Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

• Waterfall = 6 step loop
• Ascending & Descending = 45 step or 4 step loop
• Drawing hands = 2 step loop

• Print Gallery = 1 step loop

  Loops imply Inifinity

aka Liar's Paradox

"All Cretans are liars."

is a One Step Loop

To every ω-consistend recursive class κ of formulae there correspond recursive class-signs r, such that neighter ν Gen r nor Neg (ν Gen r) belogs to Flg(κ) (where ν is the free variable of r)

Actually, it was in German, and perhaps you feel that it might as well be in German anyway. So, here is a paraphrase in more normal English:

All consistent axiomatic formulation of number theory include undecidable prepositions.

The discovery of non-Euclidean geometry deeply challenges the idea tat mathematics studies the real world.

Two types of adjectives:

• Class A: Self descriptive adjective: pentasyllabic, awkwardnessfull
• Class B: Non-self descriptive adjectives: edible, bisyllabic, incomplete

"non-self descriptive" is an adjective but is it calss A or B? This is a two step loop.

• M : Mechanical Mode
• I : Intelligent Mode
• U : Un-mode (Zen)

A test of theroemhood that terminates in finite time.

For a formal system, the set of axioms must have a decision procedure while the set of theorem may not.

The axioms & rules of inference 'implicity' characterize the theorems and even more 'implicity' characterize the non-theorems.

Isomorphisms induce meaning.

pq-system is isomorphic to addition. e.g.

• xp-qx- is an axiom where x is a string of hypens only
• xpy-qz- is a theorem if xpygz is a theorem (x,y,z are strings of hypesn only)

There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.

i.e.

There exist recursively enumerable sets which are not recursive.

which implies:

There exist formal systems for which there is no typographical decision procedure.

All formal systmes are recursively enumerable i.e. theorems can be generated using typographic rules recursively. But to be recursive, the non-theormes must also be recusively enumerable.

Contrapunctus is the last fugue bach wrote

B A C H

• B = B-flast
• A
• C
• H = B (in Germany)

Elliptic Geometry = Sphere

Elliptic & hyperbolic geometries are called geometries because they share the core 4 postulates of Euclid's Geometry adn only differ in the 5th (the parallel axiom)

• Internal Consistency : Consisten/compatible theorems within the formal system with same interpretation of some or all the symbols
• External Consistency : Every theorem is also true with the interpretation of the symbols in an external world.

One formal system can be extended from another then it inherits the passive meanings of the undefined terms.

At least logic must be. But zen embraces contradictions and non-contradictions with equal eagerness.

The core of number theory, the counterpart to absolute geometry—is called Peano arithmetic

Also, it is now well established—as a matter of fact as a direct consequence of Gédel’s Theorem—that number theory is a bifurcated theory, with standard and nonstandard versions. Unlike the situation in geometry, however, the number of “brands” of number theory is infinite, which makes the situation of number theory considerably more complex.

For practical purposes, all number theories are the same. In other words, if bridge building depended on number theory (which in a sense it does), the fact that there are different number theories would not matter, since in the aspects relevant to the real world, all number theories overlap.

• Consistency: when every theorem, upon interpretation, comes out true (in some imaginable world).
• Completeness: when all statements which are true (in some imaginable world), and which can be expressed as well-formed strings of the system, are theorems.

Backlinks

• One Step Loop
• Two Step Loop