Can you please explain where I went wrong? Thx.

BTW, the link I provided is to an article about Bailey's formula.

Parent is correct, digits of pi can be calculated independently in base 2, 4, 8, 16 or 2^n since the 1990s. So, it is possible to calculate the 2,000,000,000,000,000th number of pi without calculating the digits before that one. Now, if we want to calculate the digit in decimal (or converse the binary digit to decimal), we need to calculate all of the two-quadrillion digits. Knowing this digit is in itself not very interesting.

Communism is a "social system based on collective ownership". The word is originally French, existing since the 12th century. Victor d’Hupay was in 1785 the first (afaik) to use the word in its modern meaning. Marxism (19th century) is one form of communism, the form which redefines communism as "social/political system based on state ownership". "Pure communism" is the end goal of Marxism, a free state which they tried to reach by installing a dictatorship... (Sorry - as an anarchistic communist I had to try to be "cleaver".)

OK, you made me read your post again.

There are indeed limited mathematics which are built upon first order logic and which are consistent and complete. They can even be infinite, if you allow for an infinite number of axioms by using an axiom scheme. But they are not strong enough to express arithmetics.

Using nothing but logic, one can build two kinds of mathematics which are strong enough (i.e. being of second order) to express arithmetics:
1) consistent (but incomplete) ones,
2) inconsistent ones - you say that all mathematics is inconsistent, but that is just plain wrong. Unless if one uses a paraconsistent logic to prevent the ex falso sequitur quodlibet, inconsistent mathematics is trivially complete, because all well-formed formulas would be true.
3) Gödel proves that the third kind, mathematics which are complete AND consistent do not exist.
3') One could consider mathematical theories of which we do not know if they are consistent or not as a third kind of mathematics, I don't know if anybody has ever constructed a mathematical system of which it can be proven that it is undecidable wether it is consistent or not (sounds like a nice project actually.)

You are confusing "consistency" with "completeness".

I have no clue what you are talking about.

Yes, that's why there is a movement called finitism in maths.

Of course the existence of real numbers can be proved: take a triangle with a 90 angle & with both legs on that angle having a length of 1. Then the length of the third leg is a real number, SQR(2). It is easy to proof that SQR(2) is not integer nor rational. Now, defining and constructing real numbers is harder and there are non-standard mathematics which try to address the problems you hint at.

Sorry, the second sentence of my last paragraph should read as "axioms of relativity do not lead to theorems" (in stead of 'theories'.)
The theory of Goldblatt is a "first order theory", logicians like that, because it is decidable. The theory of Schutz in categorical, the mathematician like that, even if it is undecidable.

I agree with m50d that it is not relevant for the reason he gives above.

The Gödel theorems are interesting for the study of the foundation of mathematics and more specifically for the study of the relation between logic and mathematics. Using it outside that field is at least tricky, and more often than not crackpottery.

Out of a set of axioms (or out of a set of hypothesis) you use deductive logic to prove some theorems which are true if the axioms are true. The axioms together with the theorems form a theory. The question of completeness is: can we construct a proof for every true stament in that theory, or do there exist true statements which cannot be proven. The question of (in)consistentcy is: can we construct a proof for a false statement? All this is about the internal properties of a theory.

Now back to your question: relativity theory and quantum mechanics have different sets of axioms. The axioms of relativity do not lead to theories which are in contradiction with other theorems of the same theory (I am not sure about this for general relativity, there are however several sets of axioms for special relativity which have proofs of consistency*). I guess the same holds for QM**. The problem is: theorems of relativity are in contradiction with theorems QM, so this is a problem between two theories. The problem is that both theories are very solid and well-tested on their own right. A theory which tries to combine relativity and QM on a logical level is Branching Space-Time by Nuel Belnap.

* For axioms of special relativity, check: - Optical geometry of motion, a new view of the theory of relativity by A. A. Robb, 1911
- A theory of time and space by A. A. Robb
- The absolute relations of time and space by A. A. Robb, 1921
- Geometry Of Time And Space by A. A. Robb, 1936
- Orthogonality and Spacetime Geometry by Robert Goldblatt, 1987 (this is a first oder theory, so it is both complete and consistent - however it is not categorical
- Independent axioms for Minkowski space-time by John W. Schutz, 1997. This theory is of second order, so it suffers from the problems caused by the Gödel theorems.

** Check Quantum Logic by J. von Neumann (yes, the guy of the "Von Neumann Concept") and G. Birkhoff.

Thanks you for supporting my point of view - you should create an account, then I could become a fan :)
However, I disagree with mathematics including physics. I consider mathematics as being a tool or even a language used by physiscs.
Disclaimer: I'm neighter a mathematician nor a physicist (I'm a logician.)

A hypothesis is the starting point. A theory is the hypothesis and everything which follows from it. The hypothesis can be true or false, a theory can also be true or false.

You are confusing "consistency" with "completeness".

Well, propositional logic can be proven to be consistent (there are no contradictions) AND complete (all true propositions can be proven out of the axioms), so can first order predicate logic (in the PhD dissertation of Gödel, 1929).

To construct arithmetic out of logic, we however need second order predicate logic. Gödel (1930, published 1931) showed that axiomatic systems in second order logic are either incomplete (true non-provable sentences can be constructed) OR they are inconsistent (containing contradictions).

Start playing with ARM then, its design was somewhat inspired by the 65xx series and there are plenty of affordable ARM-based systems available.

Except for the state of emergency from 1975 until 1977, elected parliament has always been in power in India. Having a treaty with the Soviet Union has nothing to do with being democratic or not.

François Rabelais wrote that already in the first half of the 16th century in his book "Gargantua", chapter LIV.

Hi, I am a Linux user and I am anti-copyright and anti-"Intellectual Property" in general. But I have never stolen anything nor raided any ships. Oh, you mean illegal copying of software... Well, since I use Linux I do not need to make illegal copies, nor do I have the time for that because free software is released at such a fast rate that I have no hope to learn to use all of it in my lifetime. (Apologies for feeding the troll.)