## What is Wrong with Cantor’s Diagonal Argument?

Cantor gave two proofs that the cardinality of the set of real numbers is greater than that of the set of natural numbers. In the better known of these proofs, his diagonal argument, Cantor randomly tabulates the real numbers in the interval [0, 1) in a square array. He asks us to assume for *reductio* purposes that this table is “complete” in the sense that it contains the totality of real numbers in that interval. He then constructs a new number from the diagonal of the table. He asserts that this constructed number is a new real number in [0, 1) which cannot be found in the putatively complete list. His famous conclusion is that no table that lists real numbers in that fashion will have the items in it being enumerable by natural numbers. Since no such list will manage to include all of the real numbers in [0, 1), it follows, according to Cantor, that there have to be more real numbers than natural numbers. Hence the size, or cardinality, of the set of real numbers is greater than that of natural numbers.

I will use a binary version of Cantor’s table for ease of exposition:

1 0.**1**01110110001…

2 0.1**1**0101110000…

3 0.11**0**100011101…

4 0.011**0**11100011…

5 0.1000**0**1000110…

6 0.00111**1**111110…

7 0.001011**1**00010…

8 0.1100011**0**1001…

9 0.11101100**1**010…

10 0.011110111**0**01…

11 0.0101010101**0**0…

** .**

** .**

** .**

** ***d* = 0.11000110100…

* i* = 0.00111001011…

Thus we begin with the Cantorian assumption that this infinitely long and random list includes all the real numbers in the interval [0, 1). The numbers on the diagonal of the list (boldfaced in the table) give us the infinitely long number *d*. From *d*, Cantor would obtain the number *i*, by substituting every 0 after the decimal point in *d* by 1, and every 1 in *d* by 0. Now, the number *i* couldn’t be found in the list, according to Cantor, because *i* is guaranteed to differ from each binary number in the list since the number’s digit that falls on the diagonal is changed. Therefore, no list of real numbers (in this example the reals in [0, 1)) can be enumerated by natural numbers, which entails that there are more real numbers than natural numbers.

The problem I see with Cantor’s argument is the following. I will call two binary numbers in [0, 1) “inverses” of each other if one number has 1 in a certain location after the decimal point, the other number has 0 in the corresponding location; and if it has 0 in that location, the other number has 1. Thus the numbers on lines 3 and 7, for instance, are inverses of each other. So are the numbers on lines 5 and 10. Crucially, *d* and *i* are also inverses of each other. If we are to assume this list is complete, every number in the list must have its inverse also included in the list. Cantor says *i* is not included in the list. But how come? The number *i*’s inverse, namely *d*, *is* in the list—it is on line 8. Since its inverse is in there *i* must also be in there, like every other pair of inverses.[*] Suppose *d*’s inverse *was* in the list, say it was on line 17. Cantor would say his diagonal trick would change the 17th digit of the number on line 17, and therefore *i* could not be in the list.

So we have a contradiction: (a) The number *i* has to be in the list, say in line 17, because its inverse *d* is. (b) Yet *i* cannot be in the list, because it differs from the number on the 17th line in its 17th digit. If one thinks (a) and (b) both sound plausible, as they might, one would find himself facing this paradox:

(i) We can make a random list of real numbers in [0, 1).

(ii) This list can be assumed to be exhaustive.

(iii) The diagonal *d* of the list and *i* can be fully formed.

(iv) The number *i* has to be in the list, since its inverse is.

(v) The number *i* cannot be in the list, since it is different from every number in the list in some *n*-th digit.

Since (iv) is impeccable, Cantor’s diagonal procedure which derives (v) can be faulted with leading to paradox, and consequently his denial of (ii) can be claimed unjustified. My view is that (iv) is definitely true and Cantor’s conclusion (v) is false. Hence actually there is no paradox, but just the false proposition (v). (Unlike Cantor’s random array, I use a tree approach to *systematically* display the totality of real numbers, which guarantees that* i* never falls outside of the set of real numbers. But I do that elsewhere.[**])

As Cantor’s argument leads to paradox, or at least to (in my opinion) the false claim (v), his diagonalization procedure is flawed and his denial of (ii) based on this defective procedure is unjustified. Therefore his claim that there are more real numbers than natural numbers is unwarranted.

As for the flaw in Cantor’s diagonalization procedure, I think it is this. The diagonal and its inverse clash at some digit. Suppose the 17th digit of the diagonal is 1; then the 17th digit of its (assumed) inverse will have to be 0. But, since the two digits overlap, the diagonal’s 17th digit being 1 rules out the inverse’s 17th digit being 0. Thus the diagonal disallows its inverse from being a member of the array. The trees, on the other hand, are not susceptible to taking diagonals, and every real number has its inverse included in the tree—as it should.

.

_______________________

[*] It would not make sense to claim that since *i* is not in the list its inverse could not be in the list. The inverse of *i* is a perfectly legitimate number (like the number on line 8 in our example) to be included in the supposedly complete list. And if *i* were not to be in the list although its inverse is, we could not assume the list was complete to begin with.

[**] My more detailed criticisms of Cantor’s arguments and my way of showing that reals cannot be more numerous than naturals can be found at: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_

## Talk at Bogazici, Jonathan Cohen (UCSD), “On the Presuppositional Behavior of Coherence-Driven Pragmatic Enrichments”

Please join us for this talk.

Consider (1) and (2):

(1) Every time the company fires an employee who comes in

late, a union complaint is lodged.

(2) If the company fires an employee who comes in late, a

union complaint will be lodged.

Now suppose there is an employee, Snodgrass, who is fired because he

was discovered to have been embezzling, and that the firing occurred

on a day on which he had happened to come in late. And suppose the

union does nothing about it. In this situation, it seems that (1) and

(2) can still be true.If so, there must be an enrichment at play that affects truth

conditions: even though Snodgrass came in late and was fired, the fact

that Snodgrass wasn’t fired *because* he was late causes the

event to sit outside of the domain restriction of (1), and, likewise,

not to satisfy the antecedent of the conditional in (2). Importantly,

this enrichment has no linguistic mandate, and therefore is clearly

pragmatic. So why does it intrude upon the truth conditions of (1) and

(2)?

We argue that this behavior arises as a result of the way in which the

enrichments interact with pragmatic presupposition. In particular, in

producing examples (1)-(2) with the intention to communicate the

enrichment, a speaker presupposes that a causal relation necessary to

make the inference is part of the common ground, and intends that

presupposition to restrict the interpretation of the

quantifier/antecedent in (1)/(2).

The analysis connects the associative inferences that underlie the

establishment of coherence relations between sentences in a discourse,

a class of intrasentential enrichments that result from the same

principles, and the manner in which presuppositions constrain the

interpretation of quantified expressions and conditionals.

## MA Philosophy at Bilkent – Round 2

## Applications for Masters in Philosophy (Round 2)

We are now accepting the second round of applications for the M.A. in Philosophy (for those starting the degree in Fall 2016).

Up to five successful applicants will have the opportunity to spend a semester at Australian National University. In addition successful applicants will receive a comprehensive scholarship (tuition waiver, accommodation and monthly stipend including private health insurance).

This is a two year masters that includes coursework and a thesis. The language of instruction for all courses is English.

Admission requirements and online application can be found here.

Philosophy *and* non-philosophy majors are encouraged to apply. We also warmly welcome applications from international students.

**Application deadline (Round 2): 18 July 2016**

Written exam (Round 2): 25 July 2016**

Interview (Round 2): 27 July 2016**

Contact: phil@bilkent.edu.tr

http://www.phil.bilkent.edu.tr

*Those planning to take the COPE exam on June 28-30 should contact us at the above email address by June 22 to reserve a place.

** International students may take the written exam remotely and complete the interview via skype.

## METU Undergraduate Students Philosophy Conference

Call for Abstracts:

The Middle East Technical University Philosophy Department’s undergraduate students are organizing a conference for undergraduate students. The conference will take place over the weekend of 5-6 November 2016. The language of the conference is Turkish. Please distribute the call for papers below to undergraduate students who might be interested in submitting their works.” (http://philevents.org/event/show/23506)

## Talk at Bilkent 22/06: Sandrine Berges on Sophie de Grouchy

“Sophie de Grouchy and the publication of Condorcet’s Sketch of Human Progress: a tale of exclusion”.

__Wednesday 22 June, 12:40 – 13:30, H235__

In this paper I examine some of the evidence for collaboration between Condorcet and Sophie de Grouchy on the writing of the *Sketch of Human Progress*, but also, uncover the ways in which the publication and reception of that text worked to exclude a woman who was a philosopher in her own right from a work she clearly contributed to.

In 1795, the Convention of the French Republic, regretting its role in bringing about Condorcet’s death, commissioned 3000 copies of his last piece, a *Sketch of Human Progress*. Daunou was chosen to edit it and wisely, he asked Condorcet’s widow and collaborator, Sophie de Grouchy, to co-edit. This same text was re-edited by Grouchy in 1802 when she brought out the complete works of her husband, but when in 1847 Arago, of the Academie Francaise, decided to publish a new edition of the complete works, he put the Daunou/Grouchy edition of the *Sketch* aside and instead ‘went back to the manuscript’ provided him by his own co-editor, the Condorcets’s daughter Eliza.

A look at the manuscript itself shows that it would have been hard to extract a clear text from it – it is hard to decipher, heavily annotated, and clearly waiting further revisions. Moreover, some of the annotations appear to be in Grouchy’s hand, suggesting that she may have collaborated with her husband on the manuscript.

There are other reasons to suppose that husband and wife may have worked together on the *Sketch*, some relating to the history of this particular work, but also because they had collaborated in the past.

If I am right that Sophie de Grouchy had a hand in the writing of the *Sketch*, it seems that we have strong reasons not to dismiss – and indeed to prefer – her edition of that same text in 1795, and again 1802, as she would have been in a much stronger position to make sense of that very messy manuscript than an editor half a century later would.

## Nazim Keven joins Bilkent Philosophy

The philosophy department at Bilkent is extremely happy to announce that Nazim Keven (PhD, WUSTL Philosophy-Neuroscience-Psychology Program) will be joining the department in Fall 2016. Nazım’s main area of research is the philosophy of mind, with a particular focus on the nature of memory, reasoning, emotions, and the self.

He has a forthcoming target article in ‘Behavioral and Brain Sciences’, as well as articles in ‘Synthese’ and ‘Hippocampus’.

Web page: http://wustl.academia.edu/NazimKeven

*Mind*,

*Philosoph*

*ical Quarte*

*rly*, and

*Ratio*.

Rina Tzinman (PhD, Miami) – Rina’s main area of research is metaphysics and philosophy of mind, with a particular emphasis on person identity.

## Conference (Boğaziçi): Kant and Moral Psychology (25/06/2016-27/06/2016)

**Kant and Moral Psychology**

# Boğaziçi University, Istanbul

# (25^{th}-27^{th } June 2016)

**Saturday, 25**^{th} June 2016

^{th}June 2016

**Venue: **TB310

10.30 – 12.00 **Paul Guyer **(Brown) “Moral Worth and Moral Motivation: Kant’s Real View.” Chair: **Saniye Vatansever** (Bilkent)

12.00 – 1.00 LUNCH

1.00 – 2.20 **Lucas Thorpe** (Boğaziçi) “Impulses, Inclinations and Maxims: A Kantian Alternative to Belief-Desire Psychology” Chair: **Evrim Emir-Sayers **(Amsterdam)

2.30 – 3.50 **Sasha Mudd** (Southampton) “Both the Law and the Good: Rethinking the Ground of Kant’s Ethics” Chair: **Umut Eldem **(Bogazici)

4.00 – 5.20 **Jamie Buckland** (IFILNOVA, The New University of Lisbon) “Kant: The Limiting Case of an Internal Reasons Theorist” Chair: **Gamze Keskin Yurdakurban **(Kırklareli)

7pm Dinner

**Sunday, 26**^{th} June 2016

^{th}June 2016

**Venue:** TB310

10.30 – 12.00 **Alix Cohen** (Edinburgh) “Kant on the Nature of Emotions” Chair: **Özlem Duva** (Dokuz Eylül)

12.00 – 1.00 LUNCH

1.00 – 2.20 **Louise Chapman** (KCL) “We Don’t Need No Noumena? Freedom Through Rational Self-Cultivation in Kant” Chair: **Taylan Susam** (Bogazici)

2.30 – 3.50 **Martin Sticker** (Göttingen) & **Joe Saunders** (Leeds) “Time to Learn: Moral Education and Transcendental Idealism” Chair: **Çağlar Çömez** (Bogazici)

4.00 – 5.20 **Carl Hildebrand** (Oxford): “Education as Character Formation in Kant’s Pedagogy and Anthropology” Chair: **Zübeyde Karadağ Thorpe** (Hacettepe)

5.30- 7pm **Daniel Mendez** (Boston): ““casuistical questions” in Kant’s *Metaphysics of Morals” *Chair: **Kaan Arıkan** (Bogazici)

7pm Dinner

**Monday, 27**^{th} June 2016

^{th}June 2016

**Venue**: Demir Demirgil Solonu

10.30 – 12.00 **Ken Westphal** (Boğaziçi) “Aristotle, Kant & Moral Integrity: Our Fidelity to Reason” Chair: **Melisa Kurtcan** (Bogazici)

12.00 – 1.00 LUNCH

1.00 – 3.00 **Local Graduate Student Papers: **Chair**: Lucas Thorpe** (Bogazici)

1:00 – 1.30 **Umut Eldem** (Boğaziçi) “Kant’s Account of Conscience: A Reply to Dean Moyar”

1.30 – 2.00 **Taylan Susam** (Boğaziçi) “Weakness and Resolution in Kant”

2.00 – 2.30 **Seniye Tilev** (Boğaziçi) “Hope in Kant”

2.30 – 3.00 **Gözde Yıldırım** (Boğaziçi) “Understanding Intelligible Evil”

Support for this conference was provided by Funding for this Workshop was provided by the joint Boğaziçi -Southampton Newton-Katip Çelebi project AF140071 “Agency and Autonomy: Kant and the Normative Foundations of Republican Self-Government” run by Lucas Thorpe (Boğaziçi) and Sasha Mudd (Southampton) and by Lucas Thorpe’s Bogazici University BAP project 9320