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GEB Discussion Chapter 8

# Gödel, Escher, Bach: An Eternal Golden Braid

This is a discussion of the themes and questions concerning the Chapter 8: Typographical Number Theory and its dialogue, A Mu Offering.

### Translation Exercises

This chapter is a surprisingly simple one conceptually despite the complex formal system being introduced. All Hofstadter is doing is explaining how the system works and it’s worth taking the time to do his exercises to increase your familiarity with TNT for the upcoming chapters. All I can provide is my own answers to some of the translations.

#### 1) TNT to N

• `~∀c:∃b:(SSO • b)=c` `<=>` It is not true that for all numbers `c` such that there exists some number `b` that 2 times `b` equals `c`. `<=>`Not every number is even. True!

• `∀c:~∃b:(SSO • b)=c` `<=>` For all numbers `c`, there doesn’t exist some number `b` such that 2 times `b` equals `c`. `<=>`Every number can’t be divided by 2. False!

• `∀c:∃b:~~~(SSO • b)=c` `<=>` For all numbers `c` such that for some number `b`, it is not true that 2 cannot be multiplied by `b` to not equal `c`. (Triple negative) `<=>` For each number, there is a number which it can’t be the double of. True!

• `~∃b:∀c:(SSO • b)=c` `<=>` There doesn’t exist some number `b` such that for all numbers `c` that 2 times `b` equals `c`. `<=>` All numbers are not the double of one number. True!

• ∃b:~∀c:(SSO • b)=c `<=>` There exists some number `b` such that not for all numbers `c` that 2 times `b` equals `c`. `<=>` There is a number `b` that exists such that not every possible number `c` satisfies `2*b=c`. True!

• `∃b:∀c:~(SSO • b)=c` `<=>` There exists some number `b` such that for all numbers `c` that 2 times `b` doesn’t equals `c`. `<=>` There is a number when doubled, is not a number. False!

#### 2) N to TNT

• All natural numbers are equal to 4. `<=>` `∀a:(SSSSO=a)`

• There is no natural number which equals its own square. `<=>` `~∀a:(a • a)=a` (Note that zero is a counter-example)

• Different natural numbers have different successors. `<=>` `∀a: ∀b:~(a=b) => ~(Sa=Sb)`

• If 1 equals 0, then every number is odd. `<=>` `(0=S0) ∀a:()`

• b is a power of 2. `<=>` `∀x:∃y:(x • y = b) & ~(x=S0) => ∃z:(SS0 • z=x))` `<=>` If `x * y = b` and `x ≠ 1`, then `x` is a multiple of 2 which makes every divisor of `b` a multiple of 2 as well.

## Rules of Formation

• 0 is a numeral

• A numeral preceded by S is also a numeral

• Axiom 1: `∀a:~Sa=0` `<=>` For all numbers, it’s successor is not zero. `<=>` There is no number where zero is its successor, or there is no negative numbers.

• Axiom 2: `∀a:(a+0)=a` `<=>` For all numbers, adding zero doesn’t change the number.

• Axiom 3: `∀a: ∀b:(a+Sb)=S(a+b)` `<=>` For all numbers a and for all numbers b, a plus the successor of b equals the successor of a plus b. (Commutability) `<=>` (a + (b + 1)) = (1 + (a + b))

• Axiom 4: `∀a:(a • 0)=0` `<=>` For all numbers, multiplying by zero equals zero.

• Axiom 5: `∀a: ∀b:(a • Sb)=((a • b)+a)` `<=>` For all numbers `a` and for all numbers `b`, `a` times the successor of `b` equals `a` times `b` plus `a`. (Distributivity) `<=>` (a • (b + 1)) = ((a • b) + a)

### The Five Peano Postulates

Djinn is “natural number”, genie is “zero”, and meta is “successor”.

1. Genie is a djinn. `<=>` Zero is a natural number.

2. Every djinn has a meta (which is also a djinn). `<=>` Every natural number has a successor (which is also a natural number).

3. Genie is not the meta of any djinn. `<=>` Zero is not the successor of any natural number.

4. Different djinns have different metas. `<=>` Different natural numbers have different successors.

5. If Genie has X, and each djinn relays X to its meta, then all djinns get X. `<=>` If zero has X, and each natural number relays X to its successor, then all natural numbers get X.

### Dialogue

This dialogue may be very confusing due to the fact that three significant subjects are being intertwined, molecular biology, detecting theoremhood, and Zen. Tortoise’s and Achilles’ entire discussion of the ritual of converting koans into strings is a thinly veiled explanation of how DNA is converted into folded proteins.

The Central Dogma of biology is DNA => RNA => Protein which can be mapped onto Koan => Strings => Meaning/Buddha nature or not Buddha nature which can be mapped onto TNT well-formed string => Exact Translation in English => Meaning.

When Achilles says that he likes to go against the arrows, it’s the same idea as asking if a certain mathematical fact can be expressed in TNT. It’s easy if it can’t be turned into a well-formed string but if it does, then we have to find out if it’s a theorem without knowing how it can be derived from the axioms.

In addition, the pair of strings where the only difference was the knot is the same idea as two theorems where one is a negation of the other and since having both theorems be true would mean a contradiction, then only one theorem/string can have Buddha nature.

Finally the string talking about the Great Tortue seems to be talking about itself. This implies that TNT may have the ability to talk about itself and is powerful enough for Gödel’s Theorem to apply, where it can prove its own consistency. However, then it turns out that the string is talking about its own negation, what does this imply?