It would be interesting to hear if there are any non-self-evident axioms outside abstractions in the field of mathematics. — Zelebg

I think that all axioms are non-self-evident, especially in mathematics.

The idea that axioms would have to be self-evident stems from classical Greek geometry, of which the axioms were eminently visible objects, i.e. points, lines, triangles, circles, and so on.

Of course, that kind of axioms were "self-evident": Just "have a look" by yourself.

This approach did not keep flying, however.

First, Euclid's Elements started getting competition from language-only mathematics in the Algorithmi's Liber Algebrae (12th century). Algorithmi did not draw anything at all. There was absolutely nothing to "see". At the same time, algebra was much, much more powerful than classical Greek geometry.

The final blow to classical Greek geometry and its straightedge and compass constructions came when Gauss algebraically expounded its fundamental limitations:

The Greeks were very adept at constructing polygons, but it took the genius of Gauss to mathematically determine which constructions were possible and which were not. As a result, Gauss determined that a series of polygons (the smallest of which has 17 sides; the heptadecagon) had constructions unknown to the Greeks. Gauss showed that the constructible polygons (several of which are illustrated above) were closely related to numbers called the Fermat primes. — Wolfram

It wasn't just Gauss' genius but especially Gauss algebra tools that allowed him to run circles around the ancient Greek. If everything you do, has to be "self-evident" -- like the ancient Greek wanted -- then you are not going to get particularly far ...

Yes, axioms are self-evident assumptions — Zelebg

Many axioms aren't self-evident in any fashion.

For example, take the axiom of regularity in ZFC set theory. The reason why it is there, is because in 1917 Dmitry Mirimanoff started writing lengthy rants on the existence of sets that are not "well founded". Imagine that you define:

A = { A }

This construction is not stable, because it is equal to:

A = { { A } }
A = { { { A } } }
and so on
...

So, it is not allowed to construct a set that looks like A = { A }. Therefore, the axiom of regularity is some kind of syntactic bug fix.

The same holds true for the axiom of specification. It prevents unrestricted comprehension which can otherwise cause Russell's paradox. Hence, it is another bug fix.

The language, i.e. the set notation itself, along with the language of first-order logic allow for specifying contradictions. Hence, one way to alleviate the problem -- but not really to solve it completely -- is to add rules (as axioms) that stamps out the most obvious ones.

So, axioms are also mere syntax restrictions for the language in which the theory is being elaborated. That type of axioms cannot be self-evident in any fashion, because you will first have to discover the problems that you will need to fix, by actually using the language.

Furthermore, what is there self-evident about the other axioms in ZFC?

For example, the axiom of power set. They happened to need power sets for Cantor's infinity theorems, but they could not guarantee that these power sets always exist. So, they ended up axiomatizing their existence. Problem solved. Is that kind of origin for an axiom, something to be considered "self-evident"? I don't think so ...

The ability to be a good programmer is something you either have or do not have. — EricH

Agreed.

I have indeed warned my brother-in-law for that problem. Programming requires some kind of talent which is probably innate. It will be up to him to discover if he has it. It is a bit unfortunate that I did not have time to guide him at an earlier age, because in that case, I would have made him discover this much earlier. Still, not everybody in IT needs to be a highly-talented programmer. There are also things like quality assurance, support, product management, consulting, and sales. Even relatively mediocre programming skills are still useful in these ancillary activities. It makes the difference between being just clueless and having some kind of understanding of what it is all about. In that sense, even mediocre programming skills are professionally still useful.

However the courses I took in data structures, programming languages, math (e.g. set theory) gave me an advantage over my compatriots. — EricH

Agreed.

Data structures and algorithms are at the core of the job. Furthermore, it also makes quite a bit of sense to pick up set theory (relational databases) and serious number theory (for cryptography). Unfortunately, they don't teach compiler construction nor mathematical logic.

At the same time, the university seems to be hellbent on getting first-year freshmen to "study" gender studies, political science, and that kind of nonsense. The first year is almost entirely unrelated to computer science. It looks much more like an employment programme for manipulative social-justice warriors to lecture the kids on culturally-Marxist, radical-left ideology.

The universities have become hotbeds of communist agitprop. I find them not only useless but nowadays even outright dangerous.

"Axiomatic" knowledge without any reference to the real world is useless. When untethered from the what we observe of the world, our knowledge is meaningless. What you call "axiomatic" knowledge is really just the rule we learned by observing the world. Some people have an issue with distinguishing between following/breaking a rule with the rule itself. Rules are meaningless without a world in which they are followed or broken. — Harry Hindu

If you come from the world of science, which is staunchly empirical, you will naturally tend to think that mathematics should be a bit like science and primarily deal with the physical universe. I can imagine that mechanical, construction, -or chemical engineers will also naturally be attracted to an empirical-constructivist take on mathematics.

If you come from the world of programming and its theoretical approach, i.e. computer science, you will not think like that. In that case, you are already used to high levels of meaningless and useless Platonicity. You should be quite used already to high-level structures that are fundamentally divorced from the senses.

Look for example at this example: AbstractObjectFactory. It is a structure-defining absurdity. To what could that structural abstraction possibly correspond in the physical universe? In fact, this source code does not even "do" anything, which is unusual for software, because the idea is that it would otherwise execute some code, but it doesn't even do that.

Absurdity is what naturally emerges out of lengthy abstraction processes. You obtain structures that mean nothing and that are essentially useless. So, yes, high-level abstract structures are naturally useless and meaningless. I am used to that. It is my professional life to deal with that kind of things. That is probably why I can appreciate the beauty of general abstract nonsense, the flagship of mathematics.

Total nonsense can be breathtakingly beautiful as long as it is consistent. It is mostly a question of developing enough intuition for that. Seriously, structural nonsense can even be pleasant to look at.

That is not the same as the simple equality/identity used in logic and math. — SophistiCat

Yes, in mathematics, the syntactic entailment of the "=" operator is defined -- axiomatized really -- for each different data type. For example, you will find an axiom for the meaning of equality in number theory and in set theory ("extensionality").

Even when you create your own custom data types, you will have to provide such definition by yourself. Otherwise, the system does not know how to evaluate equality.

This principle is reflected all the way down to practical applications in programming languages.

For example, you can turn standard tables into custom data types in the lua programming language. You can do this by attaching a metatable with functions, one of which is supposed to define what equality means.

__eq - Check for equality. This method is invoked when "myTable1 == myTable2" is evaluated, but only if both tables have the exact same metamethod for __eq. If the function returns nil or false, the result of the comparison is false; otherwise, the result is true. — Metatable Events, __eq section

Not one formal system "knows" what equality is supposed to mean, unless you explicitly tell it.

This seems a bit harsh and I do not agree. However, I will admit that working in an area may clarify and solidify the knowledge gained as an undergraduate. In the academic world the problems don't necessarily have to be practical to have this effect. — jgill

Well, I got recently confronted with how clueless education can be. It defies imagination.

I sent my brother-in-law to a supposedly practical training centre where he was supposed to do a 6-month training in mobile android (phone) development, because that should allow him to get an internship at a software company, with a view on doing a real-world, hands-on project.

For mere paperwork reasons, he will also start later this year a four-year bachelor course in computer science taught during the evenings. That curriculum is supposedly more theoretical, but it actually isn't. It is just more useless.

So, at the software training centre, they insist that he must first do a class in C/C++ because hey, "C/C++ is the core language in computing". So, yes, the operating system's kernel and a small core of libraries in any system may be written in C, but there is no way that any software company will ask a recent training graduate to write any C/C++ code for them. It is just too tricky and too bug-prone.

So, he does not need this skill for his internship in mobile development. Furthermore, very, very few libraries for Android phones are ever written in C/C++. As a developer, you will only see the Java interfaces anyway. So, it is not needed, and they are simply wasting 8 weeks of my brother-in-law's time (which is way too short to learn it properly anyway).

Next, I looked at the source code he was trying to run, and it contained the mention:

#include <windows.h>

What!?

That program is supposed to compile against the Microsoft Windows environment of libraries, which are not present on an Android phone. So, I asked my brother-in-law if he intends to deploy that joke to his phone? No. He doesn't. They just run it on their Windows desktop. It is obvious why. That thing cannot possibly run on an Android phone.

So, what does that course have to do with Android mobile software engineering?

The teacher obviously does not know how to write C/C++ libraries for Android because in that case he would use the proper compiler for that work, which is the gcc crosscompiler, and not Microsoft Visual Studio. So, I assume that he is not even capable of compiling one functioning line of code for Android. He clearly cannot do it.

Still, he insists that what he teaches is "foundational" for mobile development. He is just an arrogant retard who is unaware of his own ineptitude and incompetence.

Most teachers in the academia are like that.

They cannot solve the simplest practical problem in their field, not even to save themselves from drowning, but at the same time, they insist that the bullshit they talk, would be of any interest to their students. It isn't.

Yes. However, he implied that anyone beginning a study of a particular philosopher should read not only those works, but other's critiques as well. — jgill

Applied philosophy is much more accessible than general philosophy.

The philosophyOf(X) is straightforward if you have a reasonable understanding of X, while X could be anything, really.

So, learn something else first, i.e. some kind of applied knowledge (X). Next, figure out the philosophyOf(X). If you grok that properly, only then try to figure out general philosophy, i.e. philosophyOf(X=philosophy), i.e. self-philosophy.

The other way around does not work.

That is why most philosophy majors understand almost nothing about philosophy.

Theory emerges from practice. It never works the other way around. That is why majors in almost any subject will graduate with close to zero understanding of that subject. The only ones who understand the subject are people who have been confronted with solving practical problems in that subject. Everybody else invariably sounds like an idiot.

OK - that's new to me. — Banno

Starting from Carnap's diagonal lemma, the proof strategy for Gödel's first incompleteness theorem is really easy. Once you have established the first incompleteness theorem, the second one follows almost trivially.

In fact, it is the same for Tarski's undefinability theorem. It also follows almost trivially from Carnap's diagonal lemma.

Therefore, the hard part is the proof for Carnap's diagonal lemma. That is where all the meat is ...

Where is that from? — Banno

That is known as Gödel's second incompleteness theorem.

Gödel's second incompleteness theorem states no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent. — Wolfram

I mentioned the proof strategy for this (which is actually trivial) in a previous answer.

If it is consistent, then it is inconsistent? — Banno

If it is provably consistent, then it is inconsistent.
Yes. True.

So what? — Banno

The theory needs to be consistent to be usable, but you are not allowed to prove that it is, because in that case this theory is provably inconsistent.

So, yes, we really need consistency but we are also not allowed to prove it.

If the theory is consistent, it contains unprovable truths. If, as you say, "a legitimate knowledge claim implies the existence of an effective proof", then this could not be. — Banno

Yes, and that is a problem.

Still, the incompleteness theorem does not give you access to such unprovable truth.

The theory can obviously not prove of any particular truth that it is unprovable.

The incompleteness theorem only vaguely and very ambiguously says that such elusive, unprovable truth exists, without telling you what exactly it is, and only on the condition that the theory is not inconsistent, but that is in turn something which you are not allowed to prove.

In fact, if you omit any of the material or formal conditions for the incompleteness theorem, you will effectively be asserting a falsehood. It will just not be true.

For example, "I know for sure that there are unprovable truths" (P) implies "I know for sure that the theory is consistent" (Q), and is therefore a falsehood.

P --> false
Q --> false
~Q or P --> true
Q => P --> true

So, the correct way of stating the incompleteness theorem is:

"If the theory is consistent then its model contains unprovable truths" or
"The theory is inconsistent or its model contains unprovable truths"

In fact, this follows from Carnap's diagonal lemma. There exists a logic sentence s as such that:

s <--> isNotProvable(%s)

Which we can write as:

(~s and ~isNotProvable(%s)) OR (s and isNotProvable(%s))
---------------------------- (A) OR --------------------- (B)

If you look at (A), it says (~s and isProvable(%s)). So, it says that "s is false and s is provable", which means that the theory is inconsistent, because it proves a falsehood.

If you look at (B), it says (s and isNotProvable(%s)). So, it says that "s is true and s is not provable", which means that the theory contains an unprovable truth.

Note that the diagonal lemma does not tell you what s is. It only says that there exists at least one s in A or in B. Hence, you still don't know what s is. The lemma itself cannot tell you what it is.

Imagine that you can prove ~A. So, you can prove that the theory is consistent. In that case, B is provable. So, it means:

isProvable(B)

= isProvable(s and isNotProvable(%s))
= isProvable(%s) and isProvable(isNotProvable(%s))
= isProvable(%s) and isNotProvable(%s)

That is a contradiction.

So, if B is provable, then the theory contains a contradiction. So, in that case, it is inconsistent. Therefore, you are not allowed to prove ~A (="the theory is consistent"). If you can do that, then the theory is automatically inconsistent.

Did you mean "it is necessarily incomplete"? — Banno

It is inconsistent and/or incomplete.
Therefore, it could also be complete and inconsistent.
Hence, there may not be an unprovable truth as long as the theory is also inconsistent.

God is not the creator deity of the universe and mankind, but man's true nature and the norm of all things, in general

He clearly wants to replace religion by a simplistic and childish exercise in infinite regress, the kind of which has never produced anything of value or even worth knowing. His views are therefore worthless.

Justification/Rules comes with experience. — Harry Hindu

I certainly agree that this is the case in the empirical domain. Science certainly works like that, even though mere experience is clearly not enough as a justification. In addition, such justification will still have to satisfy the entire framework of regulations of the scientific method, i.e. paperwork.

On the other hand, justification in the axiomatic domain does not require experience. It is based solely on provability, which is different kind of paperwork.

Small point, maybe not relevant: getting a correct answer and verifying that it is correct are different procedures. — tim wood

Agreed. Most mathematicians assume that P is not NP (without proof, though).

And yet we know of unprovable truths. — Banno

Only if the theory is consistent.

If it is possible to prove that such theory is consistent, then it is necessarily inconsistent (second incompleteness theorem).

So, the first incompleteness is fundamentally ambiguous: a first-order theory resting on enough arithmetic is inconsistent and/or its model contains at least one unprovable truth.

Epistemology is broader than computability. — Banno

At first glance, yes. That is indeed my first impression too.

However, in practical terms, effective justification will always get translated into a paperwork procedure. At that point, it simply degenerates into computability. If it is not possible to create paperwork for the justification, then in all practical terms, there will be no justification. So, if epistemology does not equate computability, it will actually not work.

(My answer to the above was clearly "not", as you know now.) — god must be atheist

Yes, in the current state of affairs, they are not.

The common factor in the terms "scientific method", "axiomatic method", and "historical method", is clearly the term "method", which is a synonym for "procedure".

The core of their epistemology is their "method", i.e. their procedures. So, knowledge seems to be justified by paperwork procedures ...

Mathematics and its proofs are presented as a priori truths. Physical observation and experimentation is empirical. — god must be atheist

Both mathematics and science use their procedure to justify their knowledge. So, in both cases, it is about following the correct procedure. In that sense, in both cases, knowledge is justified by formalisms.

I wasn't looking at the difference between mathematics and science in this case.

I was rather interested in what they have in common: they both have a large bureaucracy of procedures.

So, I believe that the core of knowledge-justification always consists of "paperwork", regardless of what knowledge it is about.

If you were to start from scratch to study the fields of philosophy like epistemology, logic, metaphysics, ethics, philosophy of religion/science/mind etc., not to just know them, but being able to establish knowledge on any ground, to establish a ground you can build your beliefs on, how would your ultimate planning look like? — Monist

The "philosophy of X" is much easier to grasp than "general philosophy", especially if you already have a reasonable understanding of X itself.

In that sense, philosophy can be seen as a set of one-argument functions, with X being the argument:

ontologyOf(X) and epistemologyOf(X)

You can pick any X, really.

Say that you want to explore the philosophy of tennis (X=tennis). It would undoubtedly work, but you will need a good understanding of tennis.

Ontology (What is it?) and epistemology (How do we justify knowledge?) are in my impression the only sub-disciplines that make sense.

There are serious issues with the other sub-disciplines.

Logic is a part of mathematics. It is not philosophy. It has been successfully axiomatized now. Ethics competes and then spectacularly loses from morality, which is grounded in religious law. Metaphysics amounts to an utmost dumb exercise in infinite regress, which has has never produced anything of value or anything worth knowing.

Non-mathematical logic and non-religious ethics are childish, inept, and incompetent, while metaphysics is clueless.

Only ontology and epistemology make sense.

Aren't axioms, self-evident assumptions? — Monist

Originally, in Greek antiquity, over 2500 years ago, i.e. primarily in Euclid's Elements, axioms were meant to be self-evident. For a long time, classical Greek geometry was the core foundation of mathematics. This is no longer the case. From within mathematics, axioms are nowadays considered arbitrary starting points. Especially the formalist philosophy views it like that.

There may still be a link -- outside the realm of mathematics -- that views our most important axiomatizations, i.e. number theory and set theory, as concepts that are inspired by our innate intuition and by the nature of the universe that surrounds us. From within mathematics, however, it is wrong to view it like that, because the epistemology of mathematics does not allow us to make that kind of claims.

Hence, axioms are best viewed as arbitrary, unexplained, and unjustified beliefs.

If so, when can we accept self-evident beliefs, just when they are practical? — Monist

Mathematics does not seek to be practical. On the contrary, the desire for ever-increasing abstraction leads us to make sure that mathematics is preferably meaningless and and useless:

Abstraction in mathematics is the process of extracting the underlying structures, patterns or properties of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3][4] Two of the most highly abstract areas of modern mathematics are category theory and model theory.[4] — Wikipedia on mathematical abstraction

So, mathematics is preferably unrelated to the physical world and therefore meaningless. It can certainly not be applied directly. It must go through downstream user domains such as science, engineering, and so on, which are empirical and reintroduce the physical universe, with a view on harnessing meaningfulness and usefulness.

Consequently, in and of itself, mathematics is not just meaningless but also useless.

The ontology of mathematics is a set of arbitrary, unexplained, and unjustified core beliefs (=axioms) from which we derive new beliefs (=theorems) that are purposely meaningless and useless.

Do we have to analyse the relation between truth and practicality then? — Monist

The term "truth" in mathematics are facts (=data) in a given structure, i.e. a model, that satisfies a set of logic sentences derivable from a particular theory. It has nothing to do with truth in the physical universe. As I have written before, such mathematical model is never the physical universe, simply because we do not have a copy of the theory of everything.

So, mathematics is not real-world true. As I have mentioned above, mathematics does not seek to be practical either.

I suggest we introduce democracy in all countries in the world — god must be atheist

Well, it doesn't seem to work in even microscopically tiny Afghanistan, while you can still see the trillions of dollars sitting in the ever growing debt burden. Since the West has enthusiastically applied antinatalism to itself, it is running out of time.

Natalists do not care about dead bodies on their side, because they'll just replace them in the next generation. For antinatalists, however, every dead body irrevocably reduces their head count. Antinatalism does not work because it is just too easy to call their bluff.

(The strategy may be solid, though. I am no judge of that.) — god must be atheist

Concerning the proof strategy for Gödel's incompleteness theory, the Wikipedia page makes the following remark:

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. — Wikipedia

At the same time, I have found the following academic publication by Jeremy Avigad, "Incompleteness via the halting problem". Apparently, the Wikipedia page editors do not allow to list this particular publication as a source for the section.

The page history does not clarify why this decision was made.

In the following math stackexchange discussion I have found the following damning criticism:

I don't know of any proof of the full incompleteness theorem (the one that assumes only consistency) just from the unsolvability of the halting problem, and I doubt such a proof exists for two reasons. — Math stack exchange

The page itself says:

In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an axiomatization of the natural numbers that is both complete and sound is impossible. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers. — Wikipedia page

I suppose that the weaker system is not pressured to disprove false statements (only to prove true ones).

Hence, this proof strategy is deemed a bit controversial for Gödel's incompleteness theorem, let alone, for Tarski's undefinability. I personally like it, however, because it is much easier to understand than the alternatives.

The thing you need to get, I think, is that X=X says something about the language we use to describe the world, and not necessarily about the world itself. — ChatteringMonkey

Agreed.

Mathematical sentences, such as X=X, are constituent rules of the mathematical theory that defines them, and are true only in the model(s) for that theory. Such model is never, ever the real, physical world.

For example, if we are talking about natural numbers, we can say that the sentence is true in the standard model for number theory (=PA):

PA $\vDash$ X=X

The models for number theory (=PA) are NOT the physical universe.

The physical universe itself is a model of the elusive and inaccessible theory of everything (ToE). We do not know if the following sentence is true, because for that we would need to know the TOE:

ToE $\vDash$ X=X

We simply have no clue as to whether the sentence mentioned above is an axiom in the ToE, or otherwise a theorem provable from axioms in the ToE.

Mathematics does not prove anything, or asserts any truth, about the physical universe because the model(s) for any available mathematical theory, such as number theory, or set theory, and so on, are not the physical universe.

So this is why Chamberlain said, "Democracy is the worst possible system of government, except for all the others." — god must be atheist

I disagree with Churchill.

Democracy is doing something that even the worst communist dictatorship never did. It has taken control over marriage and divorce and instituted legal rules that make the family and generational reproduction impossible. Therefore, democracy is happily busy destroying itself.

There is simply no long-term survival possible without overthrowing the democratic regime. Unfortunately, this cannot be achieved by debating, negotiating, voting, or discussing. It has to be done through the force of arms. In other words, democracy will inevitably end in a violent blood bath.

I know that, proving 1+1=2 is hard — Monist

Well, no, it isn't.

With S the successor function, and x+1=S(x), and 1=S(0), we can see that 1+1=S(S(0))=2. You can trivially prove it by using PA's rewrite rules for addition. (PA is standard number theory)

but I do not see anyone trying to prove x=x, because it may seem so simple and obvious — Monist

PA's second axiom:

For every natural number x, x = x. That is, equality is reflexive. — PA

The fact that PA axiomatizes this rule for natural numbers means that it is not provable in PA.

In fact, you can generalize this remark: the defining rule for equality cannot be proven but only be defined in the mathematical theory.

For example in ZFC (=standard set theory), axiom 1, i.e. "the axiom of extensionality", defines the equality of sets:

Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members. The essence of this is: A set is determined uniquely by its members. — ZFC

Again, ZFC's logic for extensionality is an axiomatic starting-point rule that cannot be proven. It can only be used to prove derived rules (=theorems).

The starting-point rules, i.e. the system-wide premises, in a mathematical theory are always arbitrary, unexplained and unjustified beliefs. That is simply the essence of the axiomatic epistemology.

What I mean is - when a man falls in love with a woman, does it mean he saw someone in this woman who could provide him safety, comfort, satisfaction,...? — Craiya

You can safely assume that this man, first and foremost, wants sex. This man may not be thinking it over properly, because nowadays there are very dangerous risks associated with that behaviour, depending on what country you live in. This man should therefore learn to control his emotions and try to avoid getting dragged into risky ventures.

Is love really a good thing, or is it selfish to love somebody/something? — Craiya

Again, depending on the country this man lives in, it could be rather stupid rather than selfish. He has probably failed to make the correct cost/benefit analysis, while he may also underestimate the risks involved. This kind of behaviour is only sound in countries with a rather traditional legal system. That does not include any western country. Therefore, I would advise him not to do it, if possible.

There's too many dragons involved in having propositions range over other propositions. — Banno

A set of propositions is isomorphic under encoding to the set of the Gödel numbers of these propositions, and therefore isomorophic to a set of natural numbers, i.e. a subset of $\Bbb{N}$.

So, it would be a proposition that ranges over a set of numbers.

I think that this practice could work fine, if the set's membership function is a computable function. It could conceivably just be an exercise in standard number theory (PA) ...

What makes your pissed off religious mob any better than a government? — Pfhorrest

It is just a tool like any other tool.

Both are groups who’ll threaten to kill you if you don’t do what they say. — Pfhorrest

Not really.

I have learned to speak with utmost respect about religion. Seriously, I do not shit talk religion, and I have very good reasons for that. It is even an excellent insurance policy.

Therefore, I am not a target for religious mobs.

The secular government, however, certainly is. I am a respectful person, while the secular government is not. As you know, respect is ultimately always based on the fear for reprisals.

Seriously, I have no fear whatsoever for religious mobs.

I can even walk through them, holding the scripture in my hand, and feel completely safe, protected as I am by the powers that radiate and emanate from the scripture itself.

I can safely walk through areas that are more dangerous to atheists than the core of a nuclear reactor.

Even though I walk through the valley of the shadow of death, I will fear no evil, for you are with me; your rod and your staff, they comfort me.

Seriously, I am protected by higher powers, while the secular government is not.

Why would we need a god to exist? — Athen Goh

Religion can be as much a tool by the ruling elite as one against it. That last bit is really important, because otherwise, as a single individual, you stand powerless against the guns of the government. Religion, however, allows you to retaliate.

Imagine that the government seeks to attack you as an individual.

Are you powerless? No.

Just make the government officials publicly say or write something derogatory about religion, or even make them just deny the power of an active religion such as Islam. Since you have now manipulated them into insulting religion, you will see the hottest believers in the religion seeking to strike back, and make the severed arms and legs fly around in the air. Governments are scared to death because they know it does not take much for them to die like dogs. It happens all the time.

That is the true power of religion, and that is what I fundamentally love and admire about it.

Now you probably understand why I do not give a flying fart about any long rant that seeks to question religion. I am simply not giving up the power to strike back. Seriously, that will only happen in your dreams. Not in the real world, my friend!

My graduate degree says "philosophy" — John Gill

I came to the field in a completely different way.

I was customizing software, maintaining software, writing new programs, and otherwise doing really practical stuff for dollars, when I realized that I found myself surrounded by commercial messages that were sheer bullshit. These completely false beliefs were only meant to make other people money; people, who owned the corporations that were advertising these piles of lies.

So, what exactly was I doing? What was it? That is where ontology kicked in.

So, how else are we supposed to be doing the things that we were doing? That is where epistemology kicked in.

The beginning of the solution was to move to linux and open source. That got rid of the corporate nonsense that otherwise kept cropping up in my own work. The funny thing was that getting rid of the commercial vendors also allowed me to make more money.

Still, part of the problem remained. When people started hyping their particular choice of open-source solutions as the be-all and end-all, I understood that the battle wasn't over. We still needed better knowledge justification. Mathematics is part of the solution. The combination ontology+epistemology, aka, philosophy, is the other part.

Why shouldn't we assume that at least one valuation (call it f and even interpret it as the predicate "is false") creates a complete set of mis-matches (true-false and false-true) of s to f(s) while its opposite (~f = g) makes a complete set of matches? — bongo fury

I have finally received an answer for the issue that you have raised. The function that you describe, looks like this:

$f(\ulcorner s\urcorner) = \neg s$

I asked the question on the computer science stack exchange:

Diagonal lemma and the negation function

It is indeed possible to define it, but it is not computable, because of Tarski's Undefinability Theorem (TUT).

Consequently, my proof ultimately depends on an underlying proof for TUT. That is in fact a real problem, because TUT is usually made to depend on the diagonal lemma, creating some kind circular situation.

The only possible salvation in this situation is to use a diagonal-free proof for TUT, which also exists.

In fact, that is exactly what Saeed Salehi pointed out in the Russian workshop, "Diagonal-Free Proofs of the Diagonal Lemma", University of Tabriz & IPM WORMSHOP 2017, Moscow.

Saeed writes in page 3:

Toward a Big Surprise.Tarski’s Theorem is equivalent with the Semantic Diagonal Lemma. — Saeed Salehi

Saeed mentions three (surprisingly) short diagonal-free proofs for Tarski's Undefinability:

• Robinson (1963) on page 16
• Kotlarski (1998) on page 17
• Sereny (2004) on page 18

Deriving the diagonal lemma from any diagonal-free proof for TUT is in fact considered trivial.

Therefore, the core of my proof does not particularly do much, over and beyond the diagonal-free proof on which it rests.

Hence, my own proof is not wrong, but it is unfortunately not the essence of the matter. The hard part is proving the diagonal lemma for the one, single case (equivalent to TUT) that you pointed out!

Thanks again, for pointing out the issue!

I don't see how you are addressing anything like the same claim, e.g.,
Any symbol system that can prove all arithmetic proves at least one liar sentence. — bongo fury

[/quote]

That is not a requirement for the diagonal lemma. The statement of the lemma says:

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let F be a formula in the language with one free variable, then: There is a sentence ψ such that ψ ↔ F(°#(ψ)) is provable in T.

The liar sentence is handled in a downstream theorem that makes use of the diagonal lemma, i.e. Tarski's undefinability theorem.

We should not mix downstream theories with the lemma itself, which only says what it says, and nothing more.

Why shouldn't we assume that at least one valuation (call it f and even interpret it as the predicate "is false") creates a complete set of mis-matches (true-false and false-true) of s to f(s) while its opposite (~f = g) makes a complete set of matches? — bongo fury

Yes, agreed with this remark. (I still need to rework the argument for that problem.)

returns the godel number of the same sentence eating itself

The version in wikipedia indeed revolves around self-reference. When you look at the lemma's statement:

$s \leftrightarrow f(\ulcorner s \urcorner)$

Then, it does not necessarily say anything self-referential. What it literally says, is:

• any arbitrary logic sentence s can be true or false
• $f(\ulcorner s \urcorner)$ can be true or false, for any arbitrary f
• In those circumstances, there is at least one s for which the tuple (s,$f(\ulcorner s \urcorner)$) is (true,true) or (false,false)

It is not clear to me, what exactly would be self-referential about any of this.

Furthermore, there is another issue. Every digital representation of sentence s is always a natural number. This is generally true: every representation of language is a sequence of natural numbers and therefore again, one big natural number. In that sense, can sentence s even be anything else than a natural number? So, where does the need come from to encode sentence s as a Gödel number? The (digital) sentence s is already a Gödel number.

Therefore, I do not see the need to encode s again as a Gödel number, since it can only be represented by a Gödel number. For example, in this page, any sentence s is represented as a stream of utf8 characters, which is an entirely legitimate Gödel encoding. Hence, as far as I am concerned, the lemma can actually be simplified to:

$s \leftrightarrow f(s)$

No need to use $\ulcorner s \urcorner$, because the conversion function $\ulcorner \urcorner$ is redundant.

Maybe it was not clear in 1931, before computers existed, that a text representation is always isomorophic with a unique natural number (unique up to an isomorphism under encoding translation, of course). So, I understand that Gödel created his own utf8-style encoding. Nowadays, however, it looks quite redundant to do that.

What else than a natural number can text be? I simply cannot imagine any other representation of text that is not isomorphic to a natural number. In other words, s is already an element of N, the natural numbers. It can be represented in decimal, hexadecimal, binary, or in alphabetic text, but it is in fact always the same thing. In my opinion, there simply is no text that is not a natural number.

I also have a different interpretation for the name of the theorem itself, i.e. "diagonal" lemma. I consider the "diagonal" aspect to be very similar to the one in Cantor's diagonal argument. It is simply again a diagonal in a table:

$\begin{bmatrix}(sentence,property:f) & false & true \\ false & \textbf{(false,false)} & (false,true) \\ true & (true,false) & \textbf{(true,true)} \end{bmatrix}$

The "diagonal" lemma simply says that for every property/function f, there exists a sentence on the diagonal, i.e. (true,true) or (false,false). So, there must be sentence that is true and for which the property is true, i.e. (true,true), or a sentence that is false and for which the property is false, i.e. (false,false).

That is how I understand the term "diagonal". I really don't see the need for self-reference here.

Is your f really Γ, the "graph" predicate assumed available to "represent" (presumably like the way points on a 2d coordinate graph represent a relation of ordered pairs) any computable function and therefore f? — bongo fury

The proposition that there exists a Γ for which Γ(x,y=f(x)) -> true and Γ(x,y$\ne$f(x)) -> false, is tautological for any (computable) function f. But then again, this proposition is needed just for the original proof, and not for this one.

this somewhat bewildering and disorienting critique of wikipedia and everyone else — bongo fury

James R Meyer writes:

Because the Gödel numbering function is a function in the meta-language to the system T, there cannot be any such relation δ in the system T that contains the information of the definition of the Gödel numbering function and which can unambiguously reference formulas of its own formal system T.

My opinion is that any sentence or text is always in Gödel numbering. For example, this entire page is necessarily in Gödel numbering (utf8).

The confusion is caused by non-digital representations of text. They give the impression that text could be represented in something else than in something isomorphic with natural numbers (e.g. ink on paper), but that is not true. If someone manages to show me text that is not isomorphic with a natural number, I will of course change my mind, but first I need to see something like that. Up till now, I never have. In fact, I do not think that it even exists.

Therefore, I believe that the practice of Gödel numbering utf8 text again (which is already Gödel numbered), is simply redundant.

Hence, I fundamentally disagree with Meyer's criticism.

When you consider A, I think you can omit the s1 discussion and just note that A is an instance of the diagonal lemma that is (true,true) for function ~f. So, per the negated lemma, A can't be true. — Andrew M

Yes, agreed. That is what I wanted to say. I was still looking for a succinct way to put it. This is indeed what it is about. What you just wrote, is indeed the gist of it. I wonder if the proof can just be phrased like that? Would it still be considered a proof?

But that still leaves disjunct B that could be true. So you would also need to show a case where B fails. — Andrew M

Yes. This issue actually got introduced by your previous remark, which turned the conjunction into a disjunction. So, now there is a need to handle case B too.

Concerning the expression in B, instead of being true for all true sentences, it will be true for all false sentences. So, B is then an instance of the diagonal lemma that is (false,false) for function ~f.

The whole point of the proof is that the practice of requiring that a sentence stays off the diagonal for f will always force the sentence onto the diagonal for ~f. This is the case for both true and false sentences.

I think that the main issue left now, is to come up with a succinct way of phrasing this principle.

Shouldn't this be that it must hold true for every true sentence s? The conjunction is simply false when s is false. (Whereas B handles the case where s is false.) — Andrew M

Agreed. I have applied a fix to the proof for this problem. Do you think it works?

It is easy to show via the construction of Prov('s') that s -> Prov('s') when s is a theorem. — sime

A first remark to make is that Carnap's lemma does not need Gödel's first (syntactic) incompleteness theorem (G1SI). It is the other way around. So, in the context of the lemma's proof, we should not mention G1SI theorem, because that would make things circular.

A second remark is that Gödel's semantic completeness theorem proves that Prov('s') -> s. So, if s is provable in T then s is true in all M; with M any model of T.

The other way around is not true.

The sentence s could be true in M, but not provable in T. This could be the case, for example, when T has more than one model (T is not categorical). In that case, s could be true in M1 but false in M2. In that situation, s will not be provable in T.

Hence, we can assert:

Prov('s') -> s

However, we cannot assert:

s -> Prov('s')

The fact that there exist true statements in M that are not provable in T, is exactly what G1SI theorem proves (with T being, for example, first-order PA).

Where have you used the assumption that f is computable? — fdrake

I didn't.

By the way, the original proof does not seem to use the condition either.

I just kept the condition around, because the original proof does, and the lemma is phrased like that.

As far as I am concerned, the function only needs to map natural numbers to booleans -- I agree with your view -- but I also have no clue as to why Carnap insists that the function must be computable.

The condition may be used very implicitly in one of the steps in the proof. I was hoping that someone else would be able to point out why the condition is needed (I somehow suspect that it actually is).

Shouldn't this be that it must hold true for every true sentence s? The conjunction is simply false when s is false. (Whereas B handles the case where s is false.) — Andrew M

Yes, I think that you are right. The fix is more complex, though. Still working on it.

I think the negation of the lemma needs to contain a disjunction as follows (since for the negation every tuple must be either (true,false) or (false,true)): — Andrew M

Agreed. Fixed. Thanks for the correction.

What is the fundamental difference between these two examples? And is there a principle on deciding whether or not it is rational to accept the improbable? — Wheatley

At first glance, getting into the NBA could be just 1 chance out of 1 million or so, over the population of all Americans.

However, it is probably 1 chance out of 100 out of all Americans who train 8 hours per day on getting as far as they could get. It is possibly 1 chance out of 20 for all Americans who are 6 foot 2, train 8 hours per day, and have been doing so since they were 6 years old. It is possible 1 chance out of 5 for all Americans who also have an uncle who managed to get into the NBA.

And so on.

Most people who have been training on getting into the NBA from the age of 6 can quite accurately predict by the age of 16 if they stand a reasonable chance of getting in. That person may already be 70% likely to get in, if he keeps training 8 hours a day with a first-class coach for junior players.

And for doing next to nothing, but dominating google, they skim large amounts of money off the hotels. — Coben

The user can filter the list of hotels for a location, time period, amenities that you require, and sort the list from lowest to highest price. They save the customer an incredible amount of time.

When there are no good offers on airbnb, which sometimes happens, I always resort to booking:com. Over the years, I have spent hundreds, if not thousands, on that website. Once, I even ended up at the reception of a hotel, asking for the price, and they said $150, but at booking.com they had listed the same room for $65. So, in front of the receptionist, I booked the room on booking:com; after which he grudgingly gave me the key to the room. So, I also gave them a low rating for service.

They are parasites. — Coben

No, they are not. It is the large hotel groups that are parasites.
Anyway, I have cut down on using hotels because I strongly prefer airbnb.

With upwork, if you are earning a living through it over time, the percentage they take goes down. — Coben

I don't really care about how much their commission is. If you can sell a project for $5000 with $2000 commission (net=$3000) you are still better off than selling it for $2500 with $500 commission (net=$2000). There are lots of situations in which paying out a higher commission for a sale, makes you more money.

So these countries are selling drugs that Western research developed for low prices? — Coben

Patent protection for drugs expires after twenty years. After that, you can freely sell the medical molecule. We are no longer paying patent fees for the use of the wheel either. That particular patent expired in the stone age already.

Furthermore, most of the expense is in bribing the FDA into ignoring dangerous side effects. The FDA accept applications for new drugs only from a very small cartel of oligarchs. So, yes, very often it is western companies who originally paid the corruption fees for the fake FDA documentation of these products. The newer the product, the more likely it is really bad for your health.

This sounds like you were in a professional field, probably well educated, and your clients are seeing you are more or less an equal, not simply because of your qualities, but because of the type of contracting you were doing. — Coben

Well, you don't need a degree for software engineering. All programmers are essentially self-taught. The other ones invariably drop out. I was programming from the age of fourteen, as a hobby, together with other 14-year old classmates. I did my degree in another subject (operational research). Back then, no one else in my high school -- especially not the teachers -- knew how to program.

Often, clients did not see me as an equal. I often found them quite arrogant. I didn't see them as my equal either, because they were incapable of writing one line of code, even to save themselves from drowning. Most of the time, we mutually disrespected each other, especially at larger, corporate organizations. We tended to be mutually condescending to each other. I often ended up treating them with contempt. Things went much better when I started taking smaller-company clients online while living in SE Asia. These entrepreneurs were much more respectful, and so was I.

This is a very different situation for whatever an individual taxi contractor would be like. — Coben

They are much better off, because they do not even need to talk with customers. They just do their job, and be done with it. Thinking of it, I would have preferred that too.

Uber is neither client nor employer, it's a middleman that creates a situation where indepedendent contractors can compete,since they will not have the infrastructure to reach clients — Coben

I did have direct clients online, but I ended up using platforms like upwork.com too. Upwork is just a middleman similar to Uber. I don't see anything wrong with that. For the few percent they charged back then, they saved me a lot of time in which I didn't have to contact potential prospects. Still, eventually, I liked my startup work much better. However, doing contract work was actually not bad either. Unfortunately, in contract work, there is no possibility of cashing in on an upside. You can't suddenly make a lot of money. Startups are much better in that respect.

But it seems odd that Singapore's medication prices are 300 times cheaper, since pharmaceutical companies tend to price along national income level lines. — Coben

Just look up the molecule name for the brand that the doctor prescribed, and then order online Indian, Bangladeshi, Pakistani, or Vietnamese generics instead. Instead of paying $1000, you can often buy the medication for $3 or $4, if you do that. Very little of that medication is still under patent.

As far as health care in general a quick look I took at the Singapore system makes it sound very interesting. — Coben

I don't live in Singapore. Too much red tape over there. My favourite countries are Laos, Cambodia, and Vietnam. I also like Indonesia, but they make too much trouble about long-term visas. I can't be bothered to deal with that. Let's keep it simple, stupid!