Dr T.E. Forster:

That's what I really look like, on the right.

Here is the portrait of me that I keep in the attic.

For 21/22 I returned — with great reluctance — from my COVID-proof nuclear bunker in a secret location in New Zealand famous for its dinosaur coprolites .

Chores |
Ancestry |
Affiliations |
Movements |
Visitors |
Teaching |
A Ph.D., perhaps? |
Publications |
Other Unpublished
Materials |
NF Materials |

Email: tef10@dpmms.cam.ac.uk

For Google's benefit I record here that my name is often misspelled `Thomas Foster' and `Thomas Forester'.

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Nowadays I am a visiting professor at Mathematics at the Victoria university of Wellington.

From october 2016 to september 2022 I was the resident bye-fellow in Pure
Mathematics in Queens',
College Cambridge (where I was also Director of Studies for Part III and am now an *Emeritus Fellow*—tho' i haven't yet worked out what that involves);
now that I have reached retirement age my status as an *affiliated
lecturer* in the department of Pure Mathematics and Mathematical
Statistics in Cambridge has lapsed; I am a Life Associate of Clare Hall, Cambridge, and I
am also a Senior member of St. Edmunds,.

Outside
Cambridge my affiliations are: Adjunct Senior Fellow in the Philosophy
department at the University of Canterbury at
Christchurch; Visiting Lecturer at Queen Mary Westfield ; External
Researcher of the
Auckland Centre for Discrete Mathematics and Theoretical Computer
Science in New Zealand, and a
Fellow of the Center for
Philosophy of Science at the University of Pittsburgh.
When I was
in residence there I stayed—as many fellows do—in Carl
Hempel's flat. The Pittsburgh Center [*sic*] also has conferences. (Pittsburgh has many attractions
beyond its six (!) universities: serious weather, a wonderful
orchestra, opera houses, rivers, cable cars, the best chips in America
(at the
Original Hot Dog. Be sure to ask for a **small** portion),
steel mills, lots of history (much of it sad), and, last
but not least, an airport called—wait for it—``Tom
Foerster International Airport''! (The correct explanation for this
spelling mistake is surely the one found by Sten Vikner who pointed out
that they must have misplaced my middle initial.)

I am on the Comité de Redaction of the Belgian Centre National de Recherches de Logique;

I am a ``correspondent étranger" of Logique et Analyse;

I was the Cambridge horn (in fact the

I was Conference Director for the (now dormant) St Luke's Institute, in which capacity I organised BILAP (Buddhism in Logic and Analytic Philosophy) which held its inaugural meeting in Cambridge in November 2005, and the Logic and Rhetoric conference held in Cambridge on the last weekend of October 2006.

I was on the DPMMS Quiz team which mercilessly crushed all opposition to be the winners in the 800th anniversary Quiz contest. At the 800th anniversary garden party we went on to murder a team of alumni. I used to have a link to footage of that event, but that footage was never for the squeamish and has now been made unavailable!

Movements:

Research:

May-june 2015—and again in 2018—I hosted Randall Holmes in Cambridge, Holmes continuing his vain attempt to get the consistency of NF into my thick skull. (He tried again in 2019—with a larger hammer—but to no avail). In Michaelmas 2015/6 and again in 2018 I hosted Alice Vidrine.

In recent years I have had Olivier Esser in Cambridge as my post-doc, working on NF. Thierry Libert has been here for two visits, continuing his project to teach me some positive set theory (GPC etc: this isn't Set-Theory-by-Compte, despite the adjective and the French Connection). I had two visits from Edoardo Rivello, who wants to learn about symmetric models for stratified fragments of ZF. In Michaelmas 2019 I had the treat of an LMS-funded visit from Albert Visser .

Cambridge teaching materials (for my Cambridge students and readable only from .cam.ac.uk addresses);

You might find my essay on Julian Jaynes'

If you are a Cambridge physiology student thinking of doing my Part II project have a look at the proposal. I don't think I can support it now (I am no longer at Addenbrooke's) but I think the department there would still be interested.

The world of NF studies is being turned upside-down by the news that NF has been proved consistent. Unfortunately I am no longer allowed to supervise Ph.D. students in Cambridge, and i don't know whether my visitor status at Wellington will allow me to supervise Ph.D. students there — I haven't asked. In addition to the obvious NZ delights we all know about (hobbits behind every sofa etc etc) there is also the point that NZ fees for Ph.D. students are a lot lower than Cambridge fees!

For a while i was trying to interest people in the constructions discussed in my AMS Baltimore meeting article. Dang Vu wrote a Ph.D. thesis on this subject which he has kindly allowed me to link from here.. My hope was that these constructions would lead to a proof of Con(NF). Even tho' they have been pipped at the post, the methods used in this paper are related to those used in Randall's proof, and it could be a good place to start reading.

I am interested in taking on some more Ph.D. students to work on NF and
to this end I provide a discussion of some suitable
thesis topics. Actually not all the topics in there are NF topics...
there are some topics in philosophy of mathematics that i'd be happy to
supervise too.

There is a plan to get some money together for a Ph.D. studentship to do
a formal verification of Holmes' proof of Con(NF). It's only at the design
stage as yet, but if you are suitably qualified it would do you no harm to tell me
about yourself.

My mother came from Basel, the city that Holbein made his home, that produced Leonhard Euler, and has been home to the Bernoulli family since they were expelled by the Reyes Catolicos (isn't that enough for a small town???). We know about the mathematicians of course, (and even the tennis players: Roger Federer is a Basler) but it has also produced some wonderful painters—and not just Holbein. Sadly the creator of this little gem in the Eulerstrasse (yes, it's the same Euler) did not sign his work and his (her?) identity was not recorded for posterity. I tell everyone I am a fourth-generation academic, and this is because of her (my mother's) grandfather Otto Billeter Snr who was a fairly well-known chemist (he discovered the rearrangement of allyl thiocyanate to allyl

But in any case it's intellectual ancestry one worries about. I had two DoktorVater: Adrian Mathias in Cambridge, and Maurice Boffa in Brussels. Sadly, Boffa died in 2001: it is a terrible loss. Here are two group pictures, with three of his Ph.D. students. My other DoktorVater, Adrian Mathias, is still very much alive—for the moment at least: he does live uncomfortably near an active volcano!) According to the Mathematics Genealogy Project I am descended (thru' Mathias) from Fichte, Hegel, Kant and Leibniz!! Thru' Boffa I am descended from Poincaré!!

Like
Everyone Who Is Anyone in Silicon Fen I worked (briefly) for
Clive Sinclair, in my case in
the
Sinclair Radionics days. (He was a neighbour of my parents). My Erdős
number is 3: Forster-Truss-Shelah-Erdős ; (and Erdős many years
ago described me as ``a very strange young man''!). But there are
other metrics. My Trotsky number is 3 (I shared an office with a colleague
(Giovanna Corsi) who shared an office with a colleague
(Jean van Heijenoort) who was Trotsky's bodyguard); my Beethoven
number is 4, since I have shaken hands with a man who shook hands with
a man who shook hands with Grillparzer, who shook hands with
Beethoven, Liszt...well—everyone! I once played Garcia Lorca's
piano. I used to play chess with a woman who had been Tolstoy's
next-door neighbour, and had been taught by Alekhine
. When an infant
she had been dandled on Tchaikovsky's knee. Facebook is forever telling me i should
(``friend suggestion'') befriend Jacinda Ardern. (We have a facebook friend in common). If you are a Canadian you
may be impressed by the fact that I once had my portrait painted by Barker Fairley (he's the one with the pipe in
this group picture of the group of seven); apparently I am the only
person other than Wittgenstein to have given a course of lectures in
Cambridge gazetted under the title `Philosophy' (that is a distinction that Wittgenstein can hardly have forseen); my Ph.D. Thesis
title "N.F."
is the shortest on record; as far as I know I am the only
person to have a Cambridge Ph.D. in mathematics without having done
any undergraduate mathematics; finally I must surely be the only Logic
Ph.D. anywhere in the world who is also licenced to drive an EEG machine.
[tho' this is less remarkable than the double *coup* by my Otago colleague
Grant Gillett who had two articles accepted in the one year...one for MIND and the other for BRAIN.] My remaining distinguishing features are of less
interest. Born in the oldest house in Cambridge? And a haunted one at
that? (It's the
Abbey House). It
had been a gift from the first Baron Fairhaven to the Cambridge City Council who then
divided it into flats and let them out—so should I ever want to go
into politics I can always say I was born in council house! and I was
delivered by Alice
Roughton). The only person to have been at school with both the
erstwhile Home Secretary (Charles Clarke) and the Queen's (erstwhile)
private secretary (Robin Janvrin)? (Not to mention the—sadly—thoroughly unmemorable
Nick Drake who was in my Russian class at Marlborough. We were taught by the entirely stellar Richard Pollock, who went on to be Mrs Thatcher's personal interpreter in her conversations with Gorbachev). The
tabloids are not beating a path to my door. I do not—yet—have a
theorem named after me, (nor even an
airport, see above) tho' there is a lifeform bearing my name: not a
child—sadly—but a *midge*
called Telnatogeton Forsteri. This
lives only in New Zealand—there is a type specimen in my desk drawer in
Cambridge but it's definitely dead. This lifeform was discovered by my
friend John Leader. If you wish to name an invertebrate or a theorem—or even an airport—after me, feel free.

Here, for old times' sake, is the console of the first machine i ever wrote code for. My former student Nick Benton says that assembly language is the sound of old men crying into their beer, but i'm not ashamed.

The stuff in this section is all unpublished; for published items scroll down a wee bit.

I make available here the pdf of my Commonplace book on NF.

Cartesian-closedness fails in NF does what it says on the tin.

This essay on Realizability and the Possibility of a Consistency
Proof for the Constructive Fragment of Quine’s
Set Theory NF was accepted for the July 2019 Paris meeting on *Facets of Realizability* but i was prevented from presenting it by some surgery.

Scansets shows how to use a cute new datatype to interpret various versions of Mac Lane set theory into (various versions of) KF.

A strongly cantorian union of strongly cantorian sets is a discussion of the eponymous proposition.

Stratification mod *n* generalises (weakens) stratification in various systematic ways and proves a special case of the duality conjecture.

Two remarks on extracted models answers two questions i was asked in Wellington the other day about the extracted model construction for TZT.

More on Church-Oswald Models is my latest thinking on this amusing topic. Lots of nice stuff. My department has the matter constantly under review.

As part of the project to make historical NF literature freely available I am posting here my Ph.D. Thesis .

I supply here other material of publishable standard and interest that i can't be arsed sending off.

Supervaluationism makes a connection between supervaluationism, constructive logic and lazy evaluation.

henkin.pdf is a log of my attempt to master Henkin's two 1950's JSL papers on generalisations of the ideas of ω-consistency and ω-completeness. Readers with similar ambitions to mine may find it useful.

ramsey.pdf is, similarly, a log of my attempt to master **the** paper by Ramsey. We all know it's there of course, but nobody ever reads it, which is a shame.

I keep unbelievably exalted company....

My lectures for the Part II logic course have now come out as Logic, Induction and Sets. A list of errata for this book is to be found here.

Those works which the publishers have allowed me to make available have links on this page.

My recent talk on the Axiom of Choice wot i gave at Young Researchers in Mathematics 2014 will probably go through a few more metamorphoses before i attempt to publish it, so i am not going to post the pdf here. If you want to see the current version feel free to email me and i will send you a heavily embargoed copy. I shall expect feedback! Already in print is The Axiom of Choice and inference to the Best Explanation which has recently appeared in

The Burali-Forti Paradox offers a potentially useful take on this widely misunderstood piece of logic. It has now appeared in *The Reasoner*.

There are three translations available:

My annotated translation of Coret's seminal article

Dualität is an annotated and illustrated translation of Specker's seminal paper on duality in projective geometry, which I prepared in collaboration with Anne the widow of Harold Davenport. I prepared it for the Garland Quinefest volumes, where it finds a natural place because of the development of those ideas of duality into ideas of typical ambiguity in Type theory and NF. I have removed from this copy a typo that mars the Garland version. Chad Brown reckons there is a mistake in the original. Click here for his analysis.

Permutation Models and Stratified Formulæ, a Preservation Theorem appeared in the ZML vol 36 (1990) pp 385–388.

Asenjo's system LP does not support modus ponens has been much expanded by Jc Beall into a version which has been accepted by the NDJFL and which includes a striking strengthening of my result by Jeremy Seligman. The version posted here is essentially my original text.

A semantic characterisation of the well-typed formulae of lambda calculus is from

Asynchronous games explores a way of representing asynchonous games (Homicidal Chauffeur etc.) as alternating discrete (``clocked'') games. I would be interested in feedback. It has now appeared in the procedings of the Liverpool july 2004 meeting on knowledge and games, but the version here will always be the more up-to-date one.

The editors are allowing me to post here my introduction to the Buddhism in Logic and Analytical Philosophy volume (``Pointing at the moon'') published by Oxford University Press.

Relaxing Stratification is an answer to the top FAQ about NF: ``Can the stratification constraint be safely relaxed?''. Beginners might find this paper discouragingly thorough! It is a survey, joint work by me and Olivier Esser.

Paris-Harrington in an NF context exhibits a formulation of Paris-Harrington and one of Finite Ramsey that differ only in their quantifier prefix. It also discusses the implications of the fact that—on the face of it—the statement of the Paris-Harrington theorem is not stratified/welltyped in the Russell-Quine sense. It has appeared in volume 17 of the

Rhetorical devices in Analytic Philosophy is my contribution to the special number of

My talk at the LMS Sets-and-games meeting:

Games played on an illfounded membership relation has now appeared in the Boffa 60th birthday festschrift.

Finite-to-one maps has now appeared in the JSL. It shows, without any use of AC, that if there is a finite-to-one surjection from the power set of X to X then X is genuinely finite (its cardinal is a natural number). I proved it years ago and tho'rt nothing of it but Adrian Mathias and James Cummings couldn't prove it when I challenged them so I tho'rt it might be worth publishing and the JSL agreed, bless `em. Greg Kirmayer claims that he can improve this result, and i'm inclined to believe him.

Yablo's paradox without self-reference is in

Deterministic and Nondeterministic Strategies for Hintikka games in First-order and Branching-quantifier logic is in Logique et Analyse vol 195 pp 265-69.

Implementing Mathematical objects in Set Theory is in the special number of

AC fails in the natural analogues of V and L that model the stratified fragment of ZF is a version (cleaned of typos and minor infelicities) of the paper I gave at the Baltimore joint meeting AMS/MAA in 2003. It is a work of breathtaking fertility: read it and weep. Better still, read it and use the techniques in it to prove Con(NF).

ZF + Every set is the same size as a wellfounded set is in the March 2003 fascicule of the JSL;

BQOs and coinduction is in the dec. 2003 fascicule of Theoretical Computer Science;

The Modal Aether (

Weak Set Theories related to HOL is an improved version of my HUG 1994 paper from LNCS 859;

Why Set theory without the axiom of foundation? appeared in the Journal of Logic and Computation in 1994;

Sethood and Situations (jointly written with Cathy Rood Wyss) has appeared in Computational Linguistics

Permutations and Wellfoundedness: the true meaning of the Bizarre Arithmetic of Quine's NF is in the JSL

Church's Set Theory with a Universal Set is a revision and expansion of the last chapter of my book on set theory with a universal set, and supercedes it. It was written for the Alonzo Church festschrift. It is intended to be a comprehensive introduction to the method developed by Church and Oswald to construct models of theories like Church's. (The version here is to be preferred to the version in print, as I remove typos and mathematical errors from it as they come to my notice.) A more philosophical treatment of the same ideas is to be found in The Iterative Conception of Set. Everyone who is anyone has written an article under that title; what is distinctive about my contribution under this title is that it was voted one of the ten best philosophy articles of 2008 by the

Sharvy's

The organisers of the Riga conference on paradoxes have allowed me to post here this brief Note on Paradoxes in Ethics which will appear in their proceedings.

Erdős-Rado without Choice has appeared in the Journal of Symbolic Logic, vol 72, 2007, pp. 897-900. The version linked here contains some informative comments by a referee which readers will find very helpful; I am grateful to him/her for permission to include them here.

Mathematical entities arising from equivalence relations, and their implementation in Quine's NF is my invited talk at the Munich workshop organised by Roy Cook and Erich Reck, and is to be found in

Alice buffs might like

Three extended reviews are here:

(i) My Computer Journal review of Barwise and Moss's Vicious Circles. Readers of that review—and others with an antiquarian interest in the early literature on funny set theories—may want to see my 1982 paper on

Strong Extensionality;

(ii) my Studia Logica review of Ziegler-Booth's edition of the collected set-theoretic writings of Paul Finsler, and

(iii) my

My recent article on The Burali-Forti Paradox has come out in the Reasoner, september 2023.

The remaining material in this section is unpublished

Here are the notes for my contribution to the panel discussion organised by the Trinity Mathematical Society on 21/x/2013 with Imre Leader on

Jamie Gabbay recorded this talk i gave at Heriot-Watt.

Here is a talk about Synonymy, Stratification and the Universal Set" wot i gave on a trip to Vienna to visit Hugo Wolf's grave. It might not be the world's best talk, but the subject matter is important and I continue to work on it.

Leisure:

In my pathetically few hours of leisure I try to keep my knitting going, and strive to keep up my interest in Astronomy and Politics ....and, of course, neurology (this .gif file is called

DPMMS front page.