nLab analytic WLPO

Context

Analysis

Constructivism, Realizability, Computability

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms

Contents

Idea

In real analysis, there is a principle on the Cauchy real numbers which is equivalent in strength to the weak limited principle of omniscience for natural numbers, and states that the partial order on the Cauchy real numbers is a decidable relation, or that the Cauchy real numbers have decidable equality. This principle can be generalised from the Cauchy real numbers to all notions of real numbers, such as the Dedekind real numbers, and is hence called the analytic weak limited principle of omniscience, or the analytic WLPO. In classical mathematics, all of these are implied by excluded middle; however, in neutral constructive mathematics, the different analytic WLPO are different principles of different strength, because the Cauchy and Dedekind real numbers and any other notion of real numbers between the Cauchy and Dedekind real numbers cannot be proven to coincide.

Definition

There are many different notions of real numbers in constructive mathematics. What all these real numbers have in common are that they are an Archimedean ordered field extension RR of the field of (sequential, modulated) Cauchy real numbers R C\mathrm{R}_C; i.e. there are unique ring homomorphisms of ordered fields

CR D\mathbb{R}_C \hookrightarrow R \hookrightarrow \mathbb{R}_D

where D\mathbb{R}_D is the ordered field of Dedekind real numbers - the homomorphism into D\mathbb{R}_D always exists since D\mathbb{R}_D is the terminal Archimedean ordered field. In general the ring homomorphisms are not provably isomorphisms unless the Dedekind real numbers and Cauchy real numbers coincide. Thus, there are many different notions of the analytic weak limited principle of omniscience; the analytic weak limited principles of omniscience are defined for each Archimedean ordered field extension of the Cauchy real numbers.

Definition

Let RR be an Archimedean ordered field extension of the field of (sequential, modulated) Cauchy real numbers R C\mathrm{R}_C. The analytic weak limited principle of omniscience or analytic WLPO for RR states that RR has decidable equality, or that the partial order on RR is decidable (xyx \leq y or ¬(xy)\neg (x \leq y)).

Warning. Given an Archimedean ordered field extension RR of the field of (sequential, modulated) Cauchy real numbers, the analytic weak limited principle of omniscience for RR is not to be confused with the weak limited principle of omniscience for RR - the latter is the statement that for every function ff from RR to the boolean domain, the universal quantifier “for all elements xx in RR, f(x)=0f(x) = 0” is decidable. For RR the Cauchy real numbers, the analytic WLPO for the Cauchy real numbers is equivalent to the WLPO for the natural numbers, while the WLPO for the Cauchy real numbers is much stronger than the WLPO for the natural numbers.

Let RR be any Archimedean ordered field extension of the Cauchy real numbers. The analytic WLPO makes sense for any Archimedean ordered local Artinian R R -algebra AA as well, where the relation <\lt is in general only a strict weak order instead of a pseudo-order, the preorder \geq is not a partial order, and the equivalence relation aba \approx b derived from the preorder holds if and only if aba - b is nilpotent. In this case, the analytic WLPO for the Weil RR-algebra AA says that the preorder on AA is a decidable relation, or that the equivalence relation aba \approx b is a decidable relation, or nilpotency is decidable, and the quotient of AA by its nilradical is the field of real numbers RR satisfying the analytic WLPO.

Properties

Theorem

The analytic WLPO for the (sequential, modulated) Cauchy real numbers C\mathbb{R}_C implies the WLPO for natural numbers.

Proof

Let ff be a function from the natural numbers to the boolean domain. Define the sequence uu on the rationals as

u(n)= x=0 kf(n)2 xu(n) = \sum_{x = 0}^k \frac{f(n)}{2^x}

This sequence is a Cauchy sequence with a modulus of continuity, so taking the limit of this Cauchy sequence yields a Cauchy real number [u][u]. The analytic WLPO on the Cauchy reals implies that [u]=0[u] = 0 is decidable. Since [u]=0[u] = 0 if and only if f(n)=0f(n) = 0 for all nn, [u]=0[u] = 0 is decidable if and only if “f(n)=0f(n) = 0 for all nn” is decidable, the latter of which is the WLPO for natural numbers. Thus, analytic WLPO implies the WLPO for natural numbers.

Theorem

Suppose that there are two Archimedean ordered fields RR and RR' such that both are field extensions of the Cauchy real numbers C\mathbb{R}_C and RR' is a field extension of RR. Then, the analytic WLPO for RR' implies the analytic WLPO for RR.

Proof

Analytic WLPO for RR' implies that RR' has decidable equality. Any subset of a set with decidable equality also has decidable equality. Since RR is a subfield of RR', RR also has decidable equality, which is the analytic WLPO for RR.

Lemma

Suppose that there is an Archimedean ordered field RR which is a field extension of the Cauchy real numbers C\mathbb{R}_C. Then, the analytic WLPO for RR implies the WLPO for natural numbers.

Proof

Since the field of Cauchy real numbers is a (trivial) field extension of itself, the analytic WLPO for RR implies the analytic WLPO for the Cauchy real numbers, which implies the WLPO for the natural numbers.

Theorem

The WLPO for natural numbers implies the analytic WLPO for the (sequential, modulated) Cauchy real numbers C\mathbb{R}_C.

Proof

Lemma

The WLPO for natural numbers is equivalent to the analytic WLPO for the (sequential, modulated) Cauchy real numbers C\mathbb{R}_C.

Thus, one has a hierarchy of analytic weak limited principles of omniscience, where

  • The (sequential-, modulated-, Cantor-, Cauchy-, C\mathbb{R}_C-) analytic WLPO is the weakest and equivalent to the WLPO for the natural numbers;

  • The (Dedekind-, D\mathbb{R}_D-)analytic WLPO is the strongest.

  • For any other Archimedean ordered field extension of the Cauchy real numbers RR which is neither equivalent to the Cauchy real numbers or the Dedekind real numbers, the RR-analytic WLPO are intermediate in strength between the Cauchy-analytic WLPO and the Dedekind-analytic WLPO.

Given an Archimedean ordered field extension RR of the Cauchy real numbers, there are other statements related to the analytic WLPO:

  • That every element of RR has a choice of radix expansion in any base (e.g., a decimal expansion or binary expansion) implies that the analytic WPO\mathrm{WPO} for RR holds.

  • Subcountability of RR implies the analytic WLPO\mathrm{WLPO} because natural numbers have decidable equality and injections preserve and reflect decidable equality.

Models

References

The analytic WLPO is mentioned in the following paper, although unfortunately it uses the phrase “analytic LPO” for what should really be the analytic WLPO (the reals have decidable equality). In addition, it makes the wrong statement that LPO is equivalent to Cauchy reals having decidable equality; it is WLPO which is equivalent to Cauchy reals having decidable equality (i.e. analytic WLPO for Cauchy reals), while LPO is equivalent to Cauchy reals having decidable tight apartness.

Last revised on July 7, 2024 at 13:38:27. See the history of this page for a list of all contributions to it.