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Wikipedia / Heyting arithmetic

In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it.

Axiomatization

Heyting arithmetic can be characterized just like the first-order theory of Peano arithmetic, except that it uses the intuitionistic predicate calculus for inference. In particular, this means that the double-negation elimination principle, as well as the principle of the excluded middle-theorems-theorems.

Heyting arithmetic comprises the axioms of Peano arithmetic and the intended model is the collection of natural numbers and thus a trivial proposition.

For terms. This formal elimination of disjunctions was not possible in the quantifier-free primitive recursive arithmetic. The theory may be extended with function symbols for any primitive recursive function.

Theorems

Double negations

With explosion is provable if and only if. For a predicate expresses that it is inconsistent to rule out. A big part of the metatheoretical discussion will concern classically provable existence claims.

.

Proofs of classically equivalent statements

can classically be reversed, the classically equivalent Gödel–Gentzen negative translation The result means that all Peano arithmetic theorems have a proof that consists of a constructive proof followed by a classical logical rewriting. Roughly, the final step amounts to applications of double-negation elimination.

.

Valid principles and rules

Minimal logic. More generally, Heyting arithmetic proves this classical equivalence for any Harrop formula.

-results are well behaved as well: Markov's rule at the lowest level of the arithmetical hierarchy is an admissible rule of inference free,

Instead of speaking of quantifier-free predicates, one may equivalently formulate this for primitive recursive predicate or Kleene's T predicate.

Beware that in classifying a proposition based on its syntactic form, one ought not mistakenly assign a lower complexity based on some only classical valid equivalence.

Excluded middle

can be proven in this constructive arithmetic. By disjunction introduction from the axioms, one may validate the premise for induction. Stronger yet, as equality is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier are the free variables, the theory is closed under the rule

for all propositions. So if the theory is consistent, it never proves the negation of an excluded middle statement.

.

Conservativity

. The Friedman translation,

. Roughly, simple statements about computable relations provable classically are already provable constructively. Although in halting problems, is formally not particularly simple.

. This can be contrasted with the situation for Robinson arithmetic-theorems are already independent of it. Induction can be seen to play a crucial role in Friedman's-statements than its intuitionistic counterpart.

The discussion here is by no means exhaustive. There are various results for when a classical theorem is already entailed by the constructive theory. Also note that it can be relevant what logic was used to obtain metalogical results. For example, many results on realizability were indeed obtained in a constructive metalogic. But when no specific context is given, stated results need to be assumed to be classical.

Unprovable statements

is independent of the latter. Given some independent propositions, it is easy to define more from them, especially in a constructive framework.

Heyting arithmetic has the disjunction property,

Indeed, this and its numerical generalization are also exhibited by constructive second-order arithmetic and common set theories is independent.

, or what amounts to an {\mathrm{WLPO}} for a primitive recursive function.

Classically independent propositions

Knowledge of Gödel's incompleteness theorems-provable.

The resolution of Hilbert's tenth problem and corresponding polynomial equations, such that the claim that the latter have a solution is algorithmically undecidable. The proposition can be expressed as

prove that these propositions are equivalent to the arithmetized claim of the theories own inconsistency. Thus, such propositions can even be written down for strong classical set theories.

, by pushing a negation through the quantifier, is seen to be an independent Goldbach-type) is also independent. And any triple-negation is, in any case, already intuitionistically equivalent to a single negation.

PA violates DP

.

In an effectively axiomatized theory, one may successively perform an inspection of each proof. If a theory is indeed consistent, then there does not exist formalizes the never halting of the search by stating that all proofs are not a proof of an absurdity. And indeed, in an omega-consistent.

. In a constructive context, this use of the negation sign may be misleading nomenclature.

Friedman established another interesting unprovable statement, namely that a consistent and adequate theory never proves its arithmetized disjunction property.

Unprovable classical principles

-provable. So one of the De Morgan's laws cannot intuitionistically hold in general either.

, the least number principle, is generally valid. In light of Gödel's proof, the breakdown of these three principles can be understood as Heyting arithmetic being consistent with the provability reading of constructive logic.

. Although in the form of the corresponding rules, they are admissible, as mentioned. Similarly, the theory does not prove the independence of premise. The same holds for the version where the existential statement is replaced by a mere disjunction.

" with universal quantification, as discussed in the section on Church's thesis.

Least number principle

Making use of the order relation on the naturals, the strong induction principle reads

, a logical equivalent to induction is

:

In Peano arithmetic, where double-negation elimination is always valid, this proves the least number principle in its common formulation. In the classical reading, being non-empty is equivalent to (provably) being inhabited by some least member.

A binary relation or equivalently

.

as above), consider the predicate

is trivially true and so the class is inhabited. Further, using decidability of equality and the disjunctive syllogism is independent, then also the predicate is undecidable in the theory.

.

. This demonstrates that elusive predicates can define elusive subsets. And so also in constructive set theory, while the standard order on the class of naturals is decidable, the naturals are not well-ordered. But strong induction principles, that constructively do not imply unrealizable existence claims, are also still available.

Anti-classical extensions

, the classically trivial infinite disjunction

  1. ,

, can be read as a validation of the decidability of a decision problem.

are consistent. Again, constructively, such a negation is not.

Church's thesis

. Church's thesis principle rejects it: It implies negations such as the ones just described.

Consider the principle in the form stating that all predicates that are decidable in the logic sense above are also decidable by a total computable function for predicates defined from Kleene's T predicate of partial computable function indices with a witness describing how they halt on the diagonal, is computably enumerable but not computable is not even computably enumerable, see halting problem. This provenly undecidable problem.

. The intuitionist school of L. E. J. Brouwer.

Models

Realizability

.

is typically valid. And it is a theorem that any c.e. extension of arithmetic for which it holds also has the numerical existence property:

.

Kleene, a student of Church, introduced important realizability models of the Heyting arithmetic. In turn, his student Nels David Nelson then there is a partial recursive function-provable but do have a realization.

Typed versions of realizability have been introduced by Georg Kreisel. With it he demonstrated the independence of the classically valid Markov's principle for intuitionistic theories.

See also BHK interpretation and Dialectica interpretation.

In the effective topos is categorical. Categoricity here is reminiscent of Tennenbaum's theorem and so completeness fails in this context.

Consistency and Soundness

If a theory is consistent, then no proof is one of an absurdity. Kurt Gödel. However, Gödel's incompleteness theorems, about the incapability of certain theories to prove their own consistency, also applies to Heyting arithmetic itself.

-sound..)

as well as any of its non-standard models.

Set theory

There are also constructive set theory and its intended semantics. Relatively weak set theories suffice: They shall adopt the Axiom of infinity, the Axiom schema of predicative separation, as well as the existence of function spaces on finite domains for recursive definitions, the full axiom of separation or set induction (let alone the axiom of regularity), nor general function spaces (let alone the full axiom of power set).

furthermore is bi-interpretable, so that the collection of von Neumann naturals do not exist as a set in the theory. and the other axioms are those related to set algebra and order: Union and Binary Intersection, which is tightly related to the Predicative Separation schema, Extensionality, Pairing, and the Set induction in this set theory is as in model theory. And in the other direction, the set theoretical axioms are proven with respect to the primitive recursive relation

That small universe of sets can be understood as the ordered collection of finite binary sequences'th set contains four other sets. See BIT predicate.

Type theory

Type theoretical realizations mirroring inference rules based logic formalizations have been implemented in various languages.

Extensions

Heyting arithmetic has been discussed with potential function symbols added for primitive recursive functions. That theory proves the Ackermann function total.

have been extensively studied in proof theory would be continuous. But adopting, say, different extensionality rules, choice axioms, Markov's and independence principles and even the Kőnig's lemma-formulas. Early on, also variants with intensional equality and Brouwerian choice sequence have been investigated.

Reverse mathematics studies of constructive second-order arithmetic have been performed.

History

Formal axiomatization of the theory trace back to Heyting (1930), Herbrand and Kleene. Gödel in 1933.

Related concepts

Heyting arithmetic should not be confused with Heyting algebras, which are the intuitionistic analogue of Boolean algebras.

See also

References

  • Ulrich Kohlenbach (2008), Applied proof theory, Springer.

  • Anne S. Troelstra, ed. (1973), Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, 1973.