I don't think it's a theorem in PA, it's a theorem about PA. — fdrake
Incompleteness is not a theorem of PA, unless PA is inconsistent. — TonesInDeepFreeze
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
That page relies on '|=' which is from model theory. — TonesInDeepFreeze
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).
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
(1) If PA is consistent, then there is a true but unprovable sentence. — TonesInDeepFreeze
There are denumerably many of each. — TonesInDeepFreeze
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.
The part that requires much proof is that the standard model is a model of PA. — TonesInDeepFreeze
"If a sentence P is provable from a set of sentences G, then all models of G are models of P" — TonesInDeepFreeze
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
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⌝)
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⌝)
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)
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)
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
What we do prove (in, for example, set theory) is that PA has model thus PA is consistent. — TonesInDeepFreeze
But if the system is sound, then the second disjunct is precluded. — TonesInDeepFreeze
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
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 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]
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
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”.
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.
Somehow I would refer to these two countries as being examples of liberalism and respecting the free market. — ssu
But who owns the bitcoins? I think the people from the West. — ssu
The death of the sovereign states is in my view highly exaggerated and basically false. — ssu
And they (sovereign states) do love their central banks, just as Russia and China do. — ssu
In fact when the fiat system collapses, it won't be such a catastrophic event that society collapses. — ssu
How many actually think of the last time the US dollar defaulted as a default? — ssu
You're a funny man and obviously have no idea how taxes worked in Egypt. — Benkei
The three countries you mention have huge issues with modern slavery or human trafficking — Benkei
Also, nice false analogy showing a picture of poverty and a rich bitch on the beach. — Benkei
But in any case, you're not refusing the point that income tax has existed for millenia. — Benkei
Good luck in those failed states when you get sick. — Benkei
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
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
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
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
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
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
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
And unfortunately we do need that thing called "taxation". — ssu
Also the ruling mafia also defends their monopoly legal tender. Which cryptocurrencies aren't, but currencies of sovereign states are. — ssu
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.
Hence there were also some reasons just why legal tender was monopolized. — ssu
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.
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
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.
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
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
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
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.
The paragraph expresses a number, not an unstateable truth. — Banno
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.
Give an example of one of these unstatable true sentences... — Banno
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
Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way. — 180 Proof
They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified. — hypericin
But all of these ineffable truths seem quite irrelevant too. — hypericin
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 …
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."
Assuming this statement is true, what do you think is its philosophical significance? — 180 Proof
What is one example of a subset of the natural numbers that cannot be expressed by language? — hypericin
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.
Any thoughts on why? Is it a blow to people's egos to face the limitations of human thought? — wonderer1
I can see you and I are not going to agree on this. — T Clark
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?"
judgments of true or false only apply to propositions — T Clark
If it can't be expressed in language, it isn't a proposition — T Clark
Propositions are linguistic entities — T Clark
The presumption of innocence or correctness can be possibly disproven, but the reverse, requiring proof across unlimited time and space, renders unfair. — ucarr
the burden of proof for the truth of a defendant’s innocence is a standard too stringent? — ucarr
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.
Since there are statements true but unprovable, there seems to be a disconnection between truth and proof. — ucarr
However, isn’t ‘the Turing machine’ something that only exists in the minds of humans? — Wayfarer
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