(Sci Am, 2017)
TABLE OF CONTENTS
1. Introduction
2. The Easy Part: Why 1=0.999 …
3. The Hard Part: Why ~ (1=0.999 …) is True as Well
4. Compatibility with the Standard Real Number Axioms
5. The Archimedean Principle Objection
6. A Tangent: Refuting Cantor’s Diagonal Argument
7. Conclusion
The essay below has been published in two installments; this, the second and final one, contains sections 5-7.
But you can also download and read or share a .pdf of the complete text of the essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.
Why 1=.999… and ~ (1=.999…) are Both True! An Argument for the Inconsistency of the Reals, #2
5. The Archimedean Principle Objection
The standard proof that derives the Archimedean Principle from the completeness axiom of the real numbers—any non-empty set of numbers that is bounded above has a least upper bound —presupposes assumptions that are non-constructive, question-begging, or model-dependent. Once we step outside the framework of classical, set-theoretic real analysis, the connection between completeness and Archimedeanness becomes far less secure.
In standard real analysis, the Archimedean Principle states that for every real number x, there exists a natural number n such that n > x. This principle is often “derived” from the completeness axiom of the real numbers, also known as the least upper bound property.
The proof proceeds by reductio: assume that some x is greater than or equal to every natural number. Then the set of natural numbers is bounded above by x, so by completeness, it has a least upper bound s. But then s – 1 is not an upper bound, so there must exist a natural number n such that n > s – 1, implying n + 1 > s, which contradicts the assumption that s was the least upper bound. Therefore, such an x cannot exist.
The critical step in the proof appeals to the existence of a least upper bound for the set of natural numbers, even though this set is unbounded. The contradiction arises from assuming the existence of this bound under a false hypothesis. Any such proof already assumes that the natural numbers are unbounded in ℝ — which is logically equivalent to the Archimedean Principle.
But from a constructivist perspective, this use of the completeness axiom is invalid. Constructive mathematics, in the spirit of Brouwer or Bishop, does not permit one to assert the existence of a mathematical object unless it can be explicitly constructed or approximated. In this proof, no explicit construction is provided—only an abstract existence claim, grounded in the law of excluded middle and proof by reductio, both of which are rejected in constructive logic.
From this point of view, the standard proof is not a derivation of Archimedeanness from completeness, but a demonstration that classical logic yields certain conclusions—nothing more.
Even if we accept classical logic and the real numbers as a completed set, there is a subtler problem. The received constructions of the real numbers themselves already rely on assumptions that embed the Archimedean property.
In Dedekind’s construction, real numbers are defined as cuts in the rational numbers. But the rationals already satisfy the Archimedean property. Similarly, in Cauchy’s construction using sequences of rationals, the metric and ordering structure reflect Archimedean behavior from the outset.
Therefore, when we “derive” the Archimedean Principle from the completeness axiom, we are effectively deriving a property that was built into the structure all along. The derivation is circular: it claims to prove what was already assumed implicitly through the underlying framework.
The most striking challenge to the standard derivation comes from the existence of alternative number systems that reject the Archimedean property while still supporting forms of completeness. The best-known example is the hyperreal number system used in nonstandard analysis. Hyperreal numbers include infinitesimal and infinite elements and satisfy a model-theoretic version of completeness (e.g., saturation), yet they clearly violate the Archimedean property. For instance, there are hyperreal numbers ε such that 0 < ε < 1/n for all natural numbers n. No matter how large n is, ε remains smaller—exactly the opposite of what Archimedeanness requires.
This demonstrates that completeness, at least in the abstract, does not entail Archimedeanness. It only does so within certain constructions of the real numbers—not in all fields or ordered systems that claim a form of completeness.
A more radical critique comes from ultrafinitist or anti-Platonist perspectives, which reject the actual infinite altogether. From this viewpoint, applying the completeness axiom to the set of natural numbers is meaningless, since ℕ is not a completed totality. If we treat infinite sets as mere potential processes rather than actual objects, then applying “least upper bound” arguments to such sets becomes invalid. The contradiction that results in the classical proof then appears as a conjuring trick: it relies on reasoning about entities that don’t exist in the ontology of finitist mathematics. In this light, the Archimedean principle ceases to be a necessary logical truth and becomes instead a decree of a particular foundational framework.
The classical proof of the Archimedean Principle from completeness is compelling only within a narrow set of assumptions: classical logic, set-theoretic realism, and the traditional construction of real numbers. It presupposes what it claims to prove, and it fails to hold in constructive, nonstandard, or finitist contexts. The Archimedean Principle is thus not a universal consequence of completeness, but a reflection of the philosophical assumptions embedded in the classical model of the continuum. The more honest conclusion is that completeness and Archimedeanness are co-dependent, each making sense only in a system that has already chosen to accept the other.
From the perspective of the reworking of real numbers in this essay, which accepts infinite numbers for the purposes of a reductio ad absurdum argument in order ultimately to support finitism, we can justifiably maintain that the Archimedean Principle fails for numbers in our system, such as 0.000 … 001 and 999 … 999. That being said, it should be noted that from the table giving the construction of the reals, the first number in the line is a natural number which is in a 1-1 correlation with a real. Thus:
| Index | Real Number | Description |
| 0 | 0.000…000 | All zeros, last digit 0 at infinity |
| 1 | 0.000…001 | Zeros with last digit 1 at infinity |
| 2 | 0.000 … 002 | Zeros with last digit 2 at infinity |
| … | … | Continuing through all possible digits |
| N | 999 … 999 | All nines, last digit 9 at infinity |
Here n = 999 … 999, so the last real number, and last natural number, are the same, contrary to the Archimedean Principle. We may take this as a counter-example to the full generality of the Archimedean Principle which does not negate its applicability outside of exotic considerations such as the present discussion.
Indeed, even apart from the presentation given here there is a case to be made for a Counter Archimedean Principle (CP): There exists a real number r* such that for all n ∈ ℕ, n < r*:
(CP) ∀ n ∈ ℕ, ∃ r* ∈ ℝ such that r* > n.
This r* would behave like an infinite number, something not allowed in standard real analysis, precisely because the Archimedean Principle rules it out by definition. But there are arguments for (CP), even within the framework of standard mathematics:
(CP) Reflects the Unbounded Nature of Reals
(CP) emphasizes a fundamental property of the real numbers: they are unbounded above. For any natural number n, you can always find a real number r* > n (e.g., r* = n + 1). This captures the intuitive idea that the real number line extends infinitely, with no largest element; rejected of course in this paper. In contrast to the Archimedean Principle, which focuses on natural numbers overtaking any real number, (CP) highlights the real numbers’ capacity to always exceed any finite bound. This perspective is equally intuitive and aligns with the structure of ℝ as an ordered field with no upper bound.
Simpler Conceptual Framework
(CP) is arguably more straightforward than the Archimedean Principle. It directly expresses the idea that the real numbers are “inexhaustible” in terms of magnitude. You don’t need to invoke the interplay between two sets (ℕ and ℝ) as AP does; (CP) requires only understanding that for any natural number, a larger real exists. This simplicity could make (CP) a more fundamental principle for describing the real numbers’ infinite extent, especially in contexts where we prioritize the properties of ℝ over its relationship with ℕ.
Philosophical Appeal: Avoiding Commitment to Natural Numbers’ Dominance
The Archimedean Principle implicitly prioritizes the natural numbers’ ability to surpass any real number, suggesting a kind of “dominance” of ℕ over ℝ in terms of magnitude. (CP), however, treats the real numbers as primary, emphasizing their infinite extensibility without requiring natural numbers to “catch up.” If one views the real numbers as a more complete or fundamental system (e.g., for modelling continuous phenomena like time or space), (CP) could be seen as a more natural axiom, focusing on ℝ’s intrinsic properties rather than its comparison to ℕ.
Compatibility with Non-Standard Systems
While (CP) holds in standard real analysis, it also aligns well with non-standard systems like the hyperreals, where infinite numbers exist. In such systems, (CP) remains true: for any natural number n, there exists a hyperreal r* > n (including infinite hyperreals). The Archimedean Principle, however, fails in non-standard analysis because infinite hyperreals exceed all natural numbers. If one were to argue for a mathematical framework that accommodates infinitesimals or infinite numbers (e.g., for modelling physical or philosophical concepts), (CP) is more flexible and doesn’t commit to the Archimedean property, which restricts the number system to exclude such elements.
Potential for Alternative Mathematical Foundations
(CP) could serve as a starting point for constructing number systems where the focus is on unboundedness rather than the Archimedean property. For example, in certain theoretical contexts (e.g., non-standard analysis or surreal numbers), prioritizing (CP) could lead to systems that allow for more exotic elements like infinitesimals or infinite numbers. This could be advantageous in fields like physics or metaphysics, where concepts of the infinite or infinitesimal might better capture certain phenomena. By contrast, the Archimedean Principle enforces a stricter structure that excludes these possibilities, which might be seen as limiting.
We conclude that the Archimedean Principle objection can be rejected.
6. A Tangent: Refuting Cantor’s Diagonal Argument
The construction of real numbers given here can offer a refutation of Georg Cantor’s Diagonal argument. Cantor’s diagonal argument is a supposed proof that the set of real numbers between 0 and 1 is uncountably infinite, meaning it cannot be put into a one-to-one correspondence with the natural numbers (1, 2, 3, …). This implies that the real numbers are “more infinite” than the natural numbers, which are countably infinite.
Cantor’s Argument
Assume that the set of all real numbers between 0 and 1 is countable. This means we can list them in a sequence, such as: r1 = 0.d11 d12 d13 d14 … r2 = 0.d21 d22 d23 d24 … r3 = 0.d31 d32 d33 d34 … r4 = 0.d41 d42 d43 d44 … and so on, where each dij is a digit (0 through 9) in the decimal expansion of the real number ri.
The assertion is that this list contains every possible real number between 0 and 1. Cantor’s goal is to show this assumption leads to a contradiction.
Construct a new number, x, by examining the diagonal digits of this list (i.e., d11, d22, d33, d44, …). For each position i, take the digit dii and create x = 0.x1 x2 x3 x4 … by “flipping” each diagonal digit as follows:
If dii is not 5, set xi = 5.
If dii is 5, set xi = 6.
This ensures that xi differs from dii for every position i. For example:
If r1 = 0.3741…, then d11 = 3, so x1 = 5 (since 3 ≠ 5).
If r2 = 0.4512…, then d22 = 5, so x2 = 6 (since 5 = 5).
If r3 = 0.2953…, then d33 = 9, so x3 = 5 (since 9 ≠ 5).
The number x is a real number between 0 and 1 (its decimal expansion consists of 5s and 6s). However, x cannot be in the list because it differs from:
r1 in the first digit (x1 ≠ d11),
r2 in the second digit (x2 ≠ d22),
r3 in the third digit (x3 ≠ d33), and so on.
This contradicts the assumption that the list contains all real numbers between 0 and 1, since x is a real number not in the list.
Conclusion: The assumption that the real numbers between 0 and 1 are countable leads to a contradiction. Therefore, the set of real numbers is uncountable, meaning it has a larger cardinality (size) than the set of natural numbers.
However, our table above lists real numbers as infinite decimal strings terminating at an “infinite” position, starting from 0.000 … 000 to 0.999 … 999, to demonstrate that all reals are countable.
Cantor’s diagonal argument constructs a number by changing the nth digit of the nth number. Cantor’s diagonalization aims to shows the reals are uncountable by constructing a number not on any list of reals.
Cantor’s diagonalization constructs a number differing from each listed number in at least one digit. In a countable list (infinite in one direction, like a sequence), the diagonal number is guaranteed to be missing, proving uncountability.
In our system, the diagonal sequence (e.g., changing the last digit of each number) is included in the table by definition, as every possible string is listed.
Therefore the reals are countable, thereby refuting Cantor’s proof in this framework.
Our Argument
Table Construction: List all reals as infinite strings, starting from 0.000 … 000 (all zeros) to 999 … 999 (all 9s), in a systematic order (e.g., lexicographically or by increasing value). The table has natural numbers in the first column, then in the body of the table, the reals so that the natural number 0 aligns with 000 … 000, 1 with 000 … 001, and so on up to 999 … 999, aligning with 999 … 999.
Including the Diagonal: We claim the diagonal sequence (formed by taking the nth digit of the nth number and changing it) is already in the table by definition, making the reals countable. Our table is countable (has a one-to-one correspondence with the natural numbers) and includes all possible sequences, including the diagonal.
Implication: As all reals, including the diagonal, are listed, Cantor’s proof fails.
There are alternative proofs that the cardinality of the natural numbers is not equal to that of the real numbers, such as Nested Intervals Theorem (Cantor’s Nested Intervals Argument), Baire Category Theorem Approach and the Power Set Argument. These proofs, like the diagonal argument, assume the existence of infinite sets or completed infinities, which finitists reject. The nested intervals and Baire Category approaches rely on the completeness of the real numbers, while the power set argument uses Cantor’s theorem about power sets. For finitists, who may deny infinite processes or sets, these proofs are not convincing, as they inherently involve infinite constructions.
Cantor’s diagonal argument and alternative proofs (nested intervals, Baire Category, and power set arguments) supposedly establish that the real numbers are uncountable, with a cardinality greater than that of the natural numbers. While these proofs are widely accepted in standard mathematics, there are serious objections from various mathematical and philosophical perspectives, excluding finitism (which rejects completed infinities).
Constructivist Objections
Constructivism, a philosophy of mathematics associated with figures like L.E.J. Brouwer and Errett Bishop, holds that mathematical objects must be explicitly constructible. Constructivists may object to uncountability proofs because the diagonal argument constructs a real number x by flipping digits, but this process is not necessarily constructive. It assumes a completed list of all real numbers and defines x by a non-algorithmic process, which constructivists may reject as non-effective.
Similarly, the nested intervals argument relies on the completeness property of the reals, which guarantees the existence of a number in the intersection of nested intervals. Constructivists may argue that this number is not explicitly constructible, because the process of excluding each listed number rn is not a finite algorithm.
The Baire Category approach assumes the real line can be treated as a complete metric space, which constructivists question, because they require explicit constructions rather than existence proofs based on topological properties.
The power set argument involves the set of all subsets of the natural numbers (P(ℕ)), which constructivists may not accept as a well-defined object unless each subset can be explicitly described.
Constructivists do not necessarily deny the uncountability of the reals but may argue that the proofs are invalid in a constructive framework, because they rely on non-constructive principles like the law of the excluded middle (e.g., assuming a list either contains all reals or does not).
Intuitionist Objections
Intuitionism, a related philosophy developed by Brouwer, rejects the law of the excluded middle and emphasizes mathematics as a mental activity. Intuitionists may object to the diagonal argument’s assumption that a list of all real numbers can be meaningfully considered. Intuitionists argue that infinite objects must be constructed step-by-step, and a completed infinite list is not a valid object in their framework.
The use of proof by contradiction in all these arguments (diagonal, nested intervals, Baire Category, and power set), is questioned. Intuitionists reject proofs that assume a statement (e.g., the reals are countable) and derive a contradiction, as this relies on classical logic. They require a direct construction of a counterexample, which these proofs do not provide.
For example, in the diagonal argument, the construction of x depends on inspecting an infinite list, which intuitionists may see as an invalid idealization. Similarly, the nested intervals argument assumes the existence of a limit point without constructing it explicitly.
Intuitionists might not reject the conclusion that the reals are uncountable but instead demand proofs that align with their constructive, intuition-based logic.
7. Conclusion
We conclude that both 1 = 0.999 … and ~ (1 = 0.999 …), are both true, each being supported by sound, independently based arguments. But this situation is problematic even from the perspective of paraconsistent logic:
(1) 1 = 0.999 …
(2) 1 > 0.999 …
(3) 1>1 (1), (2) substitution.
From (3) we can prove that any real number is greater than itself, which would destroy real number theory as surely as a proof that 1 = 0. Thus, we are faced with options such as rejecting that 1 = 0.999 …, or its negation. We have argued that either option is acceptable because both statements are true. This leaves the finitist position, concluding that the infinite decimal account of real numbers is objectionable and should be rejected. The mathematical skeptic, however, will see this as a demonstration of the inconsistency of the real numbers, but also defend the infinite decimal account.
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