Effective topos

Preliminaries Kleene realizability The topos is based on the partial combinatory algebra given by Kleene's first algebra {\mathcal{K}}_1. In Kleene's notion of recursive realizability, any predicate is assigned realizing numbers, i.e. a subset of {\mathbb N}. The extremal propositions are \top and \bot, realized by {\mathbb N} and \{\}. However in general, this process assigns more data to a proposition than just a binary truth value. A formula with k free variables will give rise to a map in (\mathcal P{\mathbb N})^{{\mathbb N}^k} the values of which are the subsets of corresponding realizing numbers. Realizability topoi {\mathsf{Eff}} is a prime example of a realizability topos. These are a class of elementary topoi with an intuitionistic internal logic and fulfilling a form of dependent choice. They are generally not Grothendieck topoi. In particular, the effective topos is {\mathsf{RT}}({\mathcal{K}}_1). Other realizability topos constructions can be said to abstract away some aspects played by {\mathbb N} here. Description of Eff The objects are pairs \langle X, E_X\rangle consisting of a set together with a symmetric and transitive relation in (\mathcal P{\mathbb N})^{X\times X}, representing a form of equality predicate, but taking values that are subsets of \mathbb N. One writes E_X(x) with just one argument to denote the so called existence predicate, expressing how an x relates to itself. This may be empty, as the relation is not generally reflexive. Arrows amount to equivalence classes of functional relations appropriately respecting the defined equalities. The classifier amounts to \mathcal P{\mathbb N}. The pair (or, by abuse of notation, just that underlying powerset) may be denoted as \Omega. An entailment relation \vdash_X on \mathcal P{\mathbb N}^X makes it into a Heyting pre-algebra. Such a context allows to define the appropriate lattice-like logic structure, with logical operations given in terms of operations of the realizer sets, making use of pairs and computable functions. The terminal object is a singleton \langle \{*\}, E_{\{*\}}\rangle with trivial existence predicate (i.e., equal to {\mathbb N}). The finite product respects the equality appropriately. The classifier's equality E_{\mathcal P{\mathbb N}} is given through equivalences in its lattice. ==Properties==

Relation to Sets Some objects exhibit a rather trivial existence predicate depending only on the validity of the equality relation "=" of sets, so that valid equality maps to the top set \mathbb N and rejected equality maps to \{\}. This gives rise to a full and faithful functor \nabla\colon{\mathsf{Sets}}\to{\mathsf{Eff}} out of the category of sets, which has the finite-limit preserving global sections functor \Gamma as its left-adjoint. This factors through a finite-limit preserving, full and faithful embedding \omega-{\mathsf{Sets}}\to{\mathsf{Eff}}. NNO The topos has a natural numbers object N=\langle{\mathbb N}, E_{\mathbb N}\rangle with simply E_{\mathbb N}(n)=\{n\}. Sentences true about N are exactly the recursively realized sentences of Heyting arithmetic {\mathsf{HA}}. Now arrows N\to N may be understood as the total recursive functions and this also holds internally for N^N. The latter is the pair given by total recursive functions \mathrm{TR} and a relation such that E_\mathrm{TR}(f) is the set of codes e\in {\mathbb N} for f. The latter is a subset of the naturals but not just a singleton, since there are several indices computing the same recursive function. So here the second entry of the objects represent the realizing data. With N and functions from and to it, as well as with simple rules for the equality relations when forming finite products \times, one may now more broadly define the hereditarily effective operations. Again one may think of functions in N^N as given by indices and their equality is determined by the objects that compute the same function. This equality clearly puts a constraint on N^{(N^N)}, as these functions come out to be only those computable functions that also properly respect the mentioned equality in their domain. Et cetera. The situation for general \langle X, E_X\rangle\to \langle Y, E_Y\rangle, equality (in the sense of the E's) in domain and image must be respected. Properties and principles With this, one may validate Markov's principle {\mathrm{MP}} and the extended Church's principle {\mathrm{ECT}}_0 (and a second-order variant thereof), which come down to simple statement about object such as N^N or (1+1)^N. These imply {\mathrm{CT}}_0 and independence of premise {\mathrm{IP}}_0. A choice principle N^N related to Brouwerian weak continuity fails. From any object, there are only countably many arrows to N. \Omega^N fulfills a uniformity principle. N is not the countable coproduct of copies of 1. This topos is not a category of sheaves. Analysis The object \langle{\mathbb Q}^{\mathbb N}, E_{{\mathbb Q}^{\mathbb N}}\rangle is effective in a formal sense and from it one may define computable Cauchy sequences. Through a quotient, the topos has a real numbers object which has no non-trivial decidable subobject. With choice, the notion of Dedekind reals coincides with the Cauchy one. Properties and principles Analysis here corresponds to the recursive school of constructivism. It rejects the claim that x\le 0\lor 0\le x would hold for all reals x. Formulations of the intermediate value theorem fail and all functions from the reals to the reals are provenly continuous. A Specker sequence exists and then Bolzano–Weierstrass fails. ==See also==