Russell's typicality as another randomness notion

Abstract

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of "small" cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies cf(|M|) > ℵ 0 , then there exist |M| typical elements and only <|M| non-typical ones. In particular this is true for the standard model R of second-order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML-random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways).

Typicality à la Russell in Set Theory

“…Proof. (i) The proof is easy and quite similar to the proof of Theorem 1.3 above (see [9], Theorem 2.9). It follows simply from two facts: first, there are countably many properties φ(x) of L without parameters, and second, the cofinality of…”

“…For in that case a whole segment of V that strictly extends V ω should consist exclusively of nontypical sets. In particular we would have P(ω) ⊆ V ω+1 ⊆ NT κ , i.e., all reals would be nontypical, contrary to the fact that typicality was introduced in [9] (rather successfully) as a parallel notion of randomness, with a large amount of reals to be proved typical (with respect to the "external" notion of typicality employed there).…”

Typicality à la Russell in Set Theory

“…Proof. (i) The proof is easy and quite similar to the proof of Theorem 1.3 above (see [9], Theorem 2.9). It follows simply from two facts: first, there are countably many properties φ(x) of L without parameters, and second, the cofinality of…”

“…For in that case a whole segment of V that strictly extends V ω should consist exclusively of nontypical sets. In particular we would have P(ω) ⊆ V ω+1 ⊆ NT κ , i.e., all reals would be nontypical, contrary to the fact that typicality was introduced in [9] (rather successfully) as a parallel notion of randomness, with a large amount of reals to be proved typical (with respect to the "external" notion of typicality employed there).…”

Asymptotic typicality degrees of properties over finite structures

“…In [6] we set out an investigation of a notion of typicality which is originated with B. Russell. Specifically in [5, p. 89], Russell defines a typical Englishman to be one "who possesses all the properties possessed by a majority of Englishmen."…”

“…we can define typical elements of M in the spirit of Russell, provided we first define what a typical property is. Given a formula φ(x) of L without parameters, let φ(M) denote the extension of φ(x) in M, i.e., In [6], among other things, we established the existence of typical elements in many infinite structures. For example the standard structure of reals (or second-order arithmetic) contains |R|-many typical reals, while only < |R|many nontypical ones.…”

A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

“…Recall that an enumeration ST × BC × BC 2 = { T ξ , b ξ , c ξ : ξ < ω 1 }, parameter-free definable in L ω 1 , is fixed by (8) in Section 5. We suppose that the tree T α belongs to P <α .…”

Typicality à la Russell in Set Theory

“…Proof. (i) The proof is easy and quite similar to the proof of Theorem 1.3 above (see [9], Theorem 2.9). It follows simply from two facts: first, there are countably many properties φ(x) of L without parameters, and second, the cofinality of…”

“…For in that case a whole segment of V that strictly extends V ω should consist exclusively of nontypical sets. In particular we would have P(ω) ⊆ V ω+1 ⊆ NT κ , i.e., all reals would be nontypical, contrary to the fact that typicality was introduced in [9] (rather successfully) as a parallel notion of randomness, with a large amount of reals to be proved typical (with respect to the "external" notion of typicality employed there).…”

Asymptotic typicality degrees of properties over finite structures

“…In [6] we set out an investigation of a notion of typicality which is originated with B. Russell. Specifically in [5, p. 89], Russell defines a typical Englishman to be one "who possesses all the properties possessed by a majority of Englishmen."…”

“…we can define typical elements of M in the spirit of Russell, provided we first define what a typical property is. Given a formula φ(x) of L without parameters, let φ(M) denote the extension of φ(x) in M, i.e., In [6], among other things, we established the existence of typical elements in many infinite structures. For example the standard structure of reals (or second-order arithmetic) contains |R|-many typical reals, while only < |R|many nontypical ones.…”

A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

“…Recall that an enumeration ST × BC × BC 2 = { T ξ , b ξ , c ξ : ξ < ω 1 }, parameter-free definable in L ω 1 , is fixed by (8) in Section 5. We suppose that the tree T α belongs to P <α .…”

Typicality à la Russell in Set Theory

“…Proof. (i) The proof is easy and quite similar to the proof of Theorem 1.3 above (see [9], Theorem 2.9). It follows simply from two facts: first, there are countably many properties φ(x) of L without parameters, and second, the cofinality of…”

“…For in that case a whole segment of V that strictly extends V ω should consist exclusively of nontypical sets. In particular we would have P(ω) ⊆ V ω+1 ⊆ NT κ , i.e., all reals would be nontypical, contrary to the fact that typicality was introduced in [9] (rather successfully) as a parallel notion of randomness, with a large amount of reals to be proved typical (with respect to the "external" notion of typicality employed there).…”

Asymptotic typicality degrees of properties over finite structures

“…In [6] we set out an investigation of a notion of typicality which is originated with B. Russell. Specifically in [5, p. 89], Russell defines a typical Englishman to be one "who possesses all the properties possessed by a majority of Englishmen."…”

“…we can define typical elements of M in the spirit of Russell, provided we first define what a typical property is. Given a formula φ(x) of L without parameters, let φ(M) denote the extension of φ(x) in M, i.e., In [6], among other things, we established the existence of typical elements in many infinite structures. For example the standard structure of reals (or second-order arithmetic) contains |R|-many typical reals, while only < |R|many nontypical ones.…”

A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

“…Recall that an enumeration ST × BC × BC 2 = { T ξ , b ξ , c ξ : ξ < ω 1 }, parameter-free definable in L ω 1 , is fixed by (8) in Section 5. We suppose that the tree T α belongs to P <α .…”