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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 tϵR the “time” coordinate.

For every xϵ∆, tϵR 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 tϵR, xϵ∆)
• Axiom 3) A(x,t) is finite for every xϵ∆, tϵR. (then so is 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))

Observe that by axiom 3) it follows: 0 ≤ s(x,t) ≤ a(x,t) for every xϵ∆, tϵR.

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

(By axioms 4) we ensure that the numbers of attempts and sends are not decreasing with time)

• 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”)

Observation: if (k,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.

Observation: again if (x,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) | tϵR & (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…”)

Observation: if (x,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.

Observe that by axiom 3) and by definition for every xϵ∆, tϵR G(x,t) is a rational number qϵQ 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”)

As a consequence every project is really harder than any established problem which is harder than any stair. And you cannot say that any “just a piece of rock” is harder or easier than anything.

• Axiom 5) Uniqueness of attempts: for every xϵ∆, tϵR 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∆, tϵR exist an Ɛ>0, ƐϵR 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”)

Note that if an attempt occur on x at time t, then no other attempts occur on x between t-Ɛ and t+Ɛ, and if no attempts occur on x at time t then no attempts occur on x between t-Ɛ and t+Ɛ.

• 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∆, tϵR)

Now let we analyze the effect of a new fall happened on a problem x at time t :
by axiom 6, taking t “close enough” to t with t<t we can assume G(x,t)=s/a for some s,aϵN 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]

Otherwise if a send happens on a problem x at time 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,aϵN 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]

Having that said it’s easy to show (exercise n.1) that for every xϵ∆ the function G(x,t) is a “piecewise constant function” with t as a point of discontinuity where:

1. (x,t) is a stair such that t=min{tϵR | (x,t) is a problem}.
2. (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: