(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
The following essay has been published in four installments; this installment, the fourth and final one, contains section 7.
But you can also download and read or share a .pdf of the complete text of this essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.
7. Infinite Decimals and a Contradiction
Finally, we conclude with a reconsideration of the account of real numbers in terms of infinite decimals. We previously concluded that this account does survive the finitist criticisms made against it, most notably, that multiplication and addition are not well defined. However, in other papers (Smith, Smith, & Stocks, 2023a, 2023b), we argued that a contradiction can be produced with this account and the idea of supertasks, completing an infinite number of tasks in a finite time. This notion has been employed to deal with Zeno’s paradoxes, and it seems to be the status quo position in this area of philosophy of mathematical physics. Robert Hanna has argued that Zeno’s paradoxes can be solved without recourse to super tasks, and we concur (Hanna, 2024). But that being said though, as we noted in our previous paper, supertasks have been employed elsewhere in the philosophy of mathematics apart from dealing with Zeno’s paradoxes, so we can assume that their use is legitimate, at least for a reductio. Summarized, our argument starts with the standard identity:
1.000… = .999… .
Using a supertask, inspired by Hilbert’s hotel, we shift decimals and infinite number of times to produce:
1000… = 999…
numbers that violate the Archimedean property and lead to a contradiction, as from this, it is easily seen that 1=0 is provable. Since mainstream mathematics accepts supertasks, rejecting our supertask is ad hoc. The contradiction suggests the real number system is inconsistent. Thus, we force a dilemma: either reject supertasks or accept real number inconsistency. While there are as Hanna has shown, alternative ways of dealing with Zeno’s paradoxes without use of supertasks, there have been decades of work devoted to defending the notion. Here we let it stand for the sake of argument, and conclude that what falls is the account of real numbers as infinite decimals, while previously we directed our attack against supertasks.
The objections can be countered by accusing our critics of begging the question.
An Objection from Supertask Invalidity: Critics might argue our decimal- shifting supertask isn’t analogous to Zeno’s convergent series or Hilbert’s Hotel, as it produces ill-defined numbers (1000…, 999…)
Rejoinder: This begs the question by assuming supertasks are only valid in contexts that preserve real number consistency. Since mainstream mathematics accepts supertasks (e.g., summing infinite series), it must justify why our manipulation is invalid without appealing to the system we’re questioning. Hilbert’s Hotel, which allows infinite shifts, supports our case—critics can’t just cherry-pick which infinity is “okay.”
The Objection that 999… is hyperreal: One objection is that 999… is a hyperreal, not a standard real, avoiding the contradiction in the real number field.
Rejoinder: This is irrelevant and question-begging. The infinite decimal account claims to define standard reals (for example, 0.999… = 1). If 999… is a hyperreal, the account fails to consistently define all infinite decimals as reals, proving our point. Assuming 999… isn’t a real presupposes the system’s consistency, dodging the contradiction we’ve raised. And it does not matter what one calls 999…, as our argument still goes through.
An Objection from Archimedean Violation: Critics note that 1000… and 999… are non-Archimedean, so they’re not real numbers, as the real field is Archimedean by definition.
Rejoinder: We embrace this: “Sure, but since the whole system is inconsistent, what do you expect?” If supertasks produce non-Archimedean numbers, the infinite decimal account generates entities it can’t handle, exposing its flaws. Assuming the system is Archimedean begs the question, as our contradiction (1 = 0) challenges the field’s coherence.
An Objection from Practical Success: Mainstream mathematicians might argue infinite decimals work in practice (for example, approximating π), so theoretical contradictions are irrelevant.
Rejoinder: This begs the question by prioritizing utility over logical consistency. If the system is inconsistent, practical success is a house of cards. Our skeptical stance demands a foundation free of contradictions, not a pragmatic shrug. Our approach mirrors epistemological skepticism by refusing to accept assumptions (for example, real number consistency, supertask limits) without independent justification, just as according to, say, David Hume’s problem of justifying induction. This forces critics to engage with the possibility of systemic inconsistency, or simply join Wildberger and other finitists in rejecting the infinitist approach to real numbers, which amounts to embracing the skeptical position accepted by Robinson in the two quotations used as the epigrams of this paper.
As John Lennon might have said: we hope someday you will join us, and the world will be one … or at least finite, once more!
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