Constructive set theory

Axioms that are deemed uncontroversial are Extensionality, Pairing, Union and the predicative separation, i.e. Separation for bounded quantifiers only.

Basic constructive set theory BCST consists of those and Replacement. Constructive theories often have Axiom schema of replacement. However, when other axioms are dropped, this schema is often actually strengthened - but not beyond ZF, instead merely to gain back some provability strength. Replacement and Induction suffices to axiomize hereditarily finite sets constructively and this theory is studied without Infinity.

The Elementary constructive Set Theory ECST is BCST with strong Infinity.

In the program of Predicative Arithmetic, even the axiom scheme of induction for natural numbers, with its universal quantifiers ${\displaystyle \forall n}$ has been criticized as possibly being impredicative, when natural numbers are defined as the object which fulfill this scheme. ECST has infinite objects but not Induction. Despite having the Replacement Axiom, ECST does not proof the addition to be a set function and the theory does not interpret full primitive recursion yet. But note that also here in ECST, many statement can be proven per individual set (as opposed to expressions involving a universal quantifier, as e.g. available with an induction axiom) and objects of mathematical interest can be made use of at the class level on an individual bases. As such, the axioms listed so far suffice as a working theory for a good portion of basic mathematics. In the next step, induction and Function spaces relate the theory to Peano arithmetic, or, more exactly, Heyting arithmetic.

CZFE subsumes ECST.

Firstly, it has the Axiom of regularity, finally stated in the form of an Axiom schema of Set induction:

${\displaystyle [\forall y\;([\forall x\in y\;\phi (x)]\to \phi (y))]\to \forall y\;\phi (y)}$

Regularity as it is normally stated implies LEM, whereas induction does not. Conversely, LEM together with induction implies regularity.

Recall that the class of all subsets of ${\displaystyle A}$ is related to the class of functions, which may be denoted by ${\displaystyle \{0,1\}^{A}}$. As with exponential objects and subobjects in category theory, function spaces are easier to realize than classes of subsets. These sets naturally appear, for example, as the type of the currying bijection given by the adjunction ${\displaystyle ((X\times A)\to B)\cong (X\to B^{A})}$. Constructive set theories are also studied in the context of applicative axioms. So the theory CZFE adopts the Exponentiation axiom:

• Exponentiation: Given two sets ${\displaystyle A,B}$, the collection of all total functions ${\displaystyle B^{A}}$ from one to the other is also in fact a set.

Beyond Myhill's approach, CZFE strengthens the collection scheme:

• Axiom schema of strong collection: This is the constructive replacement for the Replacement scheme. It states that if φ is a binary relation between sets which is total over a certain domain set (that is, it has at least one image of every element in the domain), then there exists a set which contains at least one image under φ of every element of the domain, and only images of elements of the domain. Formally, for any formula φ:

${\displaystyle \forall a([\forall x\in a\;\exists y\;\phi (x,y)]\to [\exists b\;(\forall x\in a\;\exists y\in b\;\phi (x,y))\wedge (\forall y\in b\;\exists x\in a\;\phi (x,y))])}$

Even weaker forms of Exponentiation (restriction to finite domains) interpret Heyting arithmetic or provide tools to uniquely characterize a set of rationals, for example. Here one can reason about ${\displaystyle {\mathbb {Q} }^{\mathbb {N} }}$ and large portion of standard math can be developed in this theory. CZFE does not show the Cauchy reals to be equivalent to the Dedekind reals, which is well known to be provable from LEM or countable choice. In fact, in this context the Dedekind class construction is not leading to a set. As a rule, questions of moderate cardinality are more subtle in a constructive setting, but CZFE has dependent products, proves that the set of all subsets of natural numbers is not subcountable and also proves that countable unions of function spaces of countable sets remain countable.

This theory without LEM, unbounded separation and "naive" Power set enjoys various nice properties. For example, it has the Existence Property: If, for any property ${\displaystyle \Phi }$, the theory proves that a set exist that has that property ${\displaystyle \exists x.\Phi (x)}$, then there is also a property ${\displaystyle \Psi }$ that uniquely describes such a set instance. I.e., the theory then also proves ${\displaystyle \exists !x.\Psi (x)\land \Phi (x)}$. This can be compared to Heyting arithmetic where theorems are realized by concrete natural numbers and have these properties. In set theory, the role is played by defined sets. For contrast, recall that in ZFC, the Axiom of Choice implies the Well-ordering theorem, such that total orderings for sets like ${\displaystyle \mathbb {R} }$ are proven to exist, even if provably no such ordering can be described.

In this context of CZFE, the Axiom of Choice would already imply LEM and Power Set and lead to a theory beyond the strength of classical type theory.

One may approach Power set further without losing a type theoretical interpretation. The theory known as CZF is CZFE plus a stronger form of Exponentiation:

• Axiom schema of subset collection: This is the constructive version of the Power set axiom. Formally, for any formula φ:

${\displaystyle \forall a,b\;\exists u\;\forall z\ ([\forall x\in a\;\exists y\in b\;\phi (x,y,z)]\to \exists v\in u\;[\forall x\in a\;\exists y\in v\;\phi (x,y,z))\wedge (\forall y\in v\;\exists x\in a\;\phi (x,y,z)])}$

The latter is equivalent to a single and somewhat clearer alternative Axiom of Fullness. This states that between any two sets a and b, there is a set c which contains a total sub-relation of any total relation from a to b that can be encoded as a set of ordered pairs. Formally, using a class that is syntactic sugar, one can express this as follows.

• (Fullness)

${\displaystyle \forall a,b\;\exists C\subseteq \Sigma (a,b)\;\forall R\in \Sigma (a,b)\;\exists S\in C\;S\subseteq R}$

Here ${\displaystyle \Sigma (a,b)}$ is the class of all total relations between a and b. With usual subset relation is defined ${\displaystyle X\subseteq Y\iff \forall x\in X\;(x\in Y)}$ and a typical set encoding of the ordered pair ${\displaystyle \langle x,y\rangle }$ is assumed, this class is given as

${\displaystyle \phi \in \Sigma (a,b)\iff (\forall p\in \phi \;\exists x\in a\;\exists y\in b\;\langle x,y\rangle =p)\wedge (\forall x\in a\;\exists y\in b\;\langle x,y\rangle \in \phi ).}$

The Fullness axiom is in turn implied by the so called Presentation Axiom about sections, which can also be formulated category theoretically. Indeed theories beyond ECST are related to predicative topoi.

Linearity of ordinals is still not proven in this theory and assuming it implies Power set in this context.

This theory has modest proof theoretic strength, see IKP: Bachmann–Howard ordinal.

This theory lacks the existence property due to the Schema, but in 1977 Aczel showed that CZF can still be interpreted in Martin-Löf type theory,[4] (using the propositions-as-types approach) providing what is now seen a standard model of CZF in type theory.[5] This is done in terms of images of its functions as well as a fairly direct constructive and predicative justification, while retaining the language of set theory.

In 1989 Ingrid Lindström showed that non-well-founded sets obtained by replacing the Axiom of Foundation in CZF with Aczel's anti-foundation axiom (CZFA) can also be interpreted in Martin-Löf type theory.[6]

As with CZF, the theory IZF has the usual axioms of Extensionality, Pairing, Union, Infinity and Set Induction.

However, IZF also has the standard Separation and Power set. In place of the Axiom schema of replacement, we use the Axiom schema of collection:

${\displaystyle \forall A\;([\forall x\in A\;\exists y\;\phi (x,y)]\to \exists B\;\forall x\in A\;\exists y\in B\;\phi (x,y))}$

While the axiom of replacement requires the relation φ to be functional over the set A (as in, for every x in A there is associated exactly one y), the Axiom of Collection does not. It merely requires there be associated at least one y, and it asserts the existence of a set which collects at least one such y for each such x. LEM together with the Collection implies Replacement.

As such, IZF can be seen as the most straight forward variant of ZF without LEM.

Changing the Axiom scheme of Replacement to the Axiom scheme of Collection, the resulting theory has the Existence Property.

Even without LEM, the proof theoretic strength of IZF equals that of ZF.

While IZF is based on intuitionistic rather than classical logic, it is considered impredicative. It allows formation of sets using the Axiom of separation with any proposition, including ones which contain quantifiers which are not bounded. Thus new sets can be formed in terms of the universe of all sets. Additionally the power set axiom implies the existence of a set of truth values. In the presence of LEM, this set exists and has two elements. In the absence of it, the set of truth values is also considered impredicative.

Back in 1973, John Myhill proposed a system of set theory based on intuitionistic logic[7] taking the most common foundation, ZFC, and throwing away the Axiom of choice (AC) and the law of the excluded middle (LEM), leaving everything else as is. However, different forms of some of the ZFC axioms which are equivalent in the classical setting are inequivalent in the constructive setting, and some forms imply LEM. In those cases, the intuitionistically weaker formulations were then adopted for the constructive set theory.

Let us mention another very weak theory that has been investigated, namely Intuitionistic (or constructive) Kripke–Platek set theory IKP. It doesn't fit into the hierarchy as presented above, simply because it has Axiom schema of Set Induction from the start. The theory has not only Separation but also Collection restricted, i.e. it is similar to BCST but with Induction instead of full Replacement. It is especially weak when studied without Infinity.

As he presented it, Myhill's system CST is a constructive first-order logic with identity and three sorts, namely sets, natural numbers, functions:

• The usual Axiom of extensionality for sets, as well as one for functions, and the usual Axiom of union.
• The Axiom of restricted, or predicative, separation, which is a weakened form of the Separation axiom in classical set theory, requiring that any quantifications be bounded to another set.
• A form of the Axiom of infinity asserting that the collection of natural numbers (for which he introduces a constant N) is in fact a set.
• The Axiom of exponentiation, asserting that for any two sets, there is a third set which contains all (and only) the functions whose domain is the first set, and whose range is the second set. This is a greatly weakened form of the Axiom of power set in classical set theory, to which Myhill, among others, objected on the grounds of its impredicativity.
• An Axiom of dependent choice, which is much weaker than the usual Axiom of choice.

And furthermore:

• The usual Peano axioms for natural numbers.
• Axioms asserting that the domain and range of a function are both sets. Additionally, an Axiom of non-choice asserts the existence of a choice function in cases where the choice is already made. Together these act like the usual Replacement axiom in classical set theory.