What is an atom?

Some set theories are pure, meaning that the only objects in the theory are sets, but some other theories, perhaps we should call them impure set theories, have another kind of object called atoms. The usual definitions of atoms run into problems.

The most common definition of an atom is that it is any object that contains no elements but is not the empty set. Part of this definition is straightforward – atoms and sets are defined intentionally, they have the property that they have no elements – but part of the definition is mysterious – how are we to know that the empty set and atoms are different? In what does the set-ness of the empty set and the atom-ness of atoms reside?

A minority definition of atoms is that they contain only themselves. This is a fine definition, in so far as it goes, and it goes a very long way! There is no end to the recursion of containing oneself, so it is not possible to use atoms in a grounded set theory where everything is built up from primitive objects. The self-containing sort of atom is not primitive because it has some content – itself!

Can transmathematics do better?

The usual logic has two truth values: true, t, and false, f. But transmathematics has a four semantic values: True, T = {t}; False, F = {f}; Contradiction, C = {t f}; and Gap, G = {}. Here the contradiction is said to be both true and false and the gap is said to be neither true nor false.

For us, the membership predicate, x ∈ α, can evaluate to any one of T, F, C, G. Suppose we define that  is an atom exactly when, for all x, it is the case that x ∈ α = G. Now an atom has no elements because there is no element of truth to the claim the x is a member of α and, critically, atoms are different from the empty set because for all x, it is the case that x ∈ {} = F.

Now we have an interesting result. It follows from these intentional definitions that the empty set is unique, and the atom is unique. We choose to call the atom nullity; hence nullity has a precise definition within our totalised set theory and both sets and nullity are primitive so they can occur in grounded set theories.

We can now define sets, S, and antinomies, A, intentionally. S is a set exactly when, for all x it is the case that x ∈ S ∈ {T F G} = T. And A is an antinomy exactly when, for some x it is the case that x ∈ A = C.

Now we have exactly two primitives in our theory – the empty set and nullity – and we have exactly two kinds of compound objects – sets and antinomies.

Falling back on my earlier proposals, we can define the transnatural numbers so that the empty set is zero, nullity is the unique atom, and the Russell set is infinity. The transnatural numbers are then ordered by the set membership predicate.

Recall that the Russell set is defined non-paradoxically in our totalised set theory.

The result that nullity is unique is very strong but it does violence to our earlier proofs that depended on having an arbitrarily large number of atoms.
I have a couple questions regarding nullity and infinity. In your paper Transmathematics, on page 15 (page 37 of 148 for pdf version) you define nullity plus infinity equal to nullity (in line A4).

Is there any particular reason why you choose this as the definition? Or is it possible to define nullity plus infinity equal to infinity?
Post has attachment
The Teaching of the Transsciences has been extinguished in the UK.

My University banned me from teaching transmathematics in first-year courses, then they banned me from teaching it in second-year courses, now they have closed my final-year course in the transsciences. This extinguished the teaching of all trans sciences in the UK.
How is transmath related to homotopy type theory?
How is transmath related to type systems with global/local contexts, effects, and coeffects?
Post has attachment
Para entrar na história: aos 26 de março de 2018, foi defendida na Universidade Federal de Mato Grosso a primeira dissertação de mestrado que relaciona Filosofia com os Números Transreais. A dissertação foi orientada por mim, e o foi defendida pelo meu orientando Jonas Junior Mendes.
Mais uma vitória dos transreais!