**Unified Theory of Climbing Grades: Part 1- Axioms and Definitions**

Our theory is based on the primitive concepts of

*rock*,

*attempt*and

*send*which we won't define here and are well established in the climbing community. (Ok, that's a bullshit, but...)

Our primary goal is to build a “grade function” which will exactly tell us the grade of

*any*climbing problem at

*any*given time because we absolutely cannot sleep otherwise.

We will use the common notation

**N**,

**Q**,

**R**, for the natural, rational and real numbers’ sets respectively.

Let

**∆**be the set of the rocks, and tϵ

**R**the “time” coordinate.

For every xϵ

**∆**, tϵ

**R**we denote by

**A**(x,t) the set of

*attempts*of x up to time t, and by

**S**(x,t) the set of

*sends*of x up to time t.

- Axiom 1)
**∆**is finite. (we all wish it was not so...) - Axiom 2) every “
*send”*is also an “*attempt*”. (then**S**(x,t)⊆**A**(x,t) for every tϵ**R**, xϵ**∆**) - Axiom 3)
**A**(x,t) is finite for every xϵ**∆**, tϵ**R**.**S**(x,t))

We can now define the functions:

**a**,

**s**:

**∆**×

**R**→

**N**by:

**a**(x,t)=#**A**(x,t) (number of elements of**A**(x,t))**s**(x,t)=#**S**(x,t) (number of elements of**S**(x,t))

**s**(x,t) ≤

**a**(x,t) for every xϵ

**∆**, tϵ

**R**.

- Axiom 4) for every xϵ
**∆**both**s**(x,t) and**a**(x,t) are increasing functions of t.

- Definition: we call a
**problem**a point (x,t) of**∆**×**R**for which**a**(x,t) ≠ 0. (we say “x is a*problem*at time t”)

__t__) is a problem than (x,t) is a problem for every t>

__t__(Axiom 4)

- Definition: a problem (x,t) is said to be a
**project**if**s**(x,t) = 0. (we say “x is a*project*at time t”) - Definition: being (x,t) a project we call
**class**of (x,t) the natural number n=**a**(x,t). (we say “x is a*project**of class n*at time t”) - Definition: we call an
**established problem**a*problem*which is not a*project*.

__t__) is an established problem than (x,t) is an established problem for every t>

__t__(Axiom 4)

- Definition: being (x,
__t__) an established problem we say “x*has been a project of class n*” where n=max{**a**(x,t) | tϵ**R**& (x,t) is a project} . - Definition: we call (x,t)
**just a piece of rock**if**a**(x,t) = 0. (we say “x is*just a piece of rock*at time t, man…”)

__t__) is just a piece of rock than (x,t) is just a piece of rock for every t<

__t__. (Axiom 4)

Now we are ready to introduce the function

**G**:

**∆**×

**R**→ [0,1] which we call the “

**grade function**” by:

**G**(x,t) = 0 ---------------- if (x,t) is just a piece of rock.**G**(x,t) =**s**(x,t)/**a**(x,t) --- if (x,t) is a problem.

**∆**, tϵ

**R**

**G**(x,t) is a rational number qϵ

**Q**such that 0≤q≤1.

Before making some considerations we have to give again some definitions and axioms:

- Definition: for every problems (x,t) and (y,s) we say “(x,t) is
*harder*(or*easier*) than (y,s)” if**G**(x,t)≤**G**(y,s) (respectively**G**(x,t)≥**G**(y,s)) and*really*harder or easier if the inequalities are strict. - Definition: a problem (x,t) is called a
**stair**if**G**(x,t) = 1. (we say “x is really a*stair*at time t”)

- Axiom 5)
**Uniqueness of attempts**: for every xϵ**∆**, tϵ**R**at most one attempt can happen on x at time t. (climbers’ bodies are supposed to be made of solid matter) - Axiom 6)
**Discreteness of attempts**: for every xϵ**∆**, tϵ**R**exist an Ɛ>0, Ɛϵ**R**such that no attempts occur on x over the period (t-Ɛ,t)∪(t,t+Ɛ). (even the most fanatic climber have to take at least an “*Ɛ-rest”*)

- Definition: we call a
**fall**an*attempt*which is not a*send*. (then being**F**(x,t) the set of the falls of x up to time t, we have:**F**(x,t)=**A**(x,t)\**S**(x,t) and**f**(x,t)=#**F**(x,t)=**a**(x,t)-**s**(x,t)≥0 for every xϵ**∆**, tϵ**R**)

__t__:

by axiom 6, taking t “close enough” to

__t__with t<

__t__we can assume

**G**(x,t)=s/a for some s,aϵ

**N**with s≤a and a>0.

Then

**a**(x,

__t__)=

**a**(x,t)+1=a+1 and

**s**(x,

__t__)=

**s**(x,t)=s, now:

**G**(x,__t__) = s/(a+1) = (s/a) – (s/a)/(a+1) =**G**(x,t) – (**G**(x,t)/(a+1)) ≤**G**(x,t)- Observation: every new fall on a problem (x,t) makes the grade of (x,t)
__decrease__by the non negative quantity:**G**(x,t)/[**a**(x,t)+1]

__t__, again by axiom 6, taking t “close enough” to

__t__with t<

__t__we can assume:

**G**(x,t)=s/a for some s,aϵ

**N**with s≤a and a>0,

**a**(x,

__t__)=

**a**(x,t)+1=a+1 and

**s**(x,

__t__)=

**s**(x,t)+1=s+1, then:

**G**(x,__t__) = (s+1)/(a+1) = (s/a) + [1-(s/a)]/(a+1) =**G**(x,t) + [1-**G**(x,t)]/(a+1) ≥**G**(x,t)- Observation: every new send on a problem (x,t) makes the grade of (x,t)
__increase__by the non negative quantity [1-**G**(x,t)]/[**a**(x,t)+1]

**∆**the function

**G**(x,t) is a “piecewise constant function” with

__t__as a point of discontinuity where:

- (x,
__t__) is a*stair*such that__t__=min{tϵ**R**| (x,t) is a problem}. - (x,
__t__) is an*established problem*which is not a*stair*and such that an*attempt*occur on x at time__t__.

Now all you have to do is to ask

**8a.nu**to add the possibility to take count of all your attempts and you will finally be able to say something objective about grades… as simple as climbing a “stair”…

Further studies about limit of the grade, popularity of a problem, average climbing level and grading scales will follow… stay tuned.

The grade of a problem:

hi ilike ...U blog.nice..

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