I don't think it's a theorem in PA, it's a theorem about PA. — fdrake

It is an existence theorem about a sentence that is supposed to exist in PA.

The canonical witness is definitely a sentence in the language of PA:

G ⇔ ¬ Bew(⌜G⌝)

meaning:

"This is not provable."

Of course, the fact that this sentence exists in the language of PA says something about PA

Incompleteness is not a theorem of PA, unless PA is inconsistent. — TonesInDeepFreeze

Gödel's incompleteness theorem proves that PA is inconsistent or incomplete. That is a perfectly legitimate theorem in PA. It does not prove that PA is incomplete. That is a theorem in PA + Cons(PA).

There are not only finitely many of them, and there are not uncountably many of them (there are only countably many sentences in the language), so there are denumerably many. — TonesInDeepFreeze

Yanosky includes the Gödelian statements that cannot be expressed by language. There are uncountably many of those.

That page relies on '|=' which is from model theory. — TonesInDeepFreeze

The use of logical entailment predates model theory by decades:

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

The turnstile symbol ⊢ was originally introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935).

The page uses it without distinguishing between theory T and its model M:

A1..An ⊢ S => A1..An ⊨ S with T equivalent to A1..An

Therefore, it is not a model-theoretic explanation. It just uses "T ⊢ S " as as a synonym for "T proves S" and T ⊨S as a synonym for "S is true in T".

If PA is consistent, then there are true but unprovable sentences. So, trivially, by disjunction introduction, it follows that there are true but unprovable sentences or there are false but provable sentences.

Meanwhile, for the third time, my remark is correct: If we assume soundness, then the second disjunct is precluded. — TonesInDeepFreeze

I did not say that your remark would be wrong or that the (Raatikainen 2020) characterization would be wrong. I just said that I prefer to introduce the disjunction and avoid assuming Cons(PA). This is just a personal preference.

(1) If PA is consistent, then there is a true but unprovable sentence. — TonesInDeepFreeze

Gödel's theorem can perfectly be phrased without assuming that PA is consistent.

In fact, by introducing the assumption "If PA is consistent", Gödel's theorem is no longer a theorem in PA. In that case, it is a theorem in PA + Cons(PA). That is not the same theory as PA.

Incompleteness as a theorem provable in PA proper must be phrased as:

There is a true but unprovable statement or a false but provable statement.

There are denumerably many of each. — TonesInDeepFreeze

In "True But Unprovable", Yanofsky insists that unprovably true statements vastly outnumber provably true ones:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

True and provable statements are denumerable while true and unprovable statements are non-denumerable.

The part that requires much proof is that the standard model is a model of PA. — TonesInDeepFreeze

Yes, and once this part has been proven, there is no need to prove soundness theorem, because the model-theoretical construction already guarantees this.

"If a sentence P is provable from a set of sentences G, then all models of G are models of P" — TonesInDeepFreeze

This statement requires the use of model theory. Soundness can also be defined without using model theory:

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

I am obviously not against using model-theoretical notions to define soundness, but it obviously raises the bar in terms of accessibility.

If you look at what exactly Gödel's theorem says, There exist propositions in Russell's system that are (true and not provable) or (false and provable) — Tarskian


Where did Godel say that? — TonesInDeepFreeze

If we start from Carnap's diagonal lemma:

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

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions, and F(y) be a formula in T with one free variable. Then there exists a sentence C such that

T ⊢ C ⇔ F (⌜C⌝)

Choose F(y) to be ¬ Bew(y), with Bew(y) the provability predicate. This will morph Carnap's diagonal lemma into Gödel's incompleteness theorem:

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Then there exists a sentence G such that

T ⊢ G ⇔ ¬ Bew(⌜G⌝)

We can see that:

T ⊢ G ⇔ ¬ Bew(⌜G⌝)

is equivalent to:

T ⊢ ( G ∧¬ Bew(⌜G⌝) ) ∨( ¬ G ∧ Bew(⌜G⌝)

In plain English:

G ∧¬ Bew(⌜G⌝) means G is true and not provable

¬ G ∧ Bew(⌜G⌝ means G is false and provable

Therefore, Gödel's incompleteness theorem can be written as:

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Then there exists a sentence G such that

G is (true and not provable) or G is (false and provable)

This is somewhat equivalent to the alternative phrasing in which we assume consistency:

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

First Incompleteness Theorem: "Any consistent formal system T within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of T which can neither be proved nor disproved in T." (Raatikainen 2020)

The result that follows straight out of Carnap's diagonal lemma does not assume consistency.That is why I do not see where the requirement comes from, to phrase it like that. I prefer to phrase it as the proof's verbatim output.

It is possible to preclude the second disjunct if we assume or prove that PA is sound. I didn't say that PA itself proves that PA is sound. Virtually every mathematician (including Godel) regards PA to be sound. — TonesInDeepFreeze

Well, if PA is not sound, then it is actually unusable. So, we have to assume that it is sound. We simply have no other choice.

However, proving soundness is even irrelevant.

Imagine that we prove soundness theorem. Does that make soundness theorem true? No, because the proposition that proof implies truth is exactly what we are trying to prove. So, that would just be a silly exercise in circular reasoning.

Proving soundness is therefore both irrelevant and self-defeating.

What we do prove (in, for example, set theory) is that PA has model thus PA is consistent. — TonesInDeepFreeze

Proving PA's soundness from set theory amounts to moving the goal post. How do you prove set theory's soundness? It is simply the same problem all over again.

But if the system is sound, then the second disjunct is precluded. — TonesInDeepFreeze

Soundness implies consistency.

So, if you manage to prove soundness theorem from PA then you have also managed to prove PA's consistency from PA.

Gödel's second incompleteness theorem prevents PA from proving its own consistency. If PA proves its own consistency, then PA is necessarily inconsistent.

Hence, it is not possible to preclude the second disjunct. PA puts up a lot of resistance to doing that by making the attempt self-defeating.

Anyway, this thread was mostly about why Wittgenstein or what Wittgenstein could have meant by claiming that Godel's Incompleteness Theorems are logical tricks. — Shawn

((everything below is in the context of PA or similar))

Gödel's own witness is certainly a corner case, but so are all the witnesses for his incompleteness theorem. Goodstein's theorem, another example, is also a corner case.

It is actually not difficult to see why every example for Gödel's theorem will always be a weird corner case.

We usually know that a proposition is true because it is provable, aka, as a consequence of the soundness theorem ("provable implies true"). Otherwise, without proof, the proposition is not a theorem but just a hypothesis. Pure reason, and therefore, mathematics, is blind. Unlike in physical reality, we cannot discover truth in mathematics by somehow observing it. We have to discover truth by discovering its proof. So, at first glance, it even looks impossible to discover examples of true but unprovable propositions. It is still possible, though. But then again, it certainly means that the discovery of a Gödelian proposition can only be achieved by means of the one or the other contorted hack.

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

Particularly controversial in the Remarks was Wittgenstein's "notorious paragraph", which contained an unusual commentary on Gödel's incompleteness theorems. Multiple commentators read Wittgenstein as misunderstanding Gödel. In 2000 Juliet Floyd and Hilary Putnam suggested that the majority of commentary misunderstands Wittgenstein but their interpretation[8] has not been met with approval.[9][10]

Wittgenstein's notorious paragraph is certainly confused:

Wittgenstein wrote:

I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable." Just as we can ask, " 'Provable' in what system?," so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false" in Russell's system means the opposite has been proved in Russell's system.—Now, what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence.—If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)[11]

In his "notorious paragraph", Wittgenstein over-complicates the matter. Concerning Gödel's canonical witness, "P is not provable":

If the proposition is true, then it is not provable.
If the proposition is false, then it is provable.

Hence, the proposition is (true and not provable) or (false and provable).

If you look at what exactly Gödel's theorem says, There exist propositions in Russell's system that are (true and not provable) or (false and provable), then you can see that Gödel's canonical witness is exactly such proposition.

Of course, it is a weird corner case.

As I have argued above, pure reason is blind, and therefore, discovering truth normally requires discovering proof. Hence, any true but unprovable proposition will necessarily be a weird corner case that was hard to discover.

Because of the difficulty of discovering Gödelian statements, the impression may arise that Gödelian statements are not the norm but the odd exception, i.e. that they are some kind of deviant abnormality. This is not the case. The overwhelmingly vast majority of true propositions in true arithmetic are Gödelian. True but unprovable statements vastly outnumber the true and provable ones (cfr. True But Unprovable by Noson S. Yanofsky). This does indeed mean that the overwhelmingly vast majority of mathematical truth is simply invisible to us. Pure reason is blind.

And why talk about the possibility of hyperinflation as an achievement? You think the US will be better afterwards? And in the end there's many ways the US can do this. — ssu

You see, I do not want KYC (Know Your Customer), because it allows the ruling mafia to target bank accounts based on nationality and other criteria. It allows them to single you out.

KYC is what allows them to expropriate your account balance, prevent incoming and outgoing payments, prevent you to open bank accounts, and wholesale attack you through financial censorship. I want an end to the financial bullying.

The US (and EU) force other countries to implement KYC:

https://www.fatf-gafi.org/en/the-fatf/what-we-do.html

Identifying high-risk jurisdictions

The FATF holds countries to account that do not comply with the FATF Standards. If a country repeatedly fails to implement FATF Standards then it can be named a Jurisdiction under Increased Monitoring or a High Risk Jurisdiction. These are often externally referred to as “the grey and black lists”.

If these countries do not implement KYC or not sufficiently, through the FATF, the USA (and EU) starts bullying them by restricting their access to the global financial network. This system needs to be destroyed. We must kill it because the bullying has to stop.

That is why the complete destruction and wholesale eradication of the fiat bankstering system is a global necessity.

The USA will be better afterwards because it will no longer be able to bully other countries. Seriously, no more KYC. This page gives an overview of cryptocurrency-related services that resolutely reject KYC:

https://kycnot.me

https://kycnot.me/about

why kycnot.me?

Cryptocurrencies were created to revolutionize the way we pay for goods and services, aiming to eliminate reliance on centralized entities such as banks and governments that control our economy.

Exchanges that enforce KYC (Know Your Customer) operate similarly to traditional banks.

With KYCNOT.ME, I hope to provide people with trustworthy alternatives for buying, exchanging, trading, and using cryptocurrencies without having to disclose their identity, thus preserving the right to privacy.

The truth is that KYC is a direct attack on our privacy and puts us in disadvantage against the governments. True criminals don't care about KYC policies. True criminals know perfectly how to avoid such policies. In fact, they normally use the FIAT system and don't even need to use cryptocurrencies.

KYC only affects small individuals like you and me. It is an annoying procedure that forces us to hand our personal information to a third party in order to buy, use or unlock our funds. We should start boycotting companies that enforce such practices. We should start using cryptocurrencies as they were intended to be used: without barriers.

I do not want to disclose any personal information that will help the ruling mafia to single me out and bully me. KYC needs to be destroyed completely. That is why the fiat banks must die, now already.

Somehow I would refer to these two countries as being examples of liberalism and respecting the free market. — ssu

I cannot comment on "liberalism" because that term has no precise definition.

But who owns the bitcoins? I think the people from the West. — ssu

The people who mine them or somehow buy them. It is the same question as "who owns all the gold?". Same answer.

The death of the sovereign states is in my view highly exaggerated and basically false. — ssu

I do not believe in the death of the sovereign states. I only believe in the death of the dollar (and the euro). That is the ultimate goal of hyperbitcoinization. We want to destroy that shit. I do not even believe in the death of all fiat currencies. I don't care about that because they do not represent an oppressive system of sanctions, confiscations, expropriations, and financial censorship. The dollar system has to die because we are seeking to kill it.

And they (sovereign states) do love their central banks, just as Russia and China do. — ssu

Yes, for all I care, let them keep their Ruble and Yuan, if they so desire. They do not try to dominate the world with their local paper fiat scrip.

In fact when the fiat system collapses, it won't be such a catastrophic event that society collapses. — ssu

Agreed. The fiat system has already collapsed in Venezuela, Zimbabwe, Lebanon, Argentina,Turkey, and countless other countries. It is a temporary annoyance, but anybody with half a brain has figured out how to deal with it.

How many actually think of the last time the US dollar defaulted as a default? — ssu

The dollar cannot "default". How can the dollar even "default"? If they do not have enough dollars, the Fed simply prints some more. The dollar can only hyper-inflate, i.e. become worthless. That is what we want to achieve. That shit has to go, now already.

You're a funny man and obviously have no idea how taxes worked in Egypt. — Benkei

We were talking about the taxes at the time of the Pharaohs. They were necessarily simple.

The three countries you mention have huge issues with modern slavery or human trafficking — Benkei

"huge issues", "modern slavery", "human trafficking" ... blah blah ... woke bullshit.

Every country has its problems. That is not a reason to import any of the hated woke nonsense from the West. Seriously, these people don't need this kind of word salads. They are not victims. They do not need your help. They are absolutely fine and there is nothing that they can learn from your propaganda.

I love SE Asia and I hate the West.
That is why I live in SE Asia.
I'd rather die than ever going back.

Also, nice false analogy showing a picture of poverty and a rich bitch on the beach. — Benkei

The locals are generally not rich (though some are) but they are definitely beautiful.

But in any case, you're not refusing the point that income tax has existed for millenia. — Benkei

The Egyptian tax on a farmer's harvest is not the same as modern personal income tax. The farmer did not have to give any information to facilitate the collection of that tax.

Good luck in those failed states when you get sick. — Benkei

Dubai, for example, does not have personal income tax. Why would it be a "failed state"?

Furthermore, I haven't lived in the West since 2005. I have never had to deal with personal income tax ever since.

The West is only an attractive place for people who want to live off welfare benefits. That is the only kind of immigrant that the West attracts. Anybody with any serious level of money is better off elsewhere.



The skid row image above is the future of the West. I'd rather spend my days at the beach in Thailand, Vietnam, and other fantastic locations in SE Asia, in the company of beautiful locals:



Seriously, the West is a despicable environment.

I don't know what that is supposed to mean, but, to be clear, the incompleteness theorem applies also to theories in higher order logic. Indeed, Godel's own proof regarded a theory in an omega-order logic. — TonesInDeepFreeze

What I meant to refer to, was:

https://en.wikipedia.org/wiki/Second-order_logic

This corollary is sometimes expressed by saying that second-order logic does not admit a complete proof theory. In this respect second-order logic with standard semantics differs from first-order logic; Quine (1970, pp. 90–91) pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not logic, properly speaking.

Maybe. For now, we don't know why physical laws are like this. — MoK

The physical laws that we know, are not an axiomatic theory.

They are a collection of stubborn observable patterns. They just say that a particular pattern should be there, but not why it is there.

A formal axiomatic system for the physical universe would be much nicer to have, but it would still not explain itself.

Such theory would only allow to explain from it. It would be the same situation as for the theory of the natural numbers, PA. We can explain from PA but not the why of PA.

The ultimate why cannot be answered by means of rationality.

Having to deliver 15 sacks of grain you harvested is effectively income tax. It existed quite a bit longer, at least since the Egyptians. — Benkei

The Egyptian tax collector would measure the farmer's land and compute taxes based on that information. He would not ask the farmer if he somehow made some more money in other ways and try to get half of that too.

I find the practice of demanding people to fill out a tax return form to be particularly detestable. Why on earth would I give that kind of information to someone else?

Seriously, I don't want to live in a country where the ruling mafia asks me how much money I have made last year and then demands that I give them half.

I didn't mean mathematical truth when I said we may one day explain reality. I mean we may be able to explain why physical laws are like this and not the other ways. — MoK

If physical reality has a formal theory, then its model/interpretation may contain inexplicable truths in a similar way as the system of the natural numbers does.

I am talking about fundamentally inexplicable truths for which you can actually prove that they are inexplicable.

For the natural numbers, we can prove the existence of inexplicable truths, prove why they are fundamentally inexplicable, and we even have examples.

I always thought the solution to the problem of insufficient logic needed to compute certain undecidable problems is solved by appealing to greater complexity class sizes, which avoids the inherent limitations of a formal system which is incapable of decidability given its inherent limitations. — Shawn

An undecidable problem in logic is undecidable irrespective of how much time or memory you throw at the problem. The P versus NP issue only applies to problems that are at least logically decidable.

I do not like stating this in formal systems like Peano Arithmetic; but, rather in decidability. By framing the question in terms of decidability, we do away with the problem of the inherent limitations of human intuition devising formal systems. This is a question model theorists might be able to prove, in my opinion. — Shawn

The problem is logic itself:

https://en.wikipedia.org/wiki/Decidability_(logic)

In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not.

There is no incompleteness in nature — Shawn

This would imply that for every true statement about the physical universe, there exists a proof that can be derived from the supposedly canonical and categorical but unknown theory of the physical universe.

We do not know the theory of the physical universe.

We also do not know if it happens to be canonical or categorical.

In the context of the natural numbers, we know that Peano Arithmetic theory is not canonical (there are definitely alternatives) and not categorical either (it does not have a single interpretation/model).

All of this in the context of first-order logic. If you allow for higher-order logic then all odds are off and even less can be asserted about the properties of the theories involved, such as incompleteness.

And unfortunately we do need that thing called "taxation". — ssu

Certainly not to the extent that it exists in the West. For example, personal income taxation was introduced only in 1913 while the human race has been around for almost 300 000 years. Governments outside the West may also have it on the books but until this day they still do not collect it and they probably never will.

Also the ruling mafia also defends their monopoly legal tender. Which cryptocurrencies aren't, but currencies of sovereign states are. — ssu

Bitcoin is now already legal tender in El Salvador. We are also rapidly making geopolitical progress with the Russian Federation:

https://www.chainalysis.com/blog/russias-cryptocurrency-legislated-sanctions-evasion

In response to mounting financial pressures of Western sanctions, Russia enacted significant legislation legalizing cryptocurrency mining and permitting the use of cryptocurrency for international payments.

Getting the Russian Federation on our side is a significant breakthrough because now the system is also backed by an arsenal of thousands of nuclear weapons. The next step, is getting China onboard in matters of cryptocurrency. That will be hard, but I suspect that Russia will sooner or later manage to convince them.

Hence there were also some reasons just why legal tender was monopolized. — ssu

Legal tender where exactly?

Along the Russian-Chinese axis, the entire Global South is now rallying and making their preferred geopolitical choice.

In this new multipolar world order, it is the cryptocurrencies, especially Bitcoin, that will eventually reign supreme.

In the new multipolar world order, international trade and commerce will be increasingly de-dollarized. Along with the enormous debt, this process will lead to hyperinflation in the dollar and the euro. Hence, the end result of our efforts is that we will successfully bankrupt and utterly destroy these fiat currencies. i.e. the American dollar and the European euro. When everything will have been said and done, they won't be legal tender anywhere on earth. The final goal is the complete and utter eradication of the American dollar and European Euro. It is clearly a lenghty process, but it is inevitable that we will geopolitically achieve our goal.

This process is called hyperbitcoinization:

https://www.tokenmetrics.com/blog/hyperbitcoinization

Defining Hyperbitcoinization

At its core, Hyperbitcoinization envisions a future where Bitcoin supplants fiat currencies as the predominant medium of exchange, store of value, and unit of account globally.

This phenomenon transcends mere adoption; it represents a paradigm shift in which Bitcoin becomes the universal currency, facilitating transactions, settlements, and economic activities worldwide.

The concept of Hyperbitcoinization was first introduced in 2014 by Daniel Krawisz, an entrepreneur and researcher at the Satoshi Nakamoto Institute.

Hyperbitcoinization is why Bitcoin maximalists do not care much about the exchange rate or about measuring its value in dollars or in euros, as we seek to destroy the dollar and the euro:

https://www.investopedia.com/terms/h/hodl.asp

Cryptocurrencies will eventually replace government-issued fiat currencies as the basis of all economic structures. Should that occur, then the exchange rates between cryptocurrencies and fiat money would become irrelevant to crypto holders.

Predictably, a meme best captures this HODL maximalist philosophy. Neo from The Matrix asks Morpheus, "What are you trying to tell me, that I can trade my Bitcoin for millions someday?" Morpheus responds, "No Neo, I'm trying to tell you that when you're ready … you won't have to."

What is PA? Why the theory for the system of natural number is incomplete? — MoK

PA is short for Peano Arithmetic theory: https://en.wikipedia.org/wiki/Peano_axioms

It is the most common theory of the natural numbers. Most true statements about the natural numbers are not provable from PA. Therefore, PA is incomplete.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Gödel's incompleteness theorems

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.

We may be able to explain things given our intellectual power. We are evolving creatures so even if we cannot explain things now we may be able to explain things in future when we are evolved well.

If a system is incomplete, then it has unprovable truths. An unprovable truth is an inexplicable truth. The existence of such fundamentally inexplicable truths has nothing to do with our own evolution in terms of understanding. There simply does not exist a justification for such truth. The natural numbers is a system with fundamentally inexplicable truths.

To elaborate let's consider the current state of the universe to be S(t) which by the state I mean the configuration of material at a given time. The state of the universe at the former time is then S(t-1) etc. until we reach the beginning of time S(0). I claim that this state is related to the configuration of some sort of material at the beginning of time. — MoK

First of all, if S(t) can be predicted from S(t-1) ... S(0) then the theory T for this system is complete. Non-trivial systems do not have a complete theory. For example, the theory for the system of the natural numbers (PA) is not complete.

Secondly, theory T is axiomatic, which means that every single one of its rules has no further explanation in terms of deeper underlying rules. So, even if we had a copy of theory T with all its rules, we would still not understand why it is there, just by looking at it. Just like PA has no ulterior logical explanation, by its very nature, T does not have one either.

Hence, from within the universe, you cannot figure out why the universe exists. Just staring at the universe or just staring at its theory (which we do not even have) won't help.

Similarly, a fish swimming in the ocean, no matter how smart, will never be able to figure out why the ocean exists. Staring at its surroundings in the ocean or even staring at the ultimate theory of the ocean won't help either.

The idea that God created the heavens and the earth, is a belief. By merely by staring at the heavens and the earth, this belief can neither be logically justified nor logically rejected. This belief has spiritual origins that transcend logic. That is why I am not particularly fond of the Kalam Cosmological Argument either.

It's quite telling that cryptocurrencies marketed as "freedom from governments and the central banks" then will have these shady frauds etc. I think it's basically a natural result when you don't have legislative supervision. — ssu

The official ruling mafia defends their monopoly on expropriation, called "taxation". Lions also fight other lions trying to feed on prey in their territory. They won't allow it, because in their case, the expropriation is called "stealing".

I think it's something that was forgotten in the ideological fervour of liberalism, that free markets have to have institutions to keep them trustworthy and operational. Otherwise simple theft is so easy. — ssu

A market authority is usually necessary but it does not need to be the government.

For example, in darknet markets, there is always a third-party adjudicator holding the third signature in the 2-of-3 multi-signature escrow contract.

In an atomic swap, for example, there is not even a need for a human adjudicator. The two interlocking smart contracts will automatically prevent both parties from defrauding each other.

https://atomicdex.io/en/blog/atomic-swaps

An atomic swap is a trade of cryptocurrency made directly from one user to another, without any intermediary to facilitate the transaction.

These swaps are called "atomic" because either the trade is successfully completed and each trader receives the other one's funds, or nothing happens and both traders simply keep the funds they started with. Atomic swaps are made wallet-to-wallet, in a fully peer-to-peer (P2P) manner.

The basic idea is that Bob and Alice can send each other funds that are locked by the hash of a predetermined secret code. Bob publicly reveals the secret code to collect Alice's funds, which allows Alice to also see the secret code and use it to collect Bob's funds. If Bob doesn't collect Alice's funds, then Alice can never spend Bob's funds. In this case, the locktime set by the CLTV command would expire and both Bob and Alice would get their money back. That's what makes the swap atomic.

If ownership of an asset can be represented by a digital token, such as goods in a bonded warehouse or just money, then any such assets can be swapped without third-party involvement.

We are only at the very beginning of tokenizing commercial traffic. In my opinion, just 0.1% of the possibilities have currently been implemented already. It will completely change the world of commerce.

The paragraph expresses a number, not an unstateable truth. — Banno

Every property of this unstateable number is itself an unstateable truth.

Example: Number r is a real number.

If number r is unstateable then this sentence is also unstateable, no matter how true this sentence may be.

https://iai.tv/articles/most-truths-cannot-be-expressed-in-language-auid-2335

Most truths cannot be expressed in language

14th December 2022
Noson S. Yanofsky | Professor of computer science at Brooklyn College

There are more true but unprovable, or even able to be expressed, statements than we can possibly imagine, argues Noson S. Yanofsky.

I actually took the example of the subsets of the natural numbers literally from Yanofsky:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

Yanofsky argues that the fact that the sentence is ineffable automatically makes it unprovable. This is indeed the case for an individual sentence. The truth of entire set of sentences, however, is provable.

There are truths about sets of sentences that apply to each individual sentence while we do not have the ability to express by language most of such individual sentences.

Give an example of one of these unstatable true sentences... — Banno

Construct a Richardian number and map it one-to-one to a subset of the natural numbers. This subset is ineffable:

https://en.m.wikipedia.org/wiki/Richard%27s_paradox

The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.

Now, can you give an example of one those the truths? — Banno

Consider the following proposition:

The set X is a subset of the natural numbers.

This is trivially true for an example subset such as {5, 67, 257}.

There are an uncountably infinite number of such subsets. However, there are only a countably infinite number of sentences in language. Therefore, for most subsets X of the natural numbers, this true sentence cannot be expressed in language.

Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way. — 180 Proof

I don't think that JTB is inadequate.

Most truth cannot be known in terms of JTB. That is not a flaw in JTB. The nature of reality is simply like that.

If we happen to know some truth, then it is the rare exception and not the rule.

They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified. — hypericin

Then there is still the next level: the beliefs about these ineffable beliefs which are not necessarily ineffable. There is a large literature about Richardian numbers even though these numbers are undefinable.

But all of these ineffable truths seem quite irrelevant too. — hypericin

Well, it's a bit like the axiom of infinity, i.e. insisting on the existence of an ineffable cardinality. At first glance, it also looks irrelevant.

https://institucional.us.es/blogimus/en/2022/01/is-infinity-really-necessary

One of the axioms of mathematics is that there exists an infinite set. Without this axiom our mathematics would be much weaker. Many of our theorems would fall like a house of cards. Newton or Gauss would probably have hesitated to accept our axiom (although without being aware that they were using it). We have accepted it for our comfort. Faith, that some people say …

Originally, most mathematicians utterly rejected the axiom of infinity and Cantor's work in general:

As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

The ineffable sequence of infinite cardinalities is an essential axiomatic belief in contemporary mathematics, no matter how much it sounds like philosophy or theology.

Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof

If you look at the epistemic JTB account for knowledge as a justified true belief, it means that the overwhelmingly vast majority of true beliefs are ineffable and cannot possibly be justified.

Hence, most truth is not knowledge.

The fact that some truth can be justified is the rare exception and not the rule.

What is one example of a subset of the natural numbers that cannot be expressed by language? — hypericin

There is a one-to-one mapping between the subsets of the natural numbers and the real numbers. So, we can represent a subset of the natural numbers by its corresponding real number.

We construct the real number as the Ricardian number r:

https://en.m.wikipedia.org/wiki/Richard%27s_paradox

Richard's paradox

The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the nth decimal place of which is 0 if n is even and 1 if n is odd" defines the real number 17.1010101... = 1693/99, whereas the phrase "the capital of England" does not define a real number, nor the phrase "the smallest positive integer not definable in under sixty letters" (see Berry's paradox).

There is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of r[n] is 1.

The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers r[n]. However, r was constructed so that it cannot equal any of the r[n] (thus, r is an undefinable number). This is the paradoxical contradiction.

The Ricardian real number r is defined as undefinable and therefore the corresponding subset of the natural numbers cannot be expressed in language either.

Any thoughts on why? Is it a blow to people's egos to face the limitations of human thought? — wonderer1

In my opinion , it decisively divorces mathematical reality from physical reality, which is otherwise its origin.

Humans, but also animals, have quite a bit of basic arithmetic and logic built into their biological firmware, if only, for reasons of survival. To the extent that mathematics stays sufficiently close to these innate notions, people readily accept its results.

There is no notion of infinity in physical reality. In that sense, Cantor's work is rather unintuitive. You have to learn to think like that. It does not come naturally.

I can see you and I are not going to agree on this. — T Clark

The distinction between countable and uncountable infinity, originally introduced by Georg Cantor, has always been controversial.

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

Controversy over Cantor's theory

Initially, Cantor's theory was controversial among mathematicians and (later) philosophers.

As Leopold Kronecker claimed: "I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there."

Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics.

Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this "harmless little argument" (i.e. Cantor's diagonal argument) asking, "what had it done to anyone to make them angry with it?"

When first confronted with the matter, I do not think that anybody right in his mind agrees on this. It is just too controversial. The first reaction is usually, disgust. It takes quite a while before someone can actually accept this kind of thinking.

judgments of true or false only apply to propositions — T Clark

The following is a legitimate proposition:

The set {6,8,11} is a subset of the natural numbers.

It is true or false.

If it can't be expressed in language, it isn't a proposition — T Clark

The following proposition is tautologically true:

Every subset of the natural numbers is a subset of the natural numbers.

The problem is that most individual subsets of the natural numbers cannot be expressed by language. Some can but most cannot.

The ineffable propositions are still true propositions because all of them are true given the tautology mentioned above.

Propositions are linguistic entities — T Clark

Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. For example, the general case of "Subset X of the natural numbers is a subset of the natural numbers" is true, irrespective of whether X can be expressed by language or not.

The presumption of innocence or correctness can be possibly disproven, but the reverse, requiring proof across unlimited time and space, renders unfair. — ucarr

Requiring proof in science would indeed be unfair, if only, because there is no (axiomatic) theory to prove it from. So, we accept the scientific claim because of the inability to discover counterexamples.

The situation in mathematics is a bit different.

There is no observation possible in mathematics. Therefore, the only reason why we know that it is true, is the proof.

Proving an impossibility in mathematics is generally also hard and in the general case would also require omniscience, but it can still be done, by discovering some helpful structure that implies the impossibility.

For example, the impossibility to find a general solution for the quintic (Abel-Ruffini), took centuries of investigation. They strongly suspected that it was impossible before finally proving it. The helpful piece of structure that dramatically simplifies the proof is the Galois correspondence.

the burden of proof for the truth of a defendant’s innocence is a standard too stringent? — ucarr

In classical law, the burden of evidence is on the prosecutor:

https://en.m.wikipedia.org/wiki/Burden_of_proof_(law)

The burden of proof is on the prosecutor for criminal cases, and the defendant is presumed innocent. If the claimant fails to discharge the burden of proof to prove their case, the claim will be dismissed.

However, for offences newly defined in modern times, this is usually not the case.

For example, for the modern offense of money laundering, the burden of evidence is placed on the defendant to argue that the money does not originate from crime.

In classical law, the offense of money laundering did not exist, and could not exist, exactly because it reverses the burden of evidence.

Most criminal offenses that were newly and recently defined in modern times require the defendant to prove that he is innocent.

Reversing the burden of evidence is in fact exactly what allowed to define these new modern offenses.

So, if it is about a criminal offense that was defined already in classical law, the defendant can count on the presumption of innocence. If it is about a modern offense, however, there is generally a presumption of guilt.

It is indeed unreasonable to expect the defendant to prove his innocence.

So, if an offense did not exist in 1800 but it exists today, then there is almost surely something wrong with it. Newly-defined modern western law is known to be unreasonable and to be in violation of classical legal traditions.

Since there are statements true but unprovable, there seems to be a disconnection between truth and proof. — ucarr

Proof is what mathematics uses as justification. There is always a disconnection between truth and justification.

Concerning physical truth, it is perfectly possible to observe an event or a state without being able to justify it. The absence of justification does not make the observation any less true.

However, pure reason, such as mathematics, is blind, and observation is impossible.

A mathematical fact can only be confirmed to be true because there exists a justification under the form of proof. Otherwise, it remains just a hypothesis.

For the overwhelmingly vast majority of mathematical facts, however, proof cannot even exist. There are various reasons for that (Yanofsky).

Only in exceptional corner cases, it is still possible to confirm the truth of Godelian facts, i.e. mathematical facts that are known to be true but not provable.

For example, Goodstein's theorem is unprovable in arithmetic (PA). The theorem is provable, however, in set theory (ZF). Therefore, we can confirm the theorem to be true, in both arithmetic and set theory. Hence, Goodstein's theorem has the rare Godelian status in arithmetic of being true but not provable.

Goodstein's theorem is a rare and exceptional corner case. Normally, it is not possible to do that.

For example, the Continuum Hypothesis (CH) is not provable in set theory (ZF). There is also no alternative theory available in which it is provable. Therefore, we simply cannot confirm the truth of CH.

This is an important difference between physical truth and mathematical truth.

We know physical truth because we can observe it. This is not the case for mathematical truth, because mathematics is blind. Observation is simply not possible in mathematics. Therefore, we can confirm mathematical truth only because we can prove it.

On the other hand, there is also never proof for physical truth. There is no mathematical theory of the physical universe available to prove from. Therefore, physical truth can only be confirmed by means of observation and never by means of proof.

However, isn’t ‘the Turing machine’ something that only exists in the minds of humans? — Wayfarer

The thing is that computer science started almost a decade before the first computer was built.

So, in 1936, Alan Turing dreamt up a machine that could compute, studied its properties extensively, but never received a budget from the cash-strapped British government to actually build it

They could have built one later but that never happened because John Von Neumann helped designing another machine -- eventually built in 1949 -- that was much more straightforward to program, the EDVAC, sporting the first CPU. It was funded by the US Ballistic Research Laboratory.

Modern computers still use this architecture. Turing's architecture was never really built.

Theoretical computer science literature has kept referring to Turing machines, though.

However, isn’t ‘the Turing machine’ something that only exists in the minds of humans? An actual Turing machine would require infinite memory, so it is not something that could ever exist. — Wayfarer

In all practical terms, the term "Turing machine" just means "computer". For the problem of proving a PA theorem in ZF, there is no need for infinite memory.