Inconsistent set theory lives on. My paper "Transfinite Cardinals in Paraconsistent Set Theory," has just been accepted by the Review of Symbolic Logic. Here's the abstract:

"This paper develops a (non-trivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem."

## A site for the Marsden Fund projects at the University of Otago and the University of Canterbury

Investigators: Zach Weber; Maarten McKubre-Jordens

## Sunday, November 20, 2011

## Thursday, October 20, 2011

### Reflections on getting a job

In 1960, Joe Kittinger took a helium balloon up to 103,000 feet ... and jumped out. The mind boggles. Watch it on youtube.

First: Kittinger was the first man in space. This is interesting from a vagueness perspective. "In space" turns out to be a vague predicate. As I've argued elsewhere, because "in space"" is vague, there are multiple men who were the first.

Second: getting a PhD in logic, hoping for and applying for jobs, has felt like jumping from the edge of space. It was a stupid thing to do, it shouldn't have worked out ... but I've landed safely.

I've just accepted a permanent position at the University of Otago, to start next year.

All of us who are jumping: it feels like an insane thing to do. It is. And not every parachute will open. I can only thank my lucky stars to have reached the ground alive.

First: Kittinger was the first man in space. This is interesting from a vagueness perspective. "In space" turns out to be a vague predicate. As I've argued elsewhere, because "in space"" is vague, there are multiple men who were the first.

Second: getting a PhD in logic, hoping for and applying for jobs, has felt like jumping from the edge of space. It was a stupid thing to do, it shouldn't have worked out ... but I've landed safely.

I've just accepted a permanent position at the University of Otago, to start next year.

All of us who are jumping: it feels like an insane thing to do. It is. And not every parachute will open. I can only thank my lucky stars to have reached the ground alive.

## Wednesday, August 24, 2011

### New article in Plus Magazine

Maarten McKubre-Jordens (my coauthor on a recent paper on real analysis) has just published a very nice introductory article about paraconsistent mathematics --

-- in a popular magazine about mathematics!

Check it out: http://plus.maths.org/content/not-carrot

-- in a popular magazine about mathematics!

Check it out: http://plus.maths.org/content/not-carrot

## Sunday, July 31, 2011

### The Modal Last Man

Of all the interesting things I learned at our Routleyfest ("Remembering Richard Sylvan") event last week, perhaps the most striking was about the history of deep ecology:

The famous last man thought experiment was first devised and proposed by Routley, in 1973.

This very dramatic experiment asks you to imagine that you are the last person on earth. No other people can or will ever be born. In the time you have remaining, alone, you decide it would be fun to go around cutting down all the trees. Just for the hell of it. Question: what, if anything, would be wrong about this?

In his talk on Wednesday, Roger Lamb presented Routley's answer to the question. There does not need to be a valuer -- a sentient being according something value -- in order for something to have value. At least, there does not need to be any actual valuer. In an attempt to split the difference between realist and subjectivist accounts, Routley suggests that as long as there is a valuer at some possible world, then things like trees have value. And this accounts for why it would be wrong to chop them down.

Now, it seems to me that this gets the intuition behind "the last man" experiment wrong. The idea is to isolate the trees from the projects of humankind. The point is that the trees have value completely independently of the people. So invoking people -- even people somewhere else, merely possible people -- concedes that the trees have no deeply intrinsic value. Therefore I propose instead (as a counterexample to Lamb's version of Routley's solution) the modal last man:

You are the last possible person. Every other possible world is empty of people; your world, the actual world, has only you left. Since synchronizing time across worlds is problematic, we simply posit that you are the only possible person. (And, as it happens, contingently so.) Admittedly, this is a strange picture of modal space. But it seems legitimate enough to entertain, for the purposes of the experiment.

And so, again, you the last modal man choose to spend the last time any possible person will ever live going around killing all the trees. Is it wrong?

It seems to me that, if you think the original last man's decision was wrong, then you should also think the modal last man is wrong. But then, there is not -- because there cannot be, ex hypothesi -- a valuer to confer on the trees any worth over and above their own intrinsic worth. So Routley's solution misses the point.

I wonder what Routley/Sylvan would have said. That there are valuers at other impossible worlds? Perhaps...but then we could iterate my response, to the ultramodal last man--you are the only person in any world, possible or not....

In any case, it was a thrill to find out that a man I already credit with so many advances in philosophy (he was the first true dialetheic set theorist, I think), is also the mind behind one of the most famous and striking ideas in environmental philosophy, too. In so many cases, Routley was the first.

The famous last man thought experiment was first devised and proposed by Routley, in 1973.

This very dramatic experiment asks you to imagine that you are the last person on earth. No other people can or will ever be born. In the time you have remaining, alone, you decide it would be fun to go around cutting down all the trees. Just for the hell of it. Question: what, if anything, would be wrong about this?

In his talk on Wednesday, Roger Lamb presented Routley's answer to the question. There does not need to be a valuer -- a sentient being according something value -- in order for something to have value. At least, there does not need to be any actual valuer. In an attempt to split the difference between realist and subjectivist accounts, Routley suggests that as long as there is a valuer at some possible world, then things like trees have value. And this accounts for why it would be wrong to chop them down.

Now, it seems to me that this gets the intuition behind "the last man" experiment wrong. The idea is to isolate the trees from the projects of humankind. The point is that the trees have value completely independently of the people. So invoking people -- even people somewhere else, merely possible people -- concedes that the trees have no deeply intrinsic value. Therefore I propose instead (as a counterexample to Lamb's version of Routley's solution) the modal last man:

You are the last possible person. Every other possible world is empty of people; your world, the actual world, has only you left. Since synchronizing time across worlds is problematic, we simply posit that you are the only possible person. (And, as it happens, contingently so.) Admittedly, this is a strange picture of modal space. But it seems legitimate enough to entertain, for the purposes of the experiment.

And so, again, you the last modal man choose to spend the last time any possible person will ever live going around killing all the trees. Is it wrong?

It seems to me that, if you think the original last man's decision was wrong, then you should also think the modal last man is wrong. But then, there is not -- because there cannot be, ex hypothesi -- a valuer to confer on the trees any worth over and above their own intrinsic worth. So Routley's solution misses the point.

I wonder what Routley/Sylvan would have said. That there are valuers at other impossible worlds? Perhaps...but then we could iterate my response, to the ultramodal last man--you are the only person in any world, possible or not....

In any case, it was a thrill to find out that a man I already credit with so many advances in philosophy (he was the first true dialetheic set theorist, I think), is also the mind behind one of the most famous and striking ideas in environmental philosophy, too. In so many cases, Routley was the first.

## Tuesday, May 24, 2011

## Monday, April 25, 2011

### Important Announcement!

I have been inducted into the Logicians Liberation League, the shadowy organization founded by Robert K. Meyer in 1969, when he read out the Manifesto.

A list of the LLL is now maintained by Jc Beall on his website, where you will notice today a new member:

The Cardinal of Comprehension.

(That's me.)

A list of the LLL is now maintained by Jc Beall on his website, where you will notice today a new member:

The Cardinal of Comprehension.

(That's me.)

## Monday, April 11, 2011

### The Recurring Question

In the past few weeks I've been listening and speaking at a lot of different places around the northeastern United States and Scotland. And the question that keeps coming up -- about dialethism and paraconsistency -- is the following. It is simple to state and hard to answer:

How much inconsistency is too much?

Classically speaking, of course, the answer is easy. Any inconsistency at all is too much. But there are too many things we want to talk about, to know, to understand, for consistency to stand in the way: even to formulate sensible statements about all the truths, all the sets, all the properties, etc (not to mention vague predicates...) requires going over the line. Some inconsistency is needed, maybe even welcome.

But how much is too much?

Absurdly speaking, the answer is also easy. There is no limit; let us have all the inconsistency and more. But let's set such an extremum aside -- or, better, take it as an obvious limiting condition on our answer.

Some, but not all, inconsistency. That's what we're looking for in a good theory. Where to find the line, though? How much noise is too much noise? My best guess so far: It depends on what kind of music you like.

Perhaps next time someone asks me this question, a more precise answer will follow....

How much inconsistency is too much?

Classically speaking, of course, the answer is easy. Any inconsistency at all is too much. But there are too many things we want to talk about, to know, to understand, for consistency to stand in the way: even to formulate sensible statements about all the truths, all the sets, all the properties, etc (not to mention vague predicates...) requires going over the line. Some inconsistency is needed, maybe even welcome.

But how much is too much?

Absurdly speaking, the answer is also easy. There is no limit; let us have all the inconsistency and more. But let's set such an extremum aside -- or, better, take it as an obvious limiting condition on our answer.

Some, but not all, inconsistency. That's what we're looking for in a good theory. Where to find the line, though? How much noise is too much noise? My best guess so far: It depends on what kind of music you like.

Perhaps next time someone asks me this question, a more precise answer will follow....

## Monday, March 14, 2011

### And here our travels begin...

Along with Vicki and Oskar, I'm heading off on a big trip this week to talk and learn with Jc Beall and many others for the next few months.

I'm looking forward to finding out what goes on in the Northern Hemisphere these days, both philosophically and...not.

Some major dates (titles subject to change):

-March 25, University of Connecticut: "Paraconsistent Vagueness"

-March 31, St Andrews (Arche): "Paraconsistent Vagueness"

-April 1, Paradox and Logic Revision (Arche): "Between Finite and Infinite"

-April 4, NYU: "Paraconsistent Vagueness"

-April 7, MIT: "Paraconsistent Vagueness"

-April 13, UMass (Dartmouth): "Paraconsistent Vagueness"

-April 15, UConn Logic Group: "Between Finite and Infinite"

-April 22, UConn Logic Group: "Godel's Theorems"

-April 27, University of Massachusetts (Amherst): "Paraconsistent Vagueness"

-April 29, CUNY: "Consistency points in the number line"

-TBA: New England Logic and Language Colloquium: "Godel's Theorems"

I'm looking forward to finding out what goes on in the Northern Hemisphere these days, both philosophically and...not.

Some major dates (titles subject to change):

-March 25, University of Connecticut: "Paraconsistent Vagueness"

-March 31, St Andrews (Arche): "Paraconsistent Vagueness"

-April 1, Paradox and Logic Revision (Arche): "Between Finite and Infinite"

-April 4, NYU: "Paraconsistent Vagueness"

-April 7, MIT: "Paraconsistent Vagueness"

-April 13, UMass (Dartmouth): "Paraconsistent Vagueness"

-April 15, UConn Logic Group: "Between Finite and Infinite"

-April 22, UConn Logic Group: "Godel's Theorems"

-April 27, University of Massachusetts (Amherst): "Paraconsistent Vagueness"

-April 29, CUNY: "Consistency points in the number line"

-TBA: New England Logic and Language Colloquium: "Godel's Theorems"

## Monday, February 21, 2011

### Is paraconsistent arithmetic categorical?

Is naive set theory capable of describing and pinning down the intended model of arithmetic?

On the one hand, it doesn't seem very likely. With so many more sets out there to cause trouble, and easy ways to make isomorphisms fail (the totality of the ordinals isn't even isomorphic to itself, for instance), the odds of showing that any two peano relations coincide are low.

On the other hand, taking naive set theory as our base opens some intriguing possibilities. In particular, we can stay in a first order language, while at the same time, noting that the axiomatic equivalence

x is a member of X iff Xx

essentially grants us all the expressive power of second (and higher) order logic. And we know that classical second order theories can establish (classical) categoricity.

And so the question of the month is from Peter Smith's Godel book (p. 186). He's asking rhetorically, but in the paraconsistent context it's an open problem:

"If second order arithmetic does pin down the structure of the natural numbers, then -- given that any arithmetic sentence makes a determinate claim about this structure -- it apparently follows that this theory does enough to settle the truth value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic..."

Well, it's an open problem if you leave out that "consistency" part...

On the one hand, it doesn't seem very likely. With so many more sets out there to cause trouble, and easy ways to make isomorphisms fail (the totality of the ordinals isn't even isomorphic to itself, for instance), the odds of showing that any two peano relations coincide are low.

On the other hand, taking naive set theory as our base opens some intriguing possibilities. In particular, we can stay in a first order language, while at the same time, noting that the axiomatic equivalence

x is a member of X iff Xx

essentially grants us all the expressive power of second (and higher) order logic. And we know that classical second order theories can establish (classical) categoricity.

And so the question of the month is from Peter Smith's Godel book (p. 186). He's asking rhetorically, but in the paraconsistent context it's an open problem:

"If second order arithmetic does pin down the structure of the natural numbers, then -- given that any arithmetic sentence makes a determinate claim about this structure -- it apparently follows that this theory does enough to settle the truth value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic..."

Well, it's an open problem if you leave out that "consistency" part...

## Tuesday, January 25, 2011

### Upcoming Workshop: Constructive Mathematics

"What constructive mathematicians actually do"

University of Melbourne

Maarten McKubre-Jordens, University of Canterbury, New Zealand

I) Introduction to constructive mathematics

11:00am, Monday 14 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)

In the first of this two-part workshop, constructive mathematics is characterised as reasoning from positive definitions.

Several different schools (or models) of constructive mathematics are outlined, together with a common base. Some important non-constructive principles are outlined and and their impact evaluated, from the point of view of the practising mathematician. A small amount of familiarity with classical logic and mathematics will be assumed.

II) Constructive mathematics in action

11:00am, Wednesday 16 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)

What is the constructive role of examples, and counterexamples, in mathematics? What are some common techniques for proving theorems constructively? The second part of this workshop surveys how the role of examples in mathematical thought sets apart constructive thought, and will briefly discuss some typical strategies for proving theorems. Time permitting, we will investigate the idea of reverse mathematics as practised constructively.

III) Strange encounters and the importance of constructive thought

11:00am, Friday 18 February 2011, Old Quad Moot Court

Everyone has heard of some variation of the infinite monkey theorem:

Theorem. In an infinite collection of monkeys, given typewriters and a suitable amount of time, there is a monkey that with probability 1 reproduces the entire collected works of Shakespeare.

Of course this is not rigorously worded, but you get the idea. Now, how do you find the cheeky monkey that actually did it? In this seminar we take a look at some of the stranger and more intriguing aspects of constructive analysis — such as strictly increasing bounded sequences that do not converge, and well-defined sets in R^n from which we cannot calculate distances — and discuss their impact.

University of Melbourne

Maarten McKubre-Jordens, University of Canterbury, New Zealand

I) Introduction to constructive mathematics

11:00am, Monday 14 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)

In the first of this two-part workshop, constructive mathematics is characterised as reasoning from positive definitions.

Several different schools (or models) of constructive mathematics are outlined, together with a common base. Some important non-constructive principles are outlined and and their impact evaluated, from the point of view of the practising mathematician. A small amount of familiarity with classical logic and mathematics will be assumed.

II) Constructive mathematics in action

11:00am, Wednesday 16 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)

What is the constructive role of examples, and counterexamples, in mathematics? What are some common techniques for proving theorems constructively? The second part of this workshop surveys how the role of examples in mathematical thought sets apart constructive thought, and will briefly discuss some typical strategies for proving theorems. Time permitting, we will investigate the idea of reverse mathematics as practised constructively.

III) Strange encounters and the importance of constructive thought

11:00am, Friday 18 February 2011, Old Quad Moot Court

Everyone has heard of some variation of the infinite monkey theorem:

Theorem. In an infinite collection of monkeys, given typewriters and a suitable amount of time, there is a monkey that with probability 1 reproduces the entire collected works of Shakespeare.

Of course this is not rigorously worded, but you get the idea. Now, how do you find the cheeky monkey that actually did it? In this seminar we take a look at some of the stranger and more intriguing aspects of constructive analysis — such as strictly increasing bounded sequences that do not converge, and well-defined sets in R^n from which we cannot calculate distances — and discuss their impact.

## Wednesday, January 19, 2011

### 2011 Preview: Beyond the Possible

This year is set to be a busy and productive one, with visitors, workshops, conferences, and travels. And, of course, writing up some results. A quick preview:

>> July 27 - 29: Beyond the Possible: A Conference in Memorial of Richard Sylvan, who died 15 years ago. Some big guests for this one -- more info to follow on this soon.

>> Maarten McKubre-Jordens, a constructive mathematician from Christchurch and collaborator on our paraconsistency project, will visit in February to run some workshops and give a talk.

>> Colin Caret (St Andrews) will visit in July, for work on relevant information, paradox solution, and hyperintensionality.

>> We're also looking at visits from Walter Carnielli, Juliana Bueno-Soler, and John Bell.

>> And I'll be off to the University of Connecticut, USA!, from mid March until May, as the first "Scholar of Consequence," to work with Jc Beall and the logic group there. And to give some talks at other schools in the area (including attending a Paradox and Logic Revision workshop at St Andrews -- which is "in the area" relative to Melbourne).

Contact me if with questions, ideas, interest ... and stay tuned.

>> July 27 - 29: Beyond the Possible: A Conference in Memorial of Richard Sylvan, who died 15 years ago. Some big guests for this one -- more info to follow on this soon.

>> Maarten McKubre-Jordens, a constructive mathematician from Christchurch and collaborator on our paraconsistency project, will visit in February to run some workshops and give a talk.

>> Colin Caret (St Andrews) will visit in July, for work on relevant information, paradox solution, and hyperintensionality.

>> We're also looking at visits from Walter Carnielli, Juliana Bueno-Soler, and John Bell.

>> And I'll be off to the University of Connecticut, USA!, from mid March until May, as the first "Scholar of Consequence," to work with Jc Beall and the logic group there. And to give some talks at other schools in the area (including attending a Paradox and Logic Revision workshop at St Andrews -- which is "in the area" relative to Melbourne).

Contact me if with questions, ideas, interest ... and stay tuned.

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