(Scientific American, 2023)
TABLE OF CONTENTS
1. Introduction
3. B.H. Slater
5. A Benacerraf-Inspired Critique of Real Numbers
6. Critique of Other Accounts of Real Numbers
7. Infinite Decimals and a Contradiction
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5. A Benacerraf-Inspired Critique of Real Numbers
Our critique of both the Dedekind cut and Cauchy sequence definitions of real numbers adapts and updates Paul Benacerraf’s 1965 argument from “What Numbers Could Not Be” (Benacerraf, 1965), which challenges the identification of numbers with specific set-theoretic objects. Benacerraf argued that natural numbers could be defined as different set constructions (for example, von Neumann ordinals or Zermelo ordinals), but equating these sets leads to absurdities, suggesting numbers are not sets but structural entities. We extend this to real numbers, noting that a real number R can be defined as a Dedekind cut, a set D, a partition of rationals into two subsets, or as an equivalence class of Cauchy sequences, a set C. By the principle of identity, if R = D and R = C, then D = C. Equating these sets is ontologically absurd, because they are fundamentally different kinds of objects, implying that R is neither.
This argument highlights the ontological ambiguity in defining real numbers. A Dedekind cut for a real number like the square root of 2 is a pair of sets of rationals, while a Cauchy sequence is an equivalence class of infinite sequences (for example, sequences like 1, 1.4, 1.414, …). These are structurally distinct: cuts are partitions of a set, while Cauchy sequences are collections of functions from natural numbers to rationals. Equating them as identical sets is problematic, since their elements and constructions differ. For instance, the set D contains rational numbers, while C contains sequences of rational numbers, making D = C incoherent in set-theoretic terms.
This mirrors Benacerraf’s point that multiple set-theoretic reductions of a mathematical object create ontological ambiguity, we would say ontological inconsistency. If real numbers can be “reduced” to different set-theoretic objects, their identity seems arbitrary, suggesting that real numbers are not inherently sets but perhaps abstract entities defined by their structural role in a complete ordered field. This aligns with structuralist philosophies, as advocated by Benacerraf and later by Stewart Shapiro, which view numbers as positions in a system rather than specific objects (Shapiro,1997).
However, the mainstream mathematical response, based in set theory (for example, ZFC), is that Dedekind cuts and Cauchy sequences are not claimed to be identical sets but are isomorphic constructions of the real numbers. Both define systems that satisfy the axioms of a complete ordered field, and category theory shows they are equivalent up to isomorphism. The real number R is not literally the set D or C, but an element in a system defined by either construction. Thus, the identity D = C is not asserted; rather, D and C are different representations of the same abstract entity. This supposedly avoids the ontological absurdity by denying that R is identical to either set in a naive sense.
It could also be objected that our argument assumes a strict set-theoretic ontology, where mathematical objects must be specific sets. Structuralists counter that real numbers are defined by their relations (for example, ordering, and arithmetic operations), not by their implementation. On this view, the choice of Dedekind cuts or Cauchy sequences is a matter of convenience, like choosing different coordinate systems in geometry. The “absurdity” of equating D and C dissolves if real numbers are not sets but placeholders in a structure, as mathematical structuralists propose.
We counter-argue against this critique by maintaining that the claim made by mainstream mathematicians that claiming D and C are isomorphic representations, not identical sets, and that R is an element in the system they define (the complete ordered field), does not address the neo-Benacerraf critique. Using “=” suggests an ontological identity that conflicts with the claim of non-identity; if there is no identity then the classical mathematician is not justified from a logical point of view in using “=”. Multiple set-theoretic reductions (undermine the idea that numbers are Dedekind cuts or Cauchy sequences, regardless of any view of them being isomorphic representations; arguably both positions apply, and ontological inconsistency follows regardless.
Without going into much detail of the structuralism debate, we note that there are strong criticisms of the position. Structuralism, as defended by Stewart Shapiro for example, posits that mathematical objects like real numbers are positions in a structure (for example, the complete ordered field), not specific objects like D or C. But structuralism’s reliance on abstract structures is ontologically problematic (Hellman, 1989). If real numbers are merely roles in a structure, what is the structure itself? Structuralism seems to require the existence of a system (for example, the set of all Dedekind cuts), which presupposes set theory or another foundational framework, risking circularity or infinite regress. This suggests that structuralism does not resolve the ambiguity of D vs. C but shifts the problem to the ontology of structures.
Penelope Maddy, in Realism in Mathematics, notes that structuralism struggles with the identity of mathematical objects (Maddy,1990). If two structures are isomorphic (for example, Dedekind cuts and Cauchy sequences), structuralism treats their elements as identical, but this erases potential distinctions. This implies that structuralism glosses over the ontological differences between D and C, which is a flaw in equating them to R. Defining real numbers as structural roles introduces its own philosophical problems, failing to fully resolve the D = C absurdity.
Mathematical structuralism simply relabels the metaphysical problem of real number identity without resolving it. The structuralist insists that mathematics is about structures, not objects, but never adequately addresses what these structures are, or how we access them.
Structuralism trades objects for structures, but it does not eliminate ontological commitment, it transfers it. Instead of committing to the existence of sets or numbers, it now commits to the existence of vast, often infinite structures like the continuum. But for the finitist, this is precisely the problem. Such structures are neither constructible nor epistemically graspable. They are invoked in toto, as already-formed infinite systems.
But what does this mean? From a finitist perspective, you cannot assert the existence of a position without asserting the existence of what occupies it. A “point” on the real number line is meaningless if you cannot finitely specify or verify it. The structuralist posits positions without content—forms without substance.
Structuralists attempt to evade Benacerraf’s argument by denying that numbers are sets. But they retain the idea that mathematical entities are identifiable by their role in a system. Yet different set-theoretic models (for example, Dedekind cuts vs. Cauchy sequences) describe the same “role” with different underlying elements. This leads structuralists to claim that only the position in the abstract structure matters.
However, if the “role” of √2 is only ever realized by different proxies, none of which is identical to any other, then no object has been specified. This undermines the claim that structuralism explains identity at all. For the finitist, without explicit, finitely realizable identity, there is no mathematical object. There is only semantic fog.
Hence, we have an independent argument standing outside of finitism, that we can use to attack the infinitism of classical mathematics, and thus enable Wildberger’s critique to go through.
6. Critique of Other Accounts of Real Numbers
The other major accounts of real numbers—continued fractions, Eudoxus reals, metric completion, axiomatic fields, hyperreals, and choice sequences—rely on sets in classical mathematics, with intuitionism offering a partial exception.
Continued Fractions: Real numbers can be defined via continued fractions, where a number is represented as an infinite expression of the form a_0 + 1/(a_1 + 1/(a_2 + …)), with a_0 an integer and a_i positive integers for i ≥ 1. Formally, this is a sequence of integers (a_n), and the real number is the limit of the sequence of rational convergents. This construction requires set theory, as the sequence is an element of the set of all infinite sequences of integers, and equivalence classes may be used to handle different representations. While computationally elegant for algebraic numbers, it shares the set-theoretic reliance of Cauchy sequences, defining real numbers as infinite objects within a set-theoretic framework.
Eudoxus Reals (Constructive Approach): Inspired by Eudoxus of Cnidus and revived in constructive mathematics, real numbers can be defined as “almost homomorphisms” from the rationals to the integers. A real number is a function f: Q → Z such that for all rationals p, q, |f(p + q) – f(p) – f(q)| ≤ 1, with additional boundedness conditions. For example, the square root of 2 is approximated by a function mapping rationals to integers based on their proximity to square root of 2. In constructive settings this avoids equivalence classes but still uses sets, as the function f is an element of a set of functions. In classical mathematics, this is embedded in set theory, though constructive versions emphasize computability, aligning partially with our interest in decimals’ computational flavor.
Completion of the Rationals (Metric Space Approach): Real numbers can be defined as the metric completion of the rational numbers under the absolute value metric. This generalizes the Cauchy sequence construction, viewing the reals as the “points” added to make the rationals complete (every Cauchy sequence converges). Formally, this involves equivalence classes of Cauchy sequences or a topological construction, both reliant on set theory to define the space of sequences or points. For example, the square root of 2 emerges as the limit point of sequences like 1, 1.4, 1.414, …. This approach is abstract and heavily set-theoretic, as the completion process requires sets of sequences or ideals in a topological space.
Axiomatic Approach (Complete Ordered Field): Real numbers can be defined axiomatically as the unique (up to isomorphism) complete ordered field, satisfying properties like commutativity, order, and the least upper bound property. This approach avoids explicit construction, focusing on the algebraic and order structure. However, in practice, proving the existence of such a field requires a set-theoretic construction (e.g., Dedekind cuts or Cauchy sequences), as the axioms alone do not specify the objects. Even category-theoretic formulations, which emphasize morphisms, rely on sets to model the field. Thus, this account indirectly depends on set theory.
Non-Standard Analysis (Hyperreals): In non-standard analysis, developed by Abraham Robinson, real numbers are embedded in the hyperreals, a field containing infinitesimal and infinite numbers. Reals are identified as the “standard” elements of this larger structure. This requires set theory, often with additional axioms (for example, the axiom of choice), to construct the hyperreals via ultrapowers or non-standard models of the reals. While philosophically distinct, this approach still defines reals within a set-theoretic framework, as hyperreals are sets of sequences or equivalence classes.
Intuitionist Choice Sequences: In L.E.J. Brouwer’s intuitionism, real numbers are defined as choice sequences—processes generating rational approximations that converge, guided by free choices rather than predetermined rules. For example, a sequence for the square root of 2 might be constructed by successively refining rationals (e.g., 1, 1.4, 1.414, …) based on computational choices. Intuitionism avoids classical set theory’s completed infinities, treating sequences as ongoing processes. However, formalizing choice sequences often involves sets of partial sequences or lawlike rules, and even intuitionist mathematics uses a weak set-theoretic framework to describe collections. Thus, while less set-dependent, this approach still engages with set-like structures.
All of these accounts rely on sets in classical mathematics, where Zermelo-Fraenkel set theory (ZFC) is the standard foundation. Dedekind cuts, Cauchy sequences, continued fractions, Eudoxus reals, and metric completions all define real numbers as sets or elements of sets (for example, sequences, partitions, or functions). The axiomatic approach requires a set-theoretic construction to prove existence, and non-standard analysis uses advanced set-theoretic tools. Intuitionist choice sequences come closest to avoiding sets by emphasizing processes, but formal treatments often involve set-like collections. Non-set-theoretic accounts are rare, as mathematics typically requires a framework to organize infinite objects, and sets are the dominant tool. Alternatives like category theory (emphasizing morphisms) or type theory (used in homotopy type theory) still rely on set-like structures or universes to model real numbers.
Following the critics of set theory cited in this article, including Rescher and Grim, and Wildberger, we may therefore reject all these approaches to defining the real numbers as well.
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