I don't know if you've visited the 21st century at all, but women can earn money for themselves now... — Isaac

If you have ever lived in the 80% of the world which is not a welfare state -- get out of your bubble and look at the real world for a change -- you would know that female employment primarily consists of make-work jobs created or subsidized by the government itself, such as feminizing little boys in the schools, drugging them with amphetamines such as ritalin for imaginary mental diseases, and getting them to attend gender-fluidity lectures by transvestites.

It is the welfare state itself that fuels all of the nonsense and outright depravity, and which keeps it afloat by endlessly printing paper money. Live and work in a country outside the welfare-state bubble and see for yourself that there is very little of that nonsense going on here.

Young women here want to get married, because they want kids, because they will need these kids in their old age. It is too late to sell the idea to them that the $50 trillion of unfunded liabilities of social security and medicare would be a better idea.

Taken together, the combined unfunded liabilities of Social Security and Medicare are more than $50 trillion, according to official government projections. Unsettling as these estimates are, they are probably optimistic — for two reasons ...

We are now in the long run of the short-term decisions that are now leading to a complete meltdown. Any country that is not a welfare state at this point already, will simply not become one. Your welfare state is not the future. It is not the 21st century. It is a complete bankruptcy caused by ignorant and arrogant follies of the past.

Furthermore, we can suspect that the next financial crisis is right behind the corner now. How will the already bankrupt welfare state deal with that problem? By printing some more paper money?

By the way, the living standards in the former Soviet Union are now much lower than in African Botswana.

The next financial crisis will inevitably turn into the final bankruptcy crisis of the western welfare state.

Along with the implosion of the dollar and the euro, you will see with your own eyes how the living standards in the West will drop below the former Soviet Union; which is already below the level in Botswana.

The only possible advantage is that the uncontrolled flood of immigration will come to a shrieking halt. There will be no reason for anybody to try to get a job at the bottom of a welfare-state's labour market because it will no longer have a bottom.

But I grow cynical whenever I hear politicians advocating for the welfare state, especially under the guise of compassion. Paying more taxes and advocating for more government services seems to me to be the very least one can do for his fellow man. — NOS4A2

Anybody who believes in what these politicians say, is a follower of their false religion.

For a starters, they can collect money under whatever guise they want, how can you check what you do with it? So, either you believe them or you don't. I simply do not. I can obviously believe exactly what I want, as there can never be an obligation to believe anything at all.

Next, I find these redistribution policies extremely dangerous.

You will end up with women who claim that they don't need a man, because the government will give them money. And where does the government get the money from? From the men, of course. Therefore, as a man, you are throwing stones into your own windows by believing anything such government says.

As a matter of fact, I am hostile towards that kind of government. I consider them to be enemies; which means that I will pretty much always prefer to side with their enemies.

I have absolutely zero sympathy for that kind of dangerous liars.

What is really described by this theory is some form of order arising from another form of order, not order arising from disorder. — Metaphysician Undercover

That is actually a reasonable interpretation. I can live with that. It is just that the people involved in working on that theory have developed their own lingo and views. I don't feel like arguing with them over this, really.

Maybe you didn't know that Nobel prizes might be given to deceivers. — Metaphysician Undercover

Well, The Nobel Prize system is probably full of deceivers. Nassim Taleb is certainly sure of that:

Taleb has called for cancellation of the Nobel Prize in Economics, saying that the damage from economic theories can be devastating.

Taleb and Nobel laureate Myron Scholes have traded personal attacks, particularly after Taleb's paper with Espen Haug on why nobody used the Black–Scholes–Merton formula. Taleb said that Scholes was responsible for the financial crises of 2008, and suggested that "this guy should be in a retirement home doing Sudoku. His funds have blown up twice. He shouldn't be allowed in Washington to lecture anyone on risk."

Haug and Taleb (2011) listed hundreds of research documents showing the Black–Scholes formula was not Scholes' at all, and argued that the economics establishment ignored literature by practitioners and mathematicians (such as Ed Thorp), who had developed a more sophisticated version of the formula.[89]

And then there is, of course, also the acrimonious insult fests with Stiglitz and Krugman about bitcoin. These two Nobel laureates are hated by the bitcoin community for their deceptive remarks.

Furthermore, the canonical prize in mathematics is the Fields medal. It is normally not possible to get the Nobel prize for mathematics.

However this isn't chaos giving rise to order. Anyway I can now conceive of order arising from chaos but such events would be improbable and short-lived. Of course we mustn't forget the qualifier "relative" for "improbable" and "short-lived". — TheMadFool

Imagine that a thing can improve its own stability by forming an equilibrium with other things, who also improve their own stability by doing that. In that case, that equilibrium is a super-thing with these existing things as sub-things. John Nash's paper "Equilibrium points in n-player strategy games" enumerates the precise conditions in which this will happen.

So, now we have super-things. Of course, the process just repeats, because super-things may be able to improve their own stability by forming an equilibrium with other super-things. So, layer after layer, you get a composition process that yields increasingly improbable structures that are increasingly stable.

It explains why a large molecule tends to be more stable than free-floating atoms, which in turn, tend to be more stable than free-floating electrons, protons, and neutrons. This large molecule, will become even more internally stable, if it becomes part of a living body, which tends to be longer-lived than its constituent molecules (Within reason, that body has a process of replacing broken sub-things).

A human body may live for almost 80 years, but its constituent free-floating electrons, protons, and neutrons are very short-lived. If these constituent parts want to stay around longer, and they seem to do, they will have to join an equilibrium with other similar parts. The influence of these other parts through the equilibrium will substantially improve their own internal stability, and therefore, make them live longer.

And you weren't? Come now... How is it that you and you alone managed to overcome the deception and manipulation that was visited upon you, and that nobody else in the world could overcome? — Bitter Crank

I think that you purposely misunderstand what I have said. I may have been manipulated too, but I unlearned all of that, or at least most of that, by living in other parts of the world. Furthermore, I think that 80% of the world is not deceived nor manipulated. Therefore, unlike you, I believe that the majority on this planet is not deceived. They clearly think differently from you, but that is because they are right.

Kindly clarify how order may arise from chaos. I thought it was the other way around. — TheMadFool

It is also the other way around. I don't think that anybody is denying the irreversible trend to entropy. However, there are also other principles at play. The likelihood of a particular situation does not matter as much as we may think. For example, a highly-improbable situation may be an incredibly stable game-theoretical equilibrium. Many structures in nature are highly improbable, but they still exist, because they are also incredibly stable.

Then, you have at least one other principle which throws a spanner in the works: group re-normalization. Some elements are substantially more stubborn than others, even in physics. These other elements will just acquiesce to what these few stubborn elements want. That leads again to absolutely improbable outcomes. Nassim Nicholas Taleb writes about that in "The most intolerant wins":

The best example I know that gives insights into the functioning of a complex system is with the following situation. It suffices for an intransigent minority –a certain type of intransigent minorities –to reach a minutely small level, say three or four percent of the total population, for the entire population to have to submit to their preferences. Further, an optical illusion comes with the dominance of the minority: a naive observer would be under the impression that the choices and preferences are those of the majority.

This example of complexity hit me, ironically, as I was attending the New England Complex Systems institute summer barbecue. As the hosts were setting up the table and unpacking the drinks, a friend who was observant and only ate Kosher dropped by to say hello. I offered him a glass of that type of yellow sugared water with citric acid people sometimes call lemonade, almost certain that he would reject it owing to his dietary laws. He didn’t. He drank the liquid called lemonade, and another Kosher person commented: “liquids around here are Kosher”. We looked at the carton container. There was a fine print: a tiny symbol, a U inside a circle, indicating that it was Kosher. The symbol will be detected by those who need to know and look for the minuscule print. As to others, like myself, I had been speaking prose all these years without knowing, drinking Kosher liquids without knowing they were Kosher liquids.

The government will not merely TRY to make you give up resources for aged atheists, they will be successful in making you pay for the luxurious assisted living and skilled care homes we shall require. — Bitter Crank

Well, in that case, these governments will have to improve their game, because they has never managed to make me pay one dollar. You see, taxes are paid by less intelligent people to more intelligent ones. That is why I find it so insulting to pay them, and that is why I have never paid them.

Furthermore, not all governments seek to extract that many taxes.

The local minister of interior affairs has recently clarified that he thinks that foreigners should simply pay their yearly visa fee (a few hundred dollars), and that he does not expect more than that. He obviously knows that he is competing for these few hundred dollars per year with lots of other countries. Hence, the reasonable nature of his views. So, we've got a nice deal going here!

You know, "bring back our jobs" will simply not happen. Either you are reasonable, or else you get nothing.

Actually I was wondering which century you were living in. Seems to be something of a perception-distorting time warp going on here. — Bitter Crank

If you have never lived outside a western country, you do not understand 80% of the world. You will also fail to unlearn the deceptive and manipulative views that you were indoctrinated with from a young age. I do understand why you fail to see that your own views can only be wrong, because you may never have seen something else. In a sense, you will remain ignorant of the truth for the whole of your life.

There is quite a bit of evidence that affluence is a key factor in people opting to have fewer -- far fewer -- children. The theory is that with high survival rates among their children, redundant children are not necessary -- the ones they have will survive. — Bitter Crank

That may have been an interesting observation possibly a century ago.

With most people living in an urban setting nowadays, the poor also do. The reason why the urban poor fail to reproduce, is not because of the survival rate of children. It is because they cannot keep their families together for long enough.

Once a woman has a child with one man, it becomes harder for her to find another man to commit to funding a second one. He would be compelled to provide funding for the first one too, and men generally don't like doing that. Hence, rampant family breakup systematically reduces the number of children per woman.

Having a lot of children requires the same nuclear family staying together for all that time. That just does not seem to happen much outside the context of religious communities.

Further, affluent people don't have to worry about not having children to care for them when they are old and feeble. Affluent people can hire poor people to that sort of work at affordable prices. — Bitter Crank

That may be true for the really wealthy ones, but certainly not for the middle class. They depend in their old age on a unilateral transfer of resources mostly from people who may not even have a middle-class level of income. With the younger generations being increasingly of strong religious background -- otherwise they would not even be there -- the government will try to ask them to give up resources to pay to retired, middle-class atheists. It is obviously not in the interest of these younger generations to do that. For reasons of religion, they will each take care of their own parents. Religion does not suggest in any way, however, that they should fund someone else's atheist parents.

I think one can make an argument (I don't have any stats for it) that it is affluence that leads to atheism. — Bitter Crank

In the West, the economic elite may be mostly atheist, but outside the West, this is certainly not the case. Furthermore, the difference in income is shrinking rapidly. These societies are getting much, much wealthier. I can easily see that around me. Therefore, the idea that affluence leads to atheism is probably just one more western ethnocentric view. Neither the ruling elite nor the economic elite here in SE Asia is atheist.

As I said, your example of game theory starts with the existence of things, which itself implies order. So the theories you refer to do not describe order coming from disorder, only one form of order coming from another form of order. If you believe that these theories describe order coming from disorder, you have been misled. — Metaphysician Undercover

That really depends on how you define "order" versus "chaos" or "disorder". The following definition for self-organization does not seem to use your definition:

Self-organization, also called (in the social sciences) spontaneous order, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when sufficient energy is available, not needing control by any external agent. It is often triggered by seemingly random fluctuations, amplified by positive feedback. The resulting organization is wholly decentralized, distributed over all the components of the system. As such, the organization is typically robust and able to survive or self-repair substantial perturbation. Chaos theory discusses self-organization in terms of islands of predictability in a sea of chaotic unpredictability. Self-organization occurs in many physical, chemical, biological, robotic, and cognitive systems. Examples of self-organization include crystallization, thermal convection of fluids, chemical oscillation, animal swarming, neural circuits, and artificial neural networks.

As I already pointed out, this "self-organization" view in exact sciences has an important foundation in John Nash's Nobel-prize winning (1994) publication (1950), "Equilibrium points in n-player strategy games", which predicts the existence of highly improbably but very stable structure-creating equilibrium-seeking processes. You can find a copy of this theorem-cum-proof in the official database of the "Proceedings of the National Academy of Sciences of the United States of America" (PNAS).

I am obviously not familiar with the nitty-gritty details of how they investigate self-organization in various physical, chemical, and biological systems, as these are highly empirical subjects that seek to guarantee minimum standards of real-world correspondence, and therefore, spend a lot of money and effort on activities such as experimental testing.

I only peruse the mathematical foundations of why it makes sense to think like that.

So, yes, I can somehow "see" the profound implications of John Nash's Nobel-prize winning discovery, and why it strongly suggests a widespread principle of self-organization.

While I certainly agree that system-bound knowledge is based on basic beliefs that must necessarily come from elsewhere, and that the universe itself must necessarily be the result of principles that lie outside of it, I avoid applying that view to observable phenomena WITHIN the universe.

Yes, I've noticed that mathematics has made incredible progress in misleading people. Luckily I'm not one of them. You ought to learn how to read these theories more critically and free yourself from the binds of such deception. — Metaphysician Undercover

Well, we know exactly what the basic beliefs are in mathematics. The axioms are not a secret. Therefore, we strictly control the source of deception. There is no hidden deception in mathematics.

Furthermore, as I already pointed out, mathematics is not about real-world correspondence. Any such claim will first have to go through the hands of empirical disciplines, who will then take responsibility for what they say about the real, physical world.

That is also one reason for my very negative views on constructivism.

Unlike the constructivists, I do not believe that mathematics should directly link to the real, physical world, without regulating and mediating such real-world claim first by empirical, minimum standards for correspondence.

So, yes, I am critical about particular mathematical philosophies, but I also subscribe to other ones. For example, I am quite happy with Platonism, structuralism, logicism, and formalism, which each of them emphasize one aspect of mathematics, which is clearly there to me. I may not agree with all ontological views, for example, by decisively rejecting constructivism, but I also do not reject all of them.

That's true, but still...some people need special rules for dealing with almost anything! :gasp: — Janus

Given the long-term trend in atheist populations of going extinct, as we understand that their currently low birth rates are going to implode even further, the lack of special, i.e. specific rules for marriage and divorce does not look like a particularly good strategy.

Religious people transmit their religion's rules to their children, while atheist people fail to do so because they generally don't have any rules nor any children.

Given the fact that the State seeks to force non-religious rules on marriage and divorce, there is not even a need for extra provocations to fuel the growing hostility that religious demographics have for the atheists.

At the same time, as their numbers shrink and as on average they get older, atheists are understood to find it more difficult to defend themselves, and are therefore turning into increasingly easier targets. In other words, every day that passes by, the likelihood of hostile reprisals keeps increasing. In terms of cost/benefit analysis, the cost of lashing out is shrinking dramatically. It is simply getting cheaper to do that. Furthermore, why do people do what they do? Well, because they can.

Therefore, in my opinion, atheist populations will not just gradually go extinct. They will suddenly do so. In other words, atheism looks like a suicide pact.

I actually don't know how the syntactic incompleteness was proven (through the same meta-logical argument as its semantic counterpart or not), but it seems that you shouldn't need a lot of semantics in order to demonstrate that some statement is independent. If the statement's derivation involves circularity (like the Gödel sentence), it should be entirely a deductive property if this circularity can be eliminated or not. But I might be wrong. — simeonz

I am not familiar enough with Gödel's semantic completeness theorem. The explanation does not contain elaborate examples to illustrate the details of what they are talking about. It would be quite a job to inject examples in the right places.

For example:

But the ring Z of integers, which is not a field, is also a σf-structure in the same way. In fact, there is no requirement that any of the field axioms hold in a σf-structure.

A ring is "almost" a field -- it just lacks inverses (it is not closed under division) -- but that's the little I understand of why this ring is still a model (=σf-structure) of a field ... I hope that Wikipedia will manage to better elaborate on this subject.

I still don't understand the distinction you're looking for. You're obviously not seriously suggesting that there aren't any deontologists, that no one is a utilitarian... That would be absurd. So what is the distinction you're trying to make between people who have read, say, Kant, and try to follow his method, and people who have read, say, the Bible, and try to follow its methods? — Isaac

Ok, let's pick an example: marriage and divorce. Each religion has its own elaborate rules on that matter.

By the way, religious marriage is strongly resurgent. Governments across the world are increasingly losing the power to impose their views in these matters. Governments are simply not the most effective principle at using violence to get their way, as there are clearly much stronger principles at play. In that sense, it is pretty much inevitable that religious marriage will find itself completely reinstated.

What are the deontologist rules on marriage and divorce? Do you know of anybody who has entered into a deontologist marriage? Without rules on marriage and divorce, a system of morality is incomplete, say, even crippled.

Well, in the published works of the relevant philosophers, of course.

You're still not being clear here about what you mean. I'm trying to be as charitable as possible and assume that you're not so poorly educated that you don't even know that people have written books about ethics, but I'm really struggling to understand your question outside of that interpretation. — Isaac

The Bible and the Quran are also books about ethics. These books are really used by entire demographics to determine ethical questions. For example, professing Christianity means that you follow Christian determinations and rulings in ethics. From there on, you have entire communities doing that. What philosophy book has entire communities determining morality according to its text? If such community does not exist, then that book is not being used; which is pretty much the same as saying that it is "useless".

So, as you stated, if ZFC were complete it would be decidable. But since it is not, does my original question - if it is decidable or not still stand? — simeonz

I suspect that ZFC is not decidable, but then again, it really depends on the link between completeness and decidability. If there exists a procedure to solve the proof problem, the proof problem is decidable. That means that the theorem is provable from the theory. In that sense, provability is a decidability problem, because a proof is a procedure.

However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF may only be arrived at via a meta-analysis from outside the system. — Wikipedia

That is actually what they do in the Wikipedia page.

This contradicts the summary, by clearly stating that validity is not the subject of the theorem. — simeonz

I think that it is the semantic completeness theorem that throws a spanner in the works: For a statement deemed true by the system, some proof must exist within the system. Therefore, the result would be contradictory, unless the truth of the Gödel sentence is evaluated outside the system. In my impression, the Gödel sentence is true in the metasystem but not provable in the system itself. One of the difficulties in this matter is that some reasoning takes place within the system, and some about the system, from outside the system.

The problem with this analogy is that you already assume the existence of "a thing", and this implies order. "A thing" is an ordered existence. Lack of order would actually mean a lack of things. In Aristotelian terms a lack of order would simply be the "potential" for existence of a thing. So if you are describing how order comes out of non-order, you cannot start with the existence of a thing, because this is to presume the existence of order already. — Metaphysician Undercover

I don't think that Aristotle was particularly familiar with self-organizing systems or the concept of spontaneous order:

Spontaneous order, also named self-organization in the hard sciences, is the spontaneous emergence of order out of seeming chaos. The evolution of life on Earth, language, crystal structure, the Internet and a free market economy have all been proposed as examples of systems which evolved through spontaneous order.[1].

It is closely related to game theory, which is largely based on the work of John von Neumann and especially John Nash, after the second world war:

Game studies. The concept of spontaneous order is closely related with modern game studies. As early as the 1940s, historian Johan Huizinga wrote that "in myth and ritual the great instinctive forces of civilized life have their origin: law and order, commerce and profit, craft and art, poetry, wisdom and science. All are rooted in the primeval soil of play." Following on this in his book The Fatal Conceit, Hayek notably wrote that "a game is indeed a clear instance of a process wherein obedience to common rules by elements pursuing different and even conflicting purposes results in overall order."

The principle of emergent behaviour is a similar concept:

An emergent behavior or emergent property can appear when a number of simple entities (agents) operate in an environment, forming more complex behaviors as a collective. Systems with emergent properties or emergent structures may appear to defy entropic principles and the second law of thermodynamics, because they form and increase order despite the lack of command and central control. This is possible because open systems can extract information and order out of the environment.

Maybe you do believe this, but you seem to misunderstand what "chaos", or complete lack of order really entails. — Metaphysician Undercover

You seem to be unfamiliar with the concepts of "spontaneous order" and "emergent behaviour" which are quite modern, only a few decades old, actually. Maybe it would make sense for you to read some publications from after the second world war. Unlike metaphysics, mathematics has made incredible progress in the 20th century.

My understanding from the definition was that for a theory to be decidable, it is necessary to have effective enumeration of its theorems, not to have a theorem for every statement or its negation. — simeonz

Effective axiomatization. A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2005, p. 112).This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC).

Completeness. A set of axioms is (syntactically, or negation-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p. 24).

The issue of syntactic completeness only arises in theories that have effective enumeration of its 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.

Hence, only theories with effective axiomatization can possibly be incomplete. Other theories are not even being considered in this context.

defending Europe from Moslem armies — Bitter Crank

That was rather:

preventing the non-Chalcedonian Christians from inviting the Moslims in -- or before that, the Persians -- to get rid of Byzantine-Chalcedonian religious persecutions.

For example, how did the Moslims get into Spain and southern France, if not to assist the Arians in their fight against the Catholics (=Chalcedonians)?

Julian, Count of Ceuta (Spanish: Don Julián, Conde de Ceuta,[nb 1], Arabic: يليان‎, (Īlyan [nb 2]) was, according to some sources a renegade governor, possibly a former comes in Byzantine service in Ceuta and Tangiers who subsequently submitted to the king of Visigothic Spain before joining the Muslims.[3]:256 According to Arab chroniclers, Julian had an important role in the Umayyad conquest of Hispania, a key event in the history of Islam, in which al-Andalus was to play an important part, and in the subsequent history of what were to become Spain and Portugal.

Every Muslim conquest followed the same pattern. The region was inhabited by non-Chalcedonian Christians who were sick and tired of the Byzantine religious persecutions, and who were happy to invite the Muslims with a view on expelling the Byzantines; because the Muslims had promised religious freedom. Furthermore, it is because the Muslims kept their promise of religious freedom that it was so hard for the Chalcedonians to ever come back.

We can run an experiment with two rooms A and B. A is in disarray with things in no particular order and B is neat and objects have been arranged in a discernable pattern. If someone, anyone, were to be taken into the two rooms and asked which room probably had an occupant then the answer would invariably be room B. I don't think anyone will/can disagree with this deduction. — TheMadFool

Just like the earth reveals through subtle hints that it is must be rather spherical, the universe reveals through subtle hints that it must have a beginning. From there arises the idea of a first cause, God.

The fact that order appears out of chaos, however, does not strike me as particularly special, or even as being such hint.

Separate things in an otherwise chaotic system will spontaneously enter and stay in a highly-improbably game-theoretical equilibrium, when such equilibrium is very, very stable.

John Nash describes the conditions in which such equilibrium will arise in his famous 1950 publication, "Equilibrium points in n-player strategy games".

Say that a thing maximizes its own integrity. If it can enter a situation in which other things contribute to its own integrity, it may favour to stay in that situation. If these other things can also maximize their own integrity by maintaining that situation, then none of the things involved, is willing to change the situation. Such situation may be highly improbable, but once it exists, it will refuse to disappear. So, that creates a new, stable thing consisting of a game-theoretical equilibrium between sub-things.

In a next stage, that new, stable, composed thing can improve its own stability by becoming a member in yet another super-thing. If all the things involved react in the same way, you get again a new situation with a super-thing that consists of things that themselves consist of sub-things. That composition pattern just keeps going on, and creates increasingly improbable results, but that are also increasingly stable.

So, incredibly complex and orderly situations tend to arise pretty much spontaneously from chaos. As far as I am concerned, they do not necessarily point to an underlying design. They could just arbitrarily be satisfying the conditions of particular game equilibria.

I also couldn't clear up if ZFC is decidable, undecidable, or not yet established. Wikipedia indicates that there are only decidable sublanguages, while a stackexchange answer indicates that ZFC is recursively enumerable, which if mu-recursively enumerable, should mean that ZFC is decidable by Turing machines. — simeonz

ZFC is subject to the diagonal lemma, and therefore is syntactically incomplete. Hence, it must necessarily have undecidable statements that can be expressed in first-order logic.

Since the semantic incompleteness proof holds for the standard model, what is the "intended" interpretation for other theories. — simeonz

Every legitimate answer that is true, has a proof (semantic completeness), but there exist legitimate questions for which an answer cannot be determined (syntactic incompleteness).

Concerning the implications of Gödel's semantic completeness theorem for other systems than Peano arithmetic, we are talking about advanced model theory, which is an enormous subject in itself. I am not sure about the answer, really. I have never looked into it.

I find the Curry-Howard Correspondence a little strange. I'm sure it makes sense, but likening axioms to a pre-execution invariant and theorems to a post-execution invariant appears complicated. It may have something to do with formal verification processes, but for me, the relationship between proofs and computation appears to be about enumeration of proofs by turing machines in one direction, and the generation of booleans on the Turing tape for the proven theorems after every inference step in the other direction. — simeonz

In 1958 [Curry] observes that a certain kind of proof system, referred to as Hilbert-style deduction systems, coincides on some fragment to the typed fragment of a standard model of computation known as combinatory logic.[4]

In order to illustrate this understanding, it is clearly a question of producing good examples of Hilbert calculi as well as good examples of combinator calculi, along with an argument that maps an example in one formalism to an example in the other. It is a lot of work, but it should allow to develop a deeper understanding of the nitty-gritty details of what exactly Curry meant to say.

It is interesting to work on such explanation by examples, but then again, who else would be interested in reading the results? This discussion on the diagonal lemma is probably already more than what many potential readers would want to handle ...

I think that's mistaken. Modern scientific atheism, of the kind advocated by popular science commentators, is constructed from the hollowed-out shell of Christian philosophy. — Wayfarer

Well, yeah, with Pauline Christianity having crossed out half of its basic scripture and mostly abrogating the real system, i.e. "The Law and the Prophets", i.e. full Jewish Law, such hollowed-out atheist shell is a hack on something that was already a hack. It is clearly not possible to solve the problems caused by a hollowed-out system by removing even more parts. So, what's left over then?

I don’t see where you’re coming from with this whole “atheists don’t have systems” thing. For myself, my philosophy is extremely systemic, probably more so than is academically popular in Anglophone countries today, and I end up being an atheist as a consequence of that system. — Pfhorrest

A system is described by its basic rules, i.e. its basic beliefs.

Propositional logic itself is a system of 14 basic beliefs.

Every other system is necessarily an extension of a core logic system. You do that by importing a module of extra basic beliefs next to the logical core:

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.

For example, if you load an additional module with 9 extra beliefs concerning standard arithmetic, you can use the system as a number theory.

The basic logical core cannot do arithmetic. The system will have to load an additional module for that. The basic logical core can also not do morality either. It will again need to load an additional module for that. Someone who thinks that the logical core is enough to doing anything at all, does not understand how a system works.

You could load Moses' 10 commandments as an extension module, and then you would already have some kind of moral system. It is generally considered incomplete, but it would still illustrate the principle of (axiomatic) moral system.

Furthermore, it is not possible to use logic to argue against the idea that you will need to load modules of basic beliefs into the system, because logic itself is just such module.

Where can I find a copy of your extension module of basic moral beliefs?

Well yes, but we're clearly not talking about the same thing because it's absolutely obvious that there are - several brands of deontology, utilitarianism (negative utilitarianism, motive utilitarianism... ), virtue ethics (in dozensof different forms). I mean the vast majority of ethical systems don't involve God. So what is it you're getting at? — Isaac

Where is any of that documented? Where do these communities live, who actually implement it?

Utilitarianism and Kantianism both make no reference to gods and so are entirely practicable by atheists. — Pfhorrest

Concerning Kant's philosophy, he described some meta-ethical principles but did not mean to provide a new system of morality. His work on ethics was certainly not to be understood as a replacement of Christian morality but rather as a rational elucidation.

Kant was religious:

Not only do we find powerful defenses of religious belief in all three Critiques, but a considerable share of Kant's work in the 1790s is devoted to the positive side of his philosophy of religion.

Now, given the fact that Christianity does not particularly have an elaborate system of jurisprudence, i.e. a complete system of religious law, very much unlike Judaism and especially Islam, this approach will by itself not really solve the problem.

It is most likely that followers of Jesus were originally supposed to implement "the Law and the Prophets", i.e. Jewish Law in full, including the Oral Torah. The Ebionite branch of Christianity actually did that. It is by exempting non-Jewish Pauline Christians from the Law, that the jurisprudential conundrum started snowballing. Without Jewish Law, there is no guarantee that the religion is complete and can offer moral guidance in all circumstances. Kant understood the problem of questionable thinking in morality that naturally arises in an incomplete system in which half the scripture gets abrogated, but he only offered some useful meta-ethical guidance without being able to provide a complete system. Kant could not reinstate the missing parts by virtue of meta-ethics alone.

So, no, Kant is not a basis for a complete morality, atheist, Christian, or otherwise.

But if so, I still don't understand what you could possibly mean by this. To take morality (the system you alluded to) there's dozens of atheistic moral systems (moral systems which do not involve God), in fact probably more than there are religious ones. So why aren't these counting in your estimations? — Isaac

Is there one example of a documented, atheist system for morality with at least some followers?

Either it is reasoning within a system, or else about a system, because in all other cases it is just system-less bullshit.

My life experience says that everything that makes sense is in one way or another structured as a system. So, what would I pick: A religious system or the atheist non-system?

In the end, atheism does not build any system. Atheism only rejects religious systems, without building anything else instead.

You see, I also reject particular systems. I deeply resent Windows. So, I use Linux. I have contempt for the fiat banking system. So, I save my money in bitcoin. An atheist also dislikes particular things, but he does not propose or use any alternative.

• An atheist would say: Do not use operating systems. Not Windows and not Linux (nor MacOSX). Use nothing, because all operating systems are bad.
• question: But how will you run your programs in that case?
• Atheist: Well, we do not need to run programs. Using a program would cause us to use an operating system, and therefore, programs are also bad.

In fact, an atheist will eventually, and grudgingly, still try to secretly run programs (=draw moral conclusions), but without using an operating system, but then his system-less bullshit will simply fail to take off. He will never admit that, however, because he has already declared that running programs (=drawing moral questions) is bad; all of that, without actually having a system to determine what is good or bad.

If I understand you correctly, there are axiomatic systems which preserve the truth values of all statements of Peano arithmetic, but make previously undecidable statements decidable? — simeonz

By the way, there is a problem with the terminology. Decidability is about computability:

Decidability should not be confused with completeness. For example, the theory of algebraically closed fields is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable. Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym for independent statement.

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer.

While independence is about provability:

In mathematical logic, independence is the unprovability of a sentence from other sentences. A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem.

There seems to be a preference to use the term "independent" (mathematics) instead of "undecidable" (computer science) for "unprovability" of the statement or its negation. The problem is caused by the fact that these terms emerged from two different domains, but are very related. Given the Curry-Howard Correspondence, undecidability and independence overlap much stronger than suggested by these commentators:

In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs.

The Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—are in fact the same kind of mathematical objects. If one abstracts on the peculiarities of either formalism, the following generalization arises: a proof is a program, and the formula it proves is the type for the program.

The difference between "undecidable" and "independent" seems to depend mostly "on the peculiarities of either formalism" rather than on truly fundamental issues.

The canonical example of a theorem not provable in Peano arithmetic but in ZFC is:

Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory.

In fact, the commentators below argue that this does not mean that ZFC is "stronger" than PA. The fact that ZFC accepts the theorem while PA doesn't, is rather a trust issue.

So, we don't actually have an axiomatic system yet, in which all statements of Peano arithmetic are decidable? — simeonz

I have found the following (formal) thought exercise on exactly this matter. The author calls such encompassing axiomatic system a "provability oracle for Peano arithmetic":

So having a provability oracle for PA or any other consistent formal system that proves some valid arithmetic truths (like ZFC) is equivalent to having a halting oracle, and thus leads to a provability oracle for any other formal system. In other words, if you knew all about the logical implications of PA, then you would also know all about the logical implications of ZFC and all other formal systems. Hee hee.

PA can see everybody else's theorems as Gödel numbers. So, it actually "knows" everything what these other systems know.

Then, I found the following intriguing remark as a comment to the article on the provability oracle for PA:

More powerful formal systems are not more powerful because they know more stuff. They're more powerful because they have more confidence. PA proves that PA+Con(PA) proves Con(PA) (the proof, presumably, is not very long), but PA does not trust PA+Con(PA) to be accurate. ZFC is even more confident - has even fewer models - and so is even stronger. (Maybe it even has no models!)

ZFC is not more powerful than PA because it knows more stuff. In fact, that would not be possible. Such theorem (as well as its proof) would still be a Gödel number which PA can also reach. Therefore, PA actually "knows" all possible theorems (and their proofs) that can be expressed in language. Therefore, it also knows all the axioms and all theorems (including their proofs) of all the other systems. The problem is that PA does not "trust" them.

Therefore, ZFC does not "know" more than PA. It just trusts more. In some ways, that is not really "stronger" or "more powerful". It could just be that ZFC is more gullible.

Isn't this just another way of saying the Quran is miraculous - the faculty of reason being incapable of producing the Quran? — TheMadFool

The terms "miracle" and "miraculous" are loaded with all kinds of connotations that sometimes sound questionable. I prefer the phrase: "the output of other, unknown mental faculties".

Look, it is not possible to come up with the 9 unexplained, speculative beliefs of number theory, just like that, just "by reasoning". You will simply never get there by reason alone. Same for the 14 speculative beliefs that construct the system of propositional logic, or of the 10 arbitrary beliefs at the basis of set theory. I do not know what their true origin is, and nobody else does either. Otherwise they would not be axioms.

In other words, all our mathematical beliefs go back to seemingly arbitrary, speculative, basic beliefs. Why would anybody right in his mind believe that our religious systems would be any different? On what grounds?

We simply do not have any knowledge at all that is not of seemingly arbitrary, speculative nature. Our own heads are replete with mental faculties that we do not understand and that we cannot reasonably explain.

So, doesn't that prove my point that some form of miracle is necessary if to be a prophet? — TheMadFool

Depending on how you define the term "miracle", yes. The prophet of Islam is widely believed to have listened into a stream of transcendental messages of which the origin was not of this world. People generally do not know how to do that because otherwise, they would be doing that all the time ...

I personally don't accept miracles as evidence for the simple reason that advanced knowledge masquerades as miracles. — TheMadFool

Evidence is the argument/proof element in the three-tuple (premises, conclusion, argument). It is even tautological that you cannot swap these elements around. You have to find a way to discover premises. You also have to find a way to discover a conclusion, and most of all, you need to find a way to connect them. None of these activities are rational processes. The main characteristic of the greatest mathematicians and the greatest scientists was not rationality, because the process of knowledge discovery is not rational at all.

Furthermore, as I already argued, swapping around premises and argument in the three-tuple (premises, conclusion, argument) does indeed not make sense. However, it is you, yourself who are trying to do that. You want to solve the problem of (underlying premises, existing premises, unknown). As I argued previously, your request is itself not rational at all; and it is even trivial to rationally argue that.

They say the Quran is the miracle of Muhammad. Why? What's so miraculous about the Quran? — TheMadFool

The mental faculty of reason is not capable of explaining, but also not of producing, the basic beliefs of a system. These basic beliefs, i.e. system-wide premises are the output of other, unknown mental faculties. This is true for all systems, not just religious ones.

Say that reason is a predicate function that accepts three inputs (premises, conclusion, argument) and produces a yes or a no, as to whether these three inputs successfully pass the verification test, which is: we can confirm that according to the argument, the conclusion necessarily follows from the premises. We carry out this verification procedure by checking if each step necessarily follows from the previous one.

Now, can reason provide a missing argument/proof? No, we have a beautiful example in the Riemann hypothesis. We know the conclusion. We also know the basic beliefs of the construction logic of number theory. Nobody has been able to find counterexamples for this hypothesis either. Still, nobody has been able to find the path between the construction logic of number theory and the Riemann hypothesis either. If reason alone could produce the argument/proof, then we would have it already, but we don't. Reason cannot solve the following expression:

isReasonable(premises,conclusion,unknown)

It is obvious that the argument/proof is function of premises and conclusion, because it represents a path between premises and conclusion. So, can reason tell us what the underlying premises are for a particular set of existing premises?

isReasonable(underlying premises, existing premises, unknown)

No, impossible, because we had already established that reason alone cannot provide the unknown argument/proof. Therefore, not only the argument/proof is the output product of other, unknown mental faculties, but so are also the premises.

Hence, the basic beliefs, i.e. system-wide premises, of any system, including religious ones, is the output of unknown, other mental faculties, and not of the faculty of reason.

So, what exactly is there exceptional in providing system-wide premises for a religion? Well, the fact that it cannot be unexplained from anything we know. Therefore, in religious lingo, these system-wide premises are considered to be the transmission of a transcendental message from another world. Can this be further explained? No, obviously also not. We are already "reasoning" about system-wide premises, while, as argued above, reason does not play any role in providing them.

So, how do we know that such system-wide premises are truly the transmission of a transcendental message from another world? As argued above, this question is out of scope for the mental faculty of reason. Therefore, here again, you will have to use other, unknown mental faculties to determine this.

Other, unknown mental faculties generally play a much more important role in the nature of systems (which are themselves reasonable) than reason itself. That is why the average reasonable person, no matter how reasonable he may be, cannot produce something like Einstein's 1905 Nobel-prize winning publication. It is simply not a matter of reason.

This also explains why some people seem to be able to listen into transmissions of transcendental messages, while everybody else, no matter how reasonable, does not have that mental faculty. These other people still somehow seem to be able to verify that the transcendental message is sound and valid. From there on, these other people will believe the message transmitter.

In fact, most of us actually know that reason does not explain quite a bit of otherwise successful behaviour. A successful football player will carry out a spectacular manoeuvre, but in fact not be able to rationally explain why he knew how to do that. You simply cannot train professional football players by making them read lots of books on football. It simply does not work that way, and people actually know that.

Correct but the original sources (prophets and books) are supposedly verified through miracles which people seem to accept as true. The next generation of preachers rely on these primary sources for their own authenticity. Right? — TheMadFool

Well, no.

The prophet of Islam did not perform one single miracle, besides providing us with a copy of the Quran.

Sunni scholar Muhammad Asad summary on the matter. In many places the Qur'an stresses the fact that the Prophet Muhammad, despite his being the last and greatest of God's apostles, was not empowered to perform miracles similar to those with which the earlier prophets are said to have reinforced their verbal messages. His only miracle was and is the Qur'an itself - a message perfect in its lucidity and ethical comprehensiveness, destined for all times and all stages of human development, addressed not merely to the feelings but also to the minds of men, open to everyone, whatever his race or social environment, and bound to remain unchanged forever…

So, no, religion does not need miracles. That is simply a very false impression.

Therefore we cannot prove a statement that cannot be proven in this axiomatic system is semantically true, and to the extent of our logical method, this new axiomatic system can never be proven incomplete. — simeonz

I think that there are two remarks that are possibly relevant in this context.

First. These "true but unprovable" statements will appear to use as simply undecidable. Indeed, how are we even going to decide that they are "true", if we cannot possibly prove the statement nor disprove its opposite? Hence, the practical qualification will not be "true but unprovable" but "undecidable".

Second. Number theory (by default: Dedekind-Peano) is a sub-theory of set theory (by default: ZFC), meaning that every number-theoretical theorem can be proven in set theory, but not the other way around. Now, there really are number-theoretical theorems that are fundamentally undecidable in number theory (=DP), but provable in the larger set theory (=ZFC):

Undecidable statements provable in larger systems

These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.

In 1977, Paris and Harrington proved that the Paris–Harrington principle, a version of the infinite Ramsey theorem, is undecidable in (first-order) Peano arithmetic, but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic.

Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.

Now, we may not have fundamentally stronger theories than set theory (=ZFC). In other words, there may not be such larger theory of which set theory would just be a sub-theory. Therefore, when a theorem is undecidable in set theory, it may even be absolutely undecidable. One reason why we may not go higher nor further than ZFC, is because ZFC is already at the level of Turing-completeness -- can compute everything that is computable and therefore: can represent all knowledge that can be expressed in language. ZFC is possibly even stronger than that:

What is the relationship between ZFC and Turing machine?

In the other direction, a universal Turing machine can be encoded in ZFC, indeed in far weaker theories, such as relatively small fragments of number theory.
...
Suppose, for reductio, that a Turing machine could decide whether the sentence with Gödel number n
is a theorem of ZFC ... Now, by the diagonalization lemma, there would be a Gödel-style sentence γ such that ... Contradiction is immediate. [meaning: ZFC is stronger than Turing-complete]

On the other hand, Turing Completeness is already a seriously constraining limit:

All known laws of physics have consequences that are computable by a series of approximations on a digital computer. A hypothesis called digital physics states that this is no accident because the universe itself is computable on a universal Turing machine. This would imply that no computer more powerful than a universal Turing machine can be built physically.

This does not mean that there are no attempts at increasing the strength of ZFC by adding new axioms. The web is plastered with such attempts.

Still, successfully arguing that a new axiom is independent of the ten existing axioms in ZFC is a non-trivial job, given the strength of ZFC. After that, they still need to successfully prove from this extended system, a theorem that is provably undecidable in ZFC, such as for example the Continuum Hypothesis. That job is certainly not easy ...

If no true but unprovable X has been found to satisfy "X ↔ isNotProvable(%X)", then why should we consider it to be a satisifiable definition? — Andrew M

How it materializes in practice is rather that we find a statement S, of which we can prove that neither S nor the opposite of S can be proven in theory T (such as the standard theory of arithmetic). So, is this statement true or false? Well, how can you know this, since you cannot prove anything about it? So, in all practical terms, we do not run into true statements that are unprovable, but into undecidable statements. There are lots of witness statements for fundamental undecidability, some of which must indeed be true, but how can we know that?

Also, would "X ↔ isNotTrue(%X)" be considered a satisifiable definition? I assume not, but that then raises the question of the criteria for judging that a definition is satisfiable. — Andrew M

The conclusion of Tarski's undefinability theorem is that the "IsNotTrue" predicate is simply not allowed in arithmetic:

$\begin{bmatrix} sentence/\boldsymbol{\color{green}{IsNotTrue}} & false & true \\ false & \color{blue}{\boldsymbol{(false,false)}} & (false,true) \\ true & (true,false) & \color{blue}{\boldsymbol{(true,true)}} \end{bmatrix}$

Both slots on the diagonal are poisonous.

The first slot (false,false) means that there exists a false statement that is not "isNotTrue" (meaning, which is true). Hence, that would be a false statement that is true. The second slot (true,true) means that there exists a true statement that "isNotTrue". Hence, that would be a true statement that would be false. Since the diagonal lemma says that there exists a sentence that will hit the diagonal at least in one of both slots, the "isNotTrue" predicate will always lead to the existence of a liar sentence, i.e. an inconsistency.

Therefore, "isNotTrue", "isTrue", "isFalse", and "isNotFalse" cannot be defined as predicates. It is simply not allowed to put them in the table in the green spot. That is the gist of Tarski's undefinability theorem (or truth theorem):

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that arithmetical truth cannot be defined in arithmetic.The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. The undefinability theorem is conventionally attributed to Alfred Tarski. Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 publication of Tarski's work (Murawski 1998). While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann.

The formal statement of Tarski's undefinability theorem is, of course, expressed in terms of the diagonal lemma:

That is, there is no L-formula True(n) such that for every L-formula A, True(g(A)) $\leftrightarrow$ A holds.

So, there does not exist such number predicate True(%S) because there would always exist exceptions to the proposition that: S $\leftrightarrow$ True(%S). That would render the entire theory inconsistent.

OK, so my understanding is that one sentence that hits (true,true) for isNotProvable is the sentence that asserts that it is itself not provable. How is that expressed as a mathematical sentence? — Andrew M

X $\leftrightarrow$ isNotProvable(%X)

The diagonal lemma does not tell you, however, how to find that true sentence. It only guarantees that it exists.

Also, why is it thought to be true? — Andrew M

Because such true sentence exists. With "isNotProvable" being a number predicate, there is an X that satisfies it. We do not need to actually find this sentence X.

Is it simply that assuming that it is provable leads to contradiction? — Andrew M

This true sentence X has the property "isNotProvable". It is not about a contradiction. It is just about the fact that we can define the "isNotProvable" predicate as a number predicate. From there on, there simply exists a true sentence X that satisfies the diagonal lemma.

Searching for such true witness X, requires you to actually construct the "isNotProvable" predicate first. That can, however, only be done as a thought exercise. That is why we do not really search for X. We are already satisfied with the theorem that it must exists.

OK, so to go back to the third step in your initial post, K could be isNegative. And so isNegative(%M) is false. Then, per the fourth step, any false sentence will have the property isNegative. — Andrew M

Yes, with false statements already hitting the diagonal (false,false) there is no compelling reason why true statements would hit the diagonal in (true,true). It is simply not possible to make isNegative(%S) return true. So, I don't think that (true,true) could ever occur.

Given the above, it seems that there doesn't have to be a true statement that is not provable. There could instead be a false statement that is provable. So it would be a choice between incompleteness and unsoundness? — Andrew M

Yes, that is exactly what Gödel concluded. Either the system is incomplete, hitting the diagnal in (true,true) or else it is inconsistent (=unsound), hitting the diagonal in (false,false). Let me check the exact phrasing of this conclusion:

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.

The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized, these concepts being detailed below.

There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.


Now, then the commentators argue that standard number theory "seems' to be consistent:

The theory of first order Peano arithmetic seems to be consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete.

Therefore, the (false,false) box in the diagonal was originally not "considered" possible (false and provable) because arithmetic was "considered" consistent. This statement became quite questioned, until at some point it was actually "mostly" proven:

In 1958, Gödel published a method for proving the consistency of arithmetic using type theory.[21] In 1936, Gerhard Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0.[22] Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0 can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.

The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers.

So, there is not 100% agreement on the idea that Gentzen's proof of the consistency of number theory is actually finitistic, because the natural numbers themselves are an infinite collection.

So, yes, your interpretation is considered correct: Either a (number-theory-including) theory is inconsistent or it is incomplete. They argued for long after Gödel's initial lecture in 1930 over this conclusion, and the matter was barely settled in 1958.

Standard number theory is considered not to be hitting the diagonal in (false,false) (=inconsistent) but in (true,true) (=incomplete) because its consistency proofs have been accepted.

But how about the property isNegativeNumber? That property will never be true. — Andrew M

Actually, you caught me on a semantic problem. I should not have called the property "isLargeNumber". I should simply have called it "isLarge" because it becomes a property of the sentence.

Mutatis mutandis, we should deal with the property "isNegative" and not "isNegativeNumber", even though we will calculate it by checking if the number is negative (which is always false for a utf8 decimal).

Let say that S is a false sentence. isNegative(%S) is false, simply because isNegative of any %X is false. So, since they are both false, the expression:

S $\leftrightarrow$ isNegative(%S)

is on the diagonal, i.e. in the box (false,false). This means that the sentence S has the isNegative property, even though %S is a positive number.

This is the part I don't understand. How is this proved? — Andrew M

That is exactly the diagonal lemma.

Take any property, e.g. isLargeNumber. Say that isLargeNumber is true for numbers above 10^20, and false for numbers that are smaller.

Now, the diagonal lemma says that you can always find a true sentence for which isLargeNumber is true. You can also always find a false sentence for which isLargeNumber is false.

Forget about Gödel's incompleteness theorem as long as you do not completely grok the diagonal lemma. Make exercises on the lemma, until you thoroughly understand it.

How do you feel about all the preachers indoctrinators proselytizers out there, then?
4th Grade Science Quiz (David Mikkelson, Snopes, Apr 2013) — jorndoe

Those people are not a good example.

Pick a serious theologian at the Catholic University of Turin to talk about Catholic theology. Or pick an experienced Rabbi at a Sanhedrin of your choice to solve jurisprudential questions in Jewish law; or choose an experienced mufti who lectures at the Islamic University of Alexandria. These people understand their own system very, very well.

Seriously, you are going for the wrong crowd. The people whom you have chosen, are not to be taken seriously. They simply known nothing about their own subject.

But if this is just an analogy for something like a referee, then the origin of logic would still need to be explained. — Teaisnice

Logic is an axiomatic system based on 14 speculative beliefs, i.e. axioms. Logic cannot explain its own basic rules. There is simply no motivation for these basic beliefs that can be reached through reason.

Hence, the basic beliefs at the origin of logic -- pretty much like all other basic beliefs -- are supplied by other, unknown mental faculties that are not documented, and that cannot be explained reasonably.

Now look in detail at how reason functions.

Reason accepts three inputs: premises, conclusions, and an argument by which the conclusion necessarily follows from the premises. So, reason is a verification predicate function with three inputs and one yes/no output:

isReasonable(premises,conclusion,argument) = yes/no

Can reason by itself supply the argument? In other words, will reason be able to fill in the missing argument by itself?

isReasonable(premises,conclusion,unknown) = yes/no

No, because in that case we would, after 160 years, already have discovered the proof/argument for the Riemann hypothesis.

It is not because we can see a pattern that we consistently fail to falsify -- for which nobody has been able to discover a counterexample -- and it is not because we also know the premises of number theory -- its nine basic beliefs -- that we can supply the argument that will connect the Riemann conclusion to its premises.

Now the next question. Can reason discover the premises that will explain existing premises? In other words, is the following question solvable?

isReasonable(unknown premises, premises, unknown)

No, because as we can see, if the premises are unknown then the proof/argument is also unknown, and as argued in the previous case, reason by itself cannot discover the proof/argument.

Therefore, it is absolutely not possible that it would be reason that supplies the basic beliefs.

It cannot do that.

Therefore, "reasoning" about the validity of basic beliefs is simply an exercise in futility. You are then trying to use a tool of which we know that it cannot deliver what you want.

I'll let you and Dingo hash the political issues out, in the meantime and in a similar way, I think you would agree that here in America our currency suggests the merits of Deity tipping the scales in favor of Christianity. — 3017amen

Yes, but Christian rules must not be imposed onto, for example, Jews. I absolutely do not believe in doing that.

If you confess to being a Christian, then you are seeking to keep Christian rules, and therefore, you implicitly declare Christian rules to be applicable to you. That is in my opinion the reason why Christians can be held against Christian law.

As far as I am concerned, there is no other reason for being held to confessional rules than your own confession.

And if religious people control the government, wouldnt they be the ones imposing? Wouldnt they be imposing on the atheists? Wouldnt they be imposing on other religions with different practices? — DingoJones

Some religions do, but other religions absolutely do not.

In the Ottoman Empire, a millet /ˈmɪlɪt/[1][needs Turkish IPA] was an independent court of law pertaining to "personal law" under which a confessional community (a group abiding by the laws of Muslim Sharia, Christian Canon law, or Jewish Halakha) was allowed to rule itself under its own laws.

The Ottoman term specifically refers to the separate legal courts pertaining to personal law under which minorities were allowed to rule themselves (in cases not involving any Muslim) with fairly little interference from the Ottoman government.[12][13]

The millets had a great deal of power – they set their own laws and collected and distributed their own taxes.

This is a direct consequence of numerous provisions in the Quran and the Sunnah which strictly forbid applying Islamic law to non-Muslims, the most important of which is the testimony in which the prophet of Islam administered law between Jews out of the Jewish scripture:

Narrated Abdullah Ibn Umar: A group of Jews came and invited the Apostle of Allah (peace_be_upon_him) to Quff. So he visited them in their school. They said: AbulQasim, one of our men has committed fornication with a woman; so pronounce judgment upon them. They placed a cushion for the Apostle of Allah (peace_be_upon_him) who sat on it and said: Bring the Torah. It was then brought. He then withdrew the cushion from beneath him and placed the Torah on it saying: I believed in thee and in Him Who revealed thee. He then said: Bring me one who is learned among you...... Then a young man was brought. The transmitter then mentioned the rest of the tradition of stoning similar to the one transmitted by Malik from Nafi'(No. 4431).

As you can understand from this narrative, an Islamic ruler will not administer Islamic law in a Jewish dispute or concerning a Jewish criminal offence. Instead, it is the Jewish religious scholars (Rabbis) who will be habilitated to adjudicate the case.

Of course, this practice only applies if such community has a documented system of law, i.e. a "scripture". There is no requirement for any ruler to recognize undocumented law.