalized system characterized by a certain degree/class of arithmetic. Will you able to shed any light, on this matter?

'Thus: it can be proved that two plus two is not five. Can it be proved as well
that two plus two is five? It would be a real blow to math, to say the least, if
it could. If it could be proved that two plus two is five, then it could be
proved that five is not five, and then there would be no claim that could not
be proved, and math would be a lot of bunk.

So, we now want to ask, can it be proved that it can't be proved that two plus
two is five? Here's the shock: no, it can't. Or, to hedge a bit: if it can be
proved that it can't be proved that two plus two is five, then it can be proved
as well that two plus two is five, and math is a lot of bunk. In fact, if math is
not a lot of bunk, then no claim of the form "claim X can't be proved" can be
proved.'

Does this (the latter) entail the crux of Gödel incompleteness, insofar as contradictions are concerned?

Here's a simplistic, non-technical sequence of rationalizations - identical to the ones in the exposition above (it doesn't consist of any jargon - inclusive of languages, theorems, proofs or axioms).

1) Hypothetically, there exists a statement A (2+2=5).
2) A's negation can be proven.
3) Ideally, A should not be provable.
4) Unfortunately, one can't prove, that one can't prove A.
5) If 4 holds true, then one can simultaneously prove A (since A must demonstrate a specific truth value).
6) Therefore, there exists an inconsistency between 2 and 5, implying that certain truths will necessarily remain unprovable.

Have I significantly misapprehended the argument, or is it at all substantive?

'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.'

Interpreting this sentence, is harder than accruing a mastery over all of Mathematics.

Will you be able to shed any light, on this matter? — Aryamoy Mitra

No. Sorry.

Have I significantly misapprehended the argument, — Aryamoy Mitra

At (5) and (6), yes.

Interpreting this sentence, is harder than accruing a mastery over all of Mathematics. — Aryamoy Mitra

But you have undertaken to follow the copious and kind advice of @TonesInDeepFreeze, so you may be pleasantly surprised. Good luck.

Have I significantly misapprehended the argument — Aryamoy Mitra

Yes.

'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.' — Aryamoy Mitra

No.

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".

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.

https://youtu.be/O4ndIDcDSGc — Amalac

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

I think the only translation of Godel's original paper approved by Godel is the one in Jean van Heijenoort's 'From Frege To Godel'.

Some people who have not very much background in mathematical logic have been able to understand incompleteness by first directly reading Godel's paper. But my native abilities would not allow me to understand the paper without having first studied incompleteness in textbooks in undergraduate level mathematical logic, and to get to mathematical logic I needed firs to study basic symbolic logic and set theory. Here are additional reasons I think it is better to learn from textbooks than from Godel's own paper.

* Godel-Rosser strengthens Godel's original and is what people actually mean now by incompleteness.

* Godel uses old-fashioned notation and terminology that is hardly used for many decades now. The newer notation, now decades established, is more streamlined and more aesthetically pleasing, and, for me at least, easier to understand. I think one is more conversant about incompleteness with the more modern notation and exposition.

In particular, since Godel's paper, the terminological distinction between recursion and primitive recursion became the prevailing convention, so some people get tripped up by Godel's older terminology.

The second incompleteness theorem is usually only sketched at the undergraduate level. And a full proof of the second incompleteness theorem is hard to find. It is in Hilbert-Bernays but it was only recently translated from German to English. But there are still some good undergraduate level books about the second theorem. Also, Godel didn't provide much proof of the Representation theorem that is a crucial lemma for incompleteness. I've read that a proof of the Representation theorem is a lot of tedious technical detail. If I'm not mistaken, it also is in Hilbert-Bernays.

The aforementioned Peter Smith book is really good, but again, I think it is better appreciated by first studying basic symbolic logic, set theory, and mathematical logic.

And for an everyman's introduction, Torkel Franzen's book is superb and a joy to read. Though again, personally, I doubt I would have appreciated it as much as I did without having first studied mathematical logic.

I think that the incompleteness results have an effect on a wide range of things not just in the set theoretic realm and with the foundations of mathematics. We just don't want to make or are ignorant about the link to the incompleteness results.

I think the classic example of something being true but unprovable is a game theoretic situation where it's easy to show that a correct solution exists, yet there seems to be no way to get there. The existence of a correct solution can be shown...based on mathematics — ssu

As Nash demonstrated fixed-point theory is useful in game theory. Brouwer's fixed-point theorem was proven indirectly, with no simple path to its value, and this distressed Brouwer, who later turned to intuitionism. Proving a math object exists indirectly, but without a process for its construction, is still proving a theorem. This sort of thing has a superficial relation to Godel's works, but I don't think it's what he had in mind. Others here, with more knowledge of the matter can correct me if I'm in error.

Godel's second incompleteness theorem explained in words of one syllable — bongo fury

We must keep in mind that when he says 'proved' it's really 'proved in [whatever particular theory we're talking about]'.

The insightful and witty George Boolos is one of the great writers about foundations - mathematical logic and set theory.

His 'The Logic Of Provability' might be the definitive treatment of the particular topics in that book.

He is a co-author of 'Computability And Logic', which is really nice classic overview.

He is a co-author of 'Logic, Logic, And Logic' which is a wonderful set of essays. Especially welcome is his cogent explanation of the intuitive basis of the set theory axioms in terms of hierarchy.

EDIT CORRECTION: Strike "We must keep in mind that when he says 'proved' it's really 'proved in [whatever particular theory we're talking about]'. He says that he means proved by "the aid the whole of math". I would take that to mean ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC.

. I do, nonetheless, intend to learn the language of formal logic — Aryamoy Mitra

I suggest this course of readings, in order:

Logic:Techniques Of Formal Reasoning - Kalish, Montague, and Mar

Elements Of Set Theory - Enderton (perhaps supplemented with Axiomatic Set Theory - Suppes)

A Mathematical Introduction To Logic - Enderton (definitely supplemented with the chapter on mathematical definitions in Inroduction To Logic - Suppes)

The Enderton logic book will take you through a proof of Godel-Rosser.

Have I significantly misapprehended the argument,
— Aryamoy Mitra

At (5) and (6), yes. — bongo fury

Can you pinpoint how I've faltered? I ask, since (5) was at the heart of Gödel incompleteness.

'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.'
— Aryamoy Mitra

No — fishfry

As a clarity, are you refuting the original exposition? This passage, for instance, was word-for-word sourced from another, non-technical resource.

As a clarity, are you refuting the original exposition? This passage, for instance, was word-for-word sourced from another, non-technical resource. — Aryamoy Mitra

I did see the italics but I did not see a link to the source. So I couldn't tell if you were quoting someone else or quoting yourself from some other publication or forum, or just separating out your ideas into quoted form. But in that case I haven't criticized you, I've criticized whoever wrote that passage. Which was:

By the way, in case you'd like to know: yes, it can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five.' — Some Unknown Entity

I believe what they mean is this. For definiteness let's take Peano arithmetic (PA). By Gödel's second incompleteness theorem, PA can not prove its own consistency. That means PA can not prove that it can't prove that 2 + 2 = 5. Agreed so far? Then if PA can prove that it can't prove that 2 + 2 = 5, then PA must have proved its own consistency, which it can only do if it's inconsistent; and if it's inconsistent, then it can prove that 2 + 2 = 5.

Have I got that right?

Now my complaint is that you did not distinguish between "PA can prove ..." and "It can be proved ..."

Because in fact we CAN prove that PA is consistent. The easiest consistency proof is to assume Zermelo-Fraenkel set theory, ZF. In ZF we have the axiom of infinity, which gives us a model of PA. Since there's a model, PA is consistent by Gödel's completeness theorem.

There's another famous proof of the consistency of PA, by Gerhard Gentzen. As Wiki puts it:

Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal $\varepsilon_0$", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.

https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

The point is that proof and consistency are relative to given axiom systems. It's true that PA can't prove its own consistency; but we CAN prove the consistency of PA by other means.

So, to sum this all up: Using ZF or Gentzen's proof, I can indeed prove that PA is consistent, and that PA can't prove that 2 + 2 = 5, and that PA can prove that it can't prove that 2 + 2 = 5.

I can always do this as long as I'm willing to go outside PA. And this is true in general. Just because some given system can't prove its own consistency doesn't mean we can't prove its consistency.

Let me know if I've missed your point. And if you'd just say where you got the quote it would be helpful.

I should add for clarity that having used ZF to prove the consistency of PA, I now have the question of whether ZF is consistent. You can push the problem around but never nail it down. Maybe after all that's what the paragraph means, but without the surrounding context it's hard to tell. I admit to being confused as to why Gentzen's proof isn't problematic in exactly the same way.

I think by now you can see why I I originally gave a one word answer. :-)

I did see the italics but I did not see a link to the source. So I couldn't tell if you were quoting someone else or quoting yourself from some other publication or forum, or just separating out your ideas into quoted form. — fishfry

Pardon me, for not properly citing the source; here's the mystical entity, from which this passage stems.

I believe what they mean is this. For definiteness let's take Peano arithmetic (PA). By Gödel's second incompleteness theorem, PA can not prove its own consistency. That means PA can not prove that it can't prove that 2 + 2 = 5. Agreed so far? Then if PA can prove that it can't prove that 2 + 2 = 5, then PA must have proved its own consistency, which it can only do if it's inconsistent; and if it's inconsistent, then it can prove that 2 + 2 = 5. — fishfry

For everyone else's following (and that of my own), let me distill this with arrows:

A) Gödel incompleteness (2)$\longrightarrow$PA can't demonstrate its own consistency$\longrightarrow$PA can't prove, that it can't prove that 2+2=5;

B) Now, if PA can prove, that it can't prove that 2+2=5 $\longrightarrow$PA shall be consistent;

C) And, if PA can prove, that it can't prove that 2+2=5 $\longrightarrow$PA needs to be inconsistent$\longrightarrow$PA should be able to prove, that 2+2=5.

The point is that proof and consistency are relative to given axiom systems. It's true that PA can't prove its own consistency; but we CAN prove the consistency of PA by other means.

So, to sum this all up: Using ZF (which at least I understand, as opposed to Gentzen's proof, which I don't) I can indeed prove that PA is consistent, and that PA can't prove that 2 + 2 = 5, and that PA can prove that it can't prove that 2 + 2 = 5.

I can always do this as long as I'm willing to go outside PA. And this is true in general. Just because some given system can't prove its own consistency doesn't mean we can't prove its consistency. — fishfry

Thank you, for expanding. PA can't show its own consistency, but PA can be proved consistent outside itself (with other axioms) - and that's a generality that may hold for other arithmetic systems; is that the crux of the argument?

Thank you, for expanding. PA can't show its own consistency, but PA can be proved consistent outside itself (with other axioms) - and that's a generality that may hold for other arithmetic systems; is that the crux of the argument? — Aryamoy Mitra

Yes, exactly. The proof from ZF is trivial. ZF contains a model of PA; that is, a set in which, with the proper interpretation, every axiom of PA is true. Therefore PA is true if ZF is. And ZF is consistent if we assume the existence of an inaccessible cardinal, which itself is a model of ZF. You just keep adding axioms to push the inconsistency problem up the chain.

And now that I see that the author is the great George Boolos, I am unhappy, because I disagree with what he said. He starts: "First of all, when I say "proved", what I will mean is "proved with the aid of
the whole of math.""

I disagreed with that the first time you wrote it. It's wrong. Proved means, "proved in a given system of axioms." If you take the "whole of math," you lose the entire point of the subject. Given some axiom system we want to know what sentences it can prove. Those are the theorems. But "the whole of math?" I don't even know what that means. Do I get to include the entire hierarchy of the large cardinal axioms? Do I include the continuum hypothesis or not? I was hopelessly annoyed as soon as I read that remark.

Having read Boolos's article, I disagree with it entirely. Incompleteness is NOT about "the whole of math." It's about particular systems of axioms. I believe he's destroyed the essence of the subject in his effort to simplify it. And that's why when you quoted that paragraph, I instinctively said, "No." It's not true that "the whole of math" is either consistent OR inconsistent, nor is it subject to incompleteness. You have to say what the axioms are, then I can say if they're consistent or inconsistent or whether certain statements are independent of the axioms.

But then again I could be wrong. Perhaps he is making the more subtle point that even if I can use ZF to prove the consistency of PA, I still don't know if ZF is consistent. So maybe in the end he's right and I'm wrong. I admit to not being sure whether he's right or wrong. But I still have the right to be annoyed that he ignored the essence of the matter, which is that we are talking about particular axiom systems and not "the whole of math."

In fact if by "the whole of math" we might mean for example the set-theoretic multiverse of Joel David Hamkins, then "the whole of math" includes the models of set theory in which CH is true, and the models of set theory in which it's false; and even perhaps the possibility that PA is consistent, and the possibility that PA is inconsistent [that latter concept is not part of Hamkins's idea, I don't think].

So I just found "the whole of math" to be too imprecise to be correct. Didn't someone (Feynman? Einstein? ) write, Things should be made as simple as possible, but no simpler.

Einstein in fact. https://www.championingscience.com/2019/03/15/everything-should-be-made-as-simple-as-possible-but-no-simpler/

Boolos's article violates the spirit of that saying.

As Nash demonstrated fixed-point theory is useful in game theory. Brouwer's fixed-point theorem was proven indirectly, with no simple path to its value, and this distressed Brouwer, who later turned to intuitionism. Proving a math object exists indirectly, but without a process for its construction, is still proving a theorem. This sort of thing has a superficial relation to Godel's works, but I don't think it's what he had in mind. Others here, with more knowledge of the matter can correct me if I'm in error. — jgill

Yes, you got the point exactly. I would say that the issue has more than just a superficial relation, but that is just my personal view about the subject.

Perhaps it is a far more simple issue and has less to do with Gödel, but this is exactly where Oskar Morgenstern saw a problem in economic forecasting in the American Journal of Economics in the 1930's. And I have forgotten which Nobel-laureate responded to him back then 1930's that Morgenstern is wrong while there is obviously is a correct solution: because fixed point theorem proves that there exists a correct solution. Yet the whole problem is that an indirect proof, a reductio ad absurdum proof leaves things unanswered.

The most simple example (well, how simple it is remains questionable) is with Cantor's proof that there are more reals than natural numbers. The issue here is that the reductio ad absurdum proof uses negative self reference. And then we are left with the Continuum Hypothesis being unanswered. In my view, with real numbers our finite mathematical system that starts from counting natural numbers gets into some foundational problems.

I wrote some time ago in PF that our understanding of mathematics has started from a necessity, from counting and thus we lay the foundations of math to natural numbers and counting. Hence it is a bit hard to add into the picture the uncomputable / uncountable afterwards. Counting and the number system should rather be seen as part of mathematics, a very natural and obvious part, which is possible to use in nearly every part of mathematics, but not with everything and especially not being a foundation. Logic in my view should be the foundation for math. It might be that it is the historical foundation of math, but not perhaps the logical foundation of mathematics.

Isn't he just ensuring that what 2 + 2 is equal to is being discussed with respect to a system big enough for the second theorem to apply?

https://en.wikipedia.org/wiki/Hilbert%E2%80%93Bernays_provability_conditions?wprov=sfla1

my complaint is that [Boolos] did not distinguish between "PA can prove ..." and "It can be proved ..." — fishfry

Boolos says that he means proved by "the aid of the whole of math". My guess is that he means ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC.

Cantor's proof that there are more reals than natural numbers. The issue here is that the reductio ad absurdum proof [...] — ssu

Cantor's proof is not a reductio ad absurdum.

Cantor's proof can be outlined:

Show that there is no enumeration of the naturals onto the reals.

Show that any enumeration of the naturals is not onto the reals.

Let f be an enumeration of the naturals.

Blah blah blah and we've shown that f is not onto the reals.

So any enumeration of the naturals is not onto the reals.

So there is no enumeration of the naturals onto the reals.

My guess is that he means ZFC, — TonesInDeepFreeze

Cool, although it didn't matter what he meant, so long as it was bigger than e.g. Robinson arithmetic, was my point.

while there is obviously is a correct solution: because fixed point theorem proves that there exists a correct solution — ssu

I've worked with fixed points for a long time, mostly recreational now, but a few years ago one of my theorems was applied to decision making, specifically the psychology of groups involved. It surprised me. But I don't recall the title or authors, otherwise I would link it here for you to see. :cool:

Isn't he just ensuring that what 2 + 2 is equal to is being discussed with respect to a system big enough for the second theorem to apply? — bongo fury

Boolos says that he means proved by "the aid of the whole of math". My guess is that he means ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC. — TonesInDeepFreeze

My point exactly. We have people speculating on what he meant, since what he wrote was wrong.

We don't know what he meant. One must always lie a little in order to simplify, and in this case the wrong judgment was made (IMO) as to what to lie about. Einstein would say he simplified too much.

what he wrote was wrong. — fishfry

How?

what he wrote was wrong.
— fishfry

How? — bongo fury

I wrote a long post about it here ...

https://thephilosophyforum.com/discussion/comment/514425

I couldn't add anything. Ok well I'll just give the tl;dr version. Boolos is trying to simplify the issue by talking about "the whole of math." That's incorrect. Consistency and provability are always relative to a given axiom system. PA can't prove its own consistency, but ZF proves PA consistent. ZF and ZFC can't prove their own consistency, but assuming an inaccessible cardinal proves ZFC consistent. And so forth. By omitting the fact that we are always working in a particular axiomatic system, the essence of the matter is ignored. That's a simplification too far in my opinion. Because the heart of the subject is axiomatic systems, and NOT "the whole of math." My opinion is supported by the fact that Kurt Gödel was a Platonist, and believed that there was a true fact of the matter for every mathematical proposition. For example he (and Cohen, for that matter) believed that the Continuum hypothesis is false; notwithstanding the fact that it's independent of ZFC.

Consistency and provability are always relative to a given axiom system. — fishfry

Sure. I was assuming that by "the whole of math" Boolos meant simply the maximal (consistent) extension or union of all the systems you mention. Thereby saving himself the bother of translating "any system strong enough to satisfy the Hilbert-Bernays provability conditions" into words of one syllable.

By omitting the fact that we are always working in a particular axiomatic system, the essence of the matter is ignored. — fishfry

But on my assumption,

First of all, when I say "proved", what I will mean is "proved with the aid of the whole of math".

is hardly omitting the fact (of the relativity of proof to system).

the maximal (consistent) extension or union of all the systems you mention. — bongo fury

What is that?

Haha. No? Not a plausible reading of "the whole of math"?

Haha. No? Not a plausible reading of "the whole of math"? — bongo fury

You said:

the maximal (consistent) extension or union of all the systems you mention. — bongo fury

I don't know what that means or what that is. I asked in good faith for you to explain to me what that phrase means, with perhaps an example or two and some context. As it is I have no idea what this phrase means. So I asked.

No worries.

No worries. — bongo fury

What does that mean? I have no idea what "the maximal (consistent) extension or union of all the systems you mention" means. Are you saying you don't either?

what Boolos wrote was wrong. — fishfry

One source says that the bit comes from a lecture he gave. It might have been a lecture for logicians and students who would understand that he means whatever theory one might take as encompassing the branches of mathematics, whether it's ZFC or, for category theory, ZFC+inaccesible_cardinal, or whatever.

The piece is just a bit of fun, a bit of a stunt - outlining some deep mathematics with just one syllable words - obviously not pretending to include all the specifics.