Here's something I'm thinking about for a while, and would like to get feedback and some relevant references.

Say I want to define by axioms an operation that will act like exponent, without using addition, multiplication, limits etc.

Let's look at set K with binary operation { , } : KxK --> K that provides for each a,x,y,z,w in K :

  1. {{a,x},y}={{a,y},x}

  2. if {{a,x},y} = {a,w} then {{a,{x,z}},{y,z}} = {a,{w,z}}

The reason for the first axiom is clear, the second will be clarified later.

For a,x in K, if exists a unique w in K so that {a,w} = x, then write: [a,x] := w

(I want to express log operation)

We immediately get (If the following is defined): [a,{a,x}] = {a,[a,x]} = x

In addition, it can be shown that for each a,b in K (again, if defined): [a,a] = [b,b]

(so from now on I'll write just [,] to represent such expressions)

It follows that if exists [a,x] then: {x,[,]} = x

More identities that can be proven:

{[,],x} = [,]

{x,[a,[,]]} = [,]

(when translating it to exponents and logarithms, we get the known identities.)

From now on I won't pay attention to weather a specific log is defined, when I write an expression with log, the meaning is to the case it is defined.

Now I'd like to express the arithmetic operations using those operations. For that reason let's take some element e from K, and in order to get more compact expressions, I'll write them without e, for example:

[,x] := [e,x]

{y,} := {y,e}

The multiplication can be defined by: x*y := [{{x},y}]

The addition by: x+y := [[{{{x}},{y}}]]

Provides: x+y = [{x}*{y}] as required.

The first axiom gives commutativity and associativity to multiplication and addition. The second axiom gives distributivity between them.

The expressions [,] and [,[,]] act as identity element regarding to multiplication and addition.

The expressions [{,x},] and [,[{,{,x}},]] act as inverse (to x) elements regarding to multiplication and addition.

This definition pushes us to define a whole series of operations (*n) .

Define recursively (although it can be defined explicitly): x(*n+1)y := [,{,x}(*n){,y}]

and respectively: x(*n-1)y := {,[,x](*n)[,y]}

Now if we define the (*0) operation to be: x(*0)y = {x,[,y]} then we'll get the multiplication and addition as the (*-1) and the (*-2) operations, as part of a series of commutative, associative operations with distributions between each two consecutive operations, and with identity elements of the form [,[,[,...[,]...]]] or {,{,{,...{,}...}}} or e, and inverse elements.

Back to classic arithmetic, we have expressions for identity elements (0 and 1), and for multiplication and addition, which means we can express every rational number, for example:

0 = [,[,]]

1 = [,[,{,}]] = [,]

2 = [,[,{{,},}]]

3 = [,[,{{{,},},}]]

-1 = [,[{,},]]

1/2 = [[,{{,},}],]


It reminds Zermelo's construction of natural numbers...

I can continue, but I think the idea is clear. I still have a big gap related to the question when log is defined. I guess it can be defined in different ways in order to get integers, rationals, or other fields. obviously we can get roots as well... I'm curious weather I can allow existence of, for example, log(0) without getting contradiction... And what else can I get? I'm also not sure if the choosing of the second axiom is the best for this purpose.

Generally, I see this construction very natural and elegant but I don't know if it has a real value, means, if it can be connected to other mathematical fields. I'll be happy to hear any idea.


  • $\begingroup$ There have been a number of axiomatizations of the theory of ordered fields (or real-closed fields) with exponentiation. Names I can think of casually are Wilkie, Macintyre, Ressayre. There are quite a few others. All the work I know of is model-theoretic, so not directly related to your work. $\endgroup$ – André Nicolas Feb 28 '16 at 7:49

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