natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
basic constructions:
strong axioms
further
Elementary function arithmetic, or EFA, is a first-order theory of natural numbers, one of the weakest fragments of arithmetic strong enough to do nontrivial mathematics. It is strictly weaker than Peano arithmetic.
Regarding the strength of EFA, Harvey Friedman has put forth the following “grand conjecture”:
“Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, ×, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.”
Here “bounded quantifier” refers to a quantifier of shape $\forall_{m \lt n}$; more formally, if a variable $n$ does not occur in a formula $\phi$, then $\forall_{m \lt n} \phi(m)$ means $\forall_m (m \lt n) \Rightarrow \phi(m)$.
The language of EFA is one-sorted with
Two constants $0, 1$
Three binary function symbols $+, \cdot, \exp$ (where $\exp(x, y)$ is usually written $x^y$).
The axioms of EFA are
Those of Robinson arithmetic (where $s$ is translated as $1 + (-)$, i.e., as $+(1, -)$);
Exponentiation axioms, viz. $x^0 = 1$ and $x^y \cdot x = x^{y+1}$;
The induction axiom for formulas all of whose quantifiers are bounded.
“Grand conjectures” by Harvey Friedman may be found here:
A MathOverflow discussion on Friedman’s grand conjecture about EFA may be found here,