**Kurt the Penguin**

Kurt was a talking penguin aware of his special existence.

**Rules of the Clock**

#1: When the clock behind Kurt struck 12, he’d yell, “I’m a penguin!”

#2: When the clock didn’t strike 1, he’d not yell, “I’m a penguin!”

#3: When the clock struck 6, he’d yell “I’m a penguin!” twice.

#4: When the clock didn’t strike 7, he’d not yell, “I’m a penguin!” twice.

**The clock strikes!**

Once, on a fine Tuesday, the clock struck 7.

Either Kurt didn’t yell “I’m a penguin!” twice or Kurt yelled “I’m a penguin!” twice.

If Kurt yelled “I’m a penguin!” twice, then Kurt has not broken none of the Rules of the Clock.

If Kurt didn’t yell “I’m a penguin!” twice, then Kurt has broken one of the Rules of the Clock.

**What’s up with Kurt?**

Therefore, if Kurt always speaks the truth, there are some truths that Kurt doesn’t speak.

And Kurt sometimes speaks the untruth.

**Kurtlogic**

#1: [T** _{12}**, K

**] →**

_{P}

*t*#2: [NT** _{1}**, NK

**] →**

_{P}

*t*#3: [T** _{6}**, {K

**, K**

_{P}**}] →**

_{P}

*t*#4: [NT** _{7}**, N{K

**, K**

_{P}**}] →**

_{P}

*t*If Kurt yelled “I’m a penguin!” twice: [T** _{7}**, {K

**, K**

_{P}**} → N**

_{P}**]**

*f*If Kurt didn’t yell “I’m a penguin!” twice: [T** _{7}**, N{K

**, K**

_{P}**} →**

_{P}**]**

*f*Where,

[T** _{7}**, {K

**, K**

_{P}**} → N**

_{P}**] ≡ “if Kurt always speaks the truth, there are some truths that Kurt doesn’t speak”**

*f*[T** _{7}**, N{K

**, K**

_{P}**} →**

_{P}**] ≡ “Kurt sometimes speaks the untruth”**

*f**

**Gödel’s first incompleteness theorem**

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.

**Gödel’s second incompleteness theorem**

For any formal effectively generated theory

Tincluding basic arithmetical truths and also certain truths about formal provability,Tincludes a statement of its own consistency if and only ifTis inconsistent.