The key idea behind Thesa is the extension of number to the huge numbers that represent programs. These numbers do not have successors like traditional numbers. A radix representation breaks down because of scale. These huge numbers are still numbers, but we need a new way to represent them.
A number is a named set of numbers. This identifies names and sets with numbers. Let's try to tweak apart these concepts. A name is a rigid designator, i.e., given a name, I can get the thing named--always. Now you may not know the name, or you may use the name for something else, or the thing may not be accessible, but what makes something a name is this absolute quality to it.

A set is a collection, a bunch of things. We want it to be the things, i.e., its extension. We use sets instead of things because we need the empty set--the set of no things. Otherwise there is no foundation to start from. So our first numbers are named empty sets. The name can be anything. The digits '0'...'9' are a common starting point. We could have started elsewhere. For example, '0' could be the only name of the empty set, or even 'digit' could be that name. It doesn't matter, as long as we agree. Let's call these named empty sets, the 'digits'.

At the next level, we're naming sets that contain digits. Again we have complete flexibility. The only limitation is that a set can't include duplicate names, otherwise a name is no longer a rigid designator. If we were building up decimal notation, our next names could be names like 'ones place', 'tens place', 'hundredths place', and so on. So the number 327 would be: '327':{'hundredths':{'3':{}}, 'tens':{'2':{}}, 'ones':{'7':{}}}.

Isn't this just numerals and notation? Where is the number, 327? That it comes after 326 and that it's divisible by 3 and the myriad other facts that we know about it. Before we answer, let's look at the problem of huge numbers where these kinds of facts are not important. A huge number is a named set of numbers. That set can be small because the numbers may themselves be huge. The names can be anything we want, i.e., they can be a vivid description of the component numbers. So there it is, a vivid representation of huge numbers. That such a number is divisible by 3 or has a successor is irrelevant to the number and to our uses of it.

There are numbers for which successor and divisibility are important. These are the numbers we use in our daily life. Let's call them the small numbers. We've already mapped one way of representing them. It was arbitrary, but then any representation is arbitrary. What makes it work is that once we agree on the representation, the number is guaranteed.

The guarantee lets us use the numbers as rules. Turing's construction is a good one. A Turing machine qua named set of numbers is a set of state/symbol pairs naming sets of next state, new symbol, and left/right numbers. Isn't this just a representation of a Turing machine and not the Turing machine itself? But the representation is the Turing machine, or more accurately the representation with the agreement about the representation. Turing's universal machine is of interest, but it's really just duplicating the agreement that we already have. The numbers can be input to the machine, and any computational thing can be the output. So we get successor and divisibility by 3.

We also get many equivalent numbers. But we already had many equivalent numbers, it was just that we tend to talk about just one of them. For example, we have 5 and V and 101 and 47 mod 7. These are all different numbers because they have different names, and they're all the same number because I can give equivalency predicates that return true for any pair.

To sum things up, a number is a named set of numbers. A Turing machine and hence a rule is a number. A set is a number (and vice versa) and a name is a number (and vice versa). The construction of a number is arbitrary, but our knowledge about the number is absolute. This is because our knowledge is the construction. Its usefulness is based on agreement. Number, system, and computation are emergent properties of language. (cbb 4/94)