I’ve started writing notes on the expository history, and here’s the first (19 page) instalment. The notes don’t at all aim to be comprehensive, though I’d like to know about significant omissions as I go along. They have been written, as much as anything, as a rather detailed aide-memoire for myself (and a source of bits and pieces I might use). I have done some joining up of the dots to make them tolerably readable, but I certainly haven’t put in the time to spell out everything out in the way a beginning student might want. Still, you shouldn’t need much background to follow the twists and turns.

The notes come in three parts. Part 1 looks at early papers by the Founding Fathers. Part 2, to follow, will look at three pivotal works, Mostowki’s Sentences Undecidable in Formalized Arithmetic and Kleene’s Introduction to Metamathematics (both from 1952), and then Tarski, Mostowski and Robinson’s Undecidable Theories (1953). Part III will continue the story on through some sixty years of textbooks.

Make what use of these notes that you will. Though the usual warning applies in spades, as I’m no historian: caveat lector! Remember how very easy it to LaTeX your work. Just because what you write then looks very pretty doesn’t mean that it is any more authoritative …

In the previous post, I claimed that in his §35 Carnap proves the semantic Diagonalization Equivalence, which he uses in §36 to prove the semantic version of Gödel’s First Theorem. But I said he doesn’t prove the Lemma there or give the now canonical syntactic version of the Theorem (the one depending on the syntactic assumption of -consistency).

Well, no one has protested yet. So, thus emboldened, let me now stick my neck out further!

What happens later in the book? Carnap’s notation and terminology together don’t make for an easy read. But as far as I can see, when he returns to Gödelian matters later, he still is using the semantic Diagonalization Equivalence and not the syntactic Diagonalization Lemma. If the latter was going to appear anywhere, you’d expect to find it in §60 when Carnap returns to the incompleteness of arithmetics: but it isn’t there. (An indication: Carnap here talks of provability being ‘definable’ in arithmetics, and it is indeed expressible — but we know it isn’t capturable/representable by a trivial argument from the Diagonalization Lemma proper. So Carnap hereabouts is still dealing with semantic expressibility, and doesn’t seem to invoke the syntactic notion of capturing/representing needed for the Lemma.)

So, in summary: Yes, Carnap gives a nice tweak to the argument in §1 of Gödel 1931 for the semantic incompleteness theorem, by generalizing the basic idea to give the Diagonalization Equivalence. But this isn’t the modern Diagonalization Lemma. Which isn’t to say that we can’t get the Lemma easily using the wff Carnap constructs: however, as far as I can see, Carnap didn’t explicitly take the step in 1934, even though he is often credited with it.

What am I missing?

Well, in §35 Carnap notes the general recipe for taking a one-place predicate and constructing a sentence such that is true if and only if is true, where as usual is  the formal numeral for the Gödel number for .

Fine. But that claim about constructing a semantic equivalence isn’t the Diagonalization Lemma as normally understood, which is a syntactic thesis, not about truth-value equivalence but about provability. It is the claim that, in the setting of the right kind of theory , then for any one-place predicate we can construct a sentence such that . And Carnap doesn’t prove that in §35.

When we turn to §36, where Carnap proves the incompleteness of his system II, again he appeals to his semantic result not the syntactic Diagonalization Lemma. We now take the relevant predicate to be NOT-PROVABLE: so we get a sentence which is true if and only if it isn’t provable. And then Carnap appeals to the soundness of his system II to argue in the elementary way that isn’t provable. So at this point Carnap is giving a version of the semantic incompleteness argument sketched in the opening section of Gödel 1931 (the one that appeals to a soundness assumption), and not a version of Gödel’s official syntactic incompleteness argument which appeals to -consistency. Indeed, Carnap doesn’t even mention -consistency in the context of his §36 incompleteness proof. He doesn’t need to.

To summarize so far: in §§35–36, Carnap doesn’t use the theorem that his system II proves NOT-PROVABLE, i.e. he doesn’t appeal to an application of the Diagonalization Lemma in the modern sense. He doesn’t need it (yet), and he doesn’t prove it (here).

But maybe he returns to the fray later in the book … the story continues!

Anyway, now that I am working on a second edition, I want to spend the next month pausing to have a look at how others have handled the First Incompleteness Theorem. The basic shape of my book is fixed (after all, I’m doing another edition of IGT, not writing another different book, fun though that would be to do): but I might well get inspiration for how to make local improvements. So how do others state the Theorem? And how do they prove it? I’m planning to write up notes on expository themes and variations as I go along, and will post them here in due course.

The natural place to start is with Hilbert and Bernays. Slight problem: I don’t have German (yep, schoolboy Greek was all well and good, but hasn’t exactly been useful). Does anyone out there have detailed notes of how they prove the First Theorem in §5.1 of their second volume? I’d be really delighted to hear if so! Otherwise, there is a French translation, so I suppose I ought to do battle with that. Not that my French is any good these days either …

Until I can get hold of a translation of Hilbert and Bernays, then, the expository tradition for me will really have to start with Kleene’s wonderful 1952 Introduction to Metamathematics. I’m looking forward a lot to dipping into that again. Then I counted over forty other books on my shelves which give more or less detailed proofs of the First Theorem: hours of fun ahead, obviously. But I’ve started, yesterday and today, by rereading Gödel’s 1931 and his 1934 Princeton lectures. I have to blush again. I’d forgotten, for example, just what Gödel didn’t prove in 1931. ‘We shall only give an outline of the proof’ that every recursive relation is (as we would now say) representable in P;  ’the entire proof’ [that for any predicate expressing a recursive property/relation there is an equivalent arithmetical predicate] ‘can be formalized in the system P‘ [it can be, no doubt, but this isn't proved].

In fact, the situation is this. There are in fact two significantly different results stated in the 1931 paper, the ‘semantic’ and the ‘syntactic’ incompleteness theorems. The first requires that we are dealing with a theory which can express primitive recursive relations and is sound; the second version beefs up the first assumption and requires a theory which can represent p.r. relations while weakening the second assumption to require only that the theory satisfies the syntactic condition of omega-consistency. The first, ‘semantic’ theorem is sketched in Gödel’s §1, the official ‘syntactic’ version is the topic of the central §2 and §3. Look carefully, however, and while there is (enough material for) a full proof of the underplayed semantic version, the proof of the ‘syntactic’ result (the result that is normally meant when people talk now about the First Theorem) is in fact gappy, gappier than I’d really remembered.

Meanwhile, I see over 200 people have downloaded the draft of the first eight chapters of the second edition. Thanks to David Auerbach and Seamus Bradley for some amazingly speedy comments. Keep ‘em coming folks!

OK, here then is

A first instalment of revised chapters of IGT2: Chapters 1 to 8

(these replace the current Chs 1 to 7). I’ve actually rather enjoyed doing this, and am quite pleased with the results — in particular, I think I’ve much improved the too-compressed incompleteness argument in the old Ch. 5. But I’m simultaneously also a bit cheesed off to see how much those early chapters did need improving. It all just goes to show that when you’ve finished writing a book, you should really put it in a drawer for three years until you realize what you were really trying to say, and then re-write it. You can just see your research-driven university approving of that policy, eh?

All suggestions/corrections for these revised chapters will be most gratefully received. You can put them in the comments below, though it is probably better all round to email them (my email address is at the bottom of the “About Peter Smith” page). For copyright reasons, I won’t be able to make all the revised chapters so freely available when I’ve done them: but anyone who emails comments will be put on a circulation list for future tranches of the new version.

Time then to pause to draw breath for Christmas after a busy/distracting time. But then what’s next, logically speaking?

Well, I plan to blog here in the new year about two more books that I’ve been asked to review together — Leon Horsten’s The Tarskian Turn: Deflationism and Axiomatic Truth and Volker Halbach’s Axiomatic Theories of Truth. Not that I claim any special expertise about their topic: but then both books are written for a reader like me — a philosopher/logician interested in theories of truth, who wants to get a handle on recent some formal developments (to which the respective authors have been notable contributors). Should be very interesting. And since both books have been out for few months, I hope some readers of the blog will be able to chip in helpfully in comments!

But mostly, it will have to be back to Gödel. At the moment, I’m re-writing the opening chapters of the book for the second edition, and I think very much improving them — about which I have mixed feelings! On the one hand, it’s very good to feel the effort of doing a second edition is going to be worth while, but on the other hand, I’m a bit downcast to see how very far from ideal those chapters previously were. Sigh. Anyway, when I’ve got the first tranche of chapters more to my liking, I’ll post them here for comments.

And I’ve one or two other plans too … So watch this space!

In my previous comments I was talking about Maddy’s discussion of Thin Realism vs Arealism, and her claim that there in the end — for the Second Philosopher — there is nothing to choose between these positions (even though one line talks of mathematical truth and the other eschews the notion).  What we are supposed to take away from that is the rather large claim that the very notions of truth and existence are not as central to our account of mathematics as philosophers like to suppose.

The danger in downplaying ideas of truth and existence is, of course, that mathematics might come to be seen as a game without any objective anchoring at all. But surely there is something more to it than that. Maddy doesn’t disagree. Rather, she suggests that it isn’t ontology that underpins the objectivity of mathematics and provides a check on our practice (it is not ‘a remote metaphysics that we access through some rational faculty’), but instead what does the anchoring are ‘the entirely palpable facts of mathematical depth’ (p. 137). So ‘[t]he objective ‘something more’ our set-theoretic methods track is the underlying contours of mathematical depth’ (p. 82).

This, perhaps, is the key novel turn in Maddy’s thought in this book. The obvious question it raises is whether the notion of mathematical depth is robust and settled enough really to carry the weight she now gives it. She avers that ‘[a] mathematician may blanch and stammer, unsure of himself, when confronted with questions of truth and existence, but on judgements of mathematical importance and depth he brims with conviction’ (p. 117). Really? Do we in fact have a single, unified phenomenon here, and shared confident judgements about it? I wonder.

Maddy herself writes: ‘A generous variety of expressions is typically used to pick out the phenomenon I’m after here: mathematical depth, mathematical fruitfulness, mathematical effectiveness, mathematical importance, mathematical productivity, and so on.’ (p. 81) We might well pause to ask, though, whether there is one phenomenon with many names here, or in fact a family of phenomena. It becomes clear that for Maddy seeking depth/fruitfulness/productivity also goes with valuing richness or breadth in the mathematical world that emerges under the mathematicians’ probings. But does it have to be like that?

In a not very remote country, Fefermania let’s say (here I’m picking up some ideas that emerged talking to Luca Incurvati), most working mathematicians—the topologists, the algebraists, the combinatorialists and the like—carry on in very much the same way as here; it’s just that the mathematicians with ‘foundational’ interests are a pretty austere lot, who are driven to try to make do with as little as they really need (after all, that too is a very recognizable mathematical goal). Mathematicians there still value making the unexpected connections we call ‘deep’, they distinguish important mathematical results from mere ‘brilliancies’, they explore fruitful new concepts, just like us. But when they turn to questions of ‘foundations’ they find it naturally compelling to seek minimal solutions, and look for just enough to suitably unify the rest of their practice, putting a very high premium on e.g. low-cost predicative regimentations. Overall, their mathematical culture keeps free invention remote from applicable maths on a somewhat tighter rein than here, and the old hands dismiss the baroquely extravagant set theories playfully dreamt up by their graduate students as unserious recreational games. Can’t we rather easily imagine that mind-set being the locally default one? And yet their local Second Philosopher, surveying the scene without first-philosophical prejudices, reflecting on the mathematical methods deployed, may surely still see her local mathematical practice as being in intellectual good order by her lights. Why not?

Supposing that story makes sense so far (I’m certainly open to argument here, but I can’t at the moment see what’s wrong with it) let’s imagine that Maddy and the Fefermanian Second Philosopher get to meet and compare notes. Will the latter be very impressed by the former’s efforts to ‘defend the axioms’ and thereby lure her into the wilder reaches of Cantor’s paradise? I really doubt it, at least if Maddy in the end has to rely on her appeal to mathematical depth. For her Fefermanian counterpart will riposte that her local mathematicians also value real depth (and fruitfulness when that is distinguished from profligacy): it is just that they also strongly value cleaving more tightly to what is really needed by way of rounding the mainstream mathematics they share with us. Who is to say which practice is ‘right’ or even the more mathematically compelling?

Musings such as these lead me to suspect that if there is objectivity to be had in settling on our set-theoretic axioms, it will arguably need to be rooted in something less malleable, less contestable than Maddy’s frankly rather arm-waving appeals to ‘depth’.

Which isn’t to deny that may be some  depth to the phenomenon of mathematical depth: all credit to Maddy for inviting philosophers to think hard about its role in our mathematical practice. Still, I suspect she overdoes her confidence about what such reflections might deliver. But dissenting comments are most welcome!

Option (A): we introduce a class of sentence letters (as it might be, ) together with a class of predicate letters for different arities (as it might be , , ). The rule for atomic wffs is then that any sentence letter is a wff, as also is an -ary predicate letter followed by terms.

Option (B): we just have a class of predicate letters for each arity (as it might be ). The rule for atomic wffs is then that any -ary predicate letter followed by terms is a wff.

What’s to choose? In terms of resulting syntax, next to nothing. On option (B) the expressions which serve as unstructured atomic sentences are decorated with subscripted zeros, on option (A) they aren’t. Big deal. But option (B) is otherwise that bit tidier. One syntactic category, predicate letters, rather than two categories, sentence letters and predicate letters: one simpler rule. So if we have a penchant for mathematical neatness, that will encourage us to take option (B).

However, philosophically (or, if you like, conceptually) option (B) might well be thought to be unwelcome. At least by the many of us who follow Great-uncle Frege. For us, there is a very deep difference between sentences, which express complete thoughts, and sub-sentential expressions which get their content from the way they contribute to fix the content of the sentences in which they feature. Wittgenstein’s Tractatus 3.3 makes the Fregean point in characteristically gnomic form: ‘Only the proposition has sense; only in the context of a proposition has a name [or predicate] meaning’.

Now, in building the artificial languages of logic, we are aiming for ‘logically perfect’ languages which mark deep semantic differences in their syntax. Thus, in a first-order language we most certainly think we should mark in our syntax the deep semantic difference between quantifiers (playing the role of e.g. “no one” in the vernacular) and terms (playing the role of “Nemo”, which in the vernacular can usually be substituted for “no one” salve congruitate, even if not always so as myth would have it). Likewise, we should mark in syntax the difference between a sentence (apt to express a stand-alone Gedanke) and a predicate (which taken alone expresses no complete thought, but whose sense is fixed in fixing how it contributes to the sense of the complete sentences in which it appears). Option (B) doesn’t quite gloss over the distinction — after all, there’s still the difference between having subscript zero and having some other subscript. However, this doesn’t exactly point up the key distinction, but rather minimises it, and for that reason taking option (B) is arguably to be deprecated.

It is pretty common though to officially set up first-order syntax without primitive sentence letters at all, so the choice of options doesn’t arise. Look for example at Mendelson or Enderton for classic examples. (I wonder if they ever asked their students to formalise an argument involving e.g. ‘If it is raining, then everyone will go home’?). Still, there’s another analogous issue on which a choice is made in all the textbooks. For in an analogous way, in defining a first order syntax, there’s another forking path.

Option (C): we introduce a class of constants (as it might be, ); we also have a class containing function letters for each arity (as it might be , ). The rule for terms is then that any constant is a term, as also is an -ary function letter followed by terms for .

Option (D): we only have a class of function letters for each arity (as it might be ). The rule for terms is then that any -ary function letter followed by terms is a term for .

What’s to choose? In terms of resulting syntax, again next to nothing. On option (D) the expressions which serve as unstructured terms are decorated with subscripted zeros, on option (C) they aren’t. Big deal. But option (D) is otherwise that bit tidier. One syntactic category, function letters, rather than two categories, constants and function letters: one simpler rule. So mathematical neatness encourages many authors to take option (D).

But again, we might wonder about the conceptual attractiveness of this option: does it really chime with the aim of constructing a logically perfect language where deep semantic differences are reflected in syntax? Arguably not. Isn’t there, as Great-uncle Frege would insist, a very deep difference between directly referring to an object and calling a function (whose application to one or more objects then takes us to some object as value). Again, so shouldn’t a logically perfect notation sharply mark the difference in the the devices it introduces for referring to objects and calling functions respectively? Option (D), however, downplays the very distinction we should want to highlight. True, there’s still the difference between having subscript zero and having some other subscript. However, this again surely minimises a distinction that a logically perfect language should aim to highlight. That seems a good enough reason to me for deprecating option (D).

We can explicitly indicate that we are dealing with e.g. a one-place total function from natural numbers to natural numbers by using the standard notation for giving domain and codomain thus: . What about two-place total functions from numbers to numbers, like addition or multiplication?

“Easy-peasy, we indicate them thus: .”

But hold on! is standard shorthand for , the cartesian product of with itself, i.e. the set of ordered pairs of numbers: and an ordered pair is standardly regarded as one thing with two members, not two things. So a function from to is in fact a one-place function that maps one argument, an ordered pair object, to a value, not (as we wanted) a two-place function mapping two arguments to a value.

“Ah, don’t be so pernickety! Given two objects, we can find a pair-object that codes for them, and we can without loss trade in a function from two objects to a value to a related function from the corresponding pair-object to the same value.”

Yes, sure, we can eventually do that. And standard notational choices can make the trade invisible. For suppose we use as our notation for the ordered pair of with , then can be parsed either way, as representing a two-place function with arguments and or as a corresponding one-place function with the single argument . But the fact that trade between the two-place and the one-place function is glossed over doesn’t mean that it isn’t being made. And the fact that the trade can be made (even staying within arithmetic, using a pairing function) is a result and not quite a triviality. So if we are doing things from scratch — including proving that there is a pairing function that matches two things with one thing in such a way that we can then extract the two objects we started with — then we do need to talk about two-place functions, no? For example, in arithmetic, we show how to construct a pairing function from the ordinary school-room two-place addition and multiplication functions, not some surrogate one-place functions!

So what should be our canonical way of indicating the domains (plural) and codomain of e.g. a two-place numerical function? An obvious candidate notation is . But I haven’t found this used, nor anything else.

Assuming it’s not the case that I (and one or two mathmos I’ve asked) have just missed a widespread usage, this raises the question: why is there this notational gap?

But for a quicker read, my overheads give the headline idea — that’s there no implication about how we grasp the standard model to be got out of the elegant but non-trivial Tennenbaum’s Theorem that you can’t get out of the very easy theorem that every model of PA where every element has a finite number of predecessors is isomorphic to the standard model. Tennenbaum’s Theorem has no extra oomph against the Skolemite sceptic. Indeed, appealing to either model theoretic result just doesn’t touch the sceptic’s worries. (The talk timed nicely, and having Tim there to help fend questions made giving it a lot more fun!)

The current temporal parts of Walter Dean and Leon Horsten were agreed, contra earlier parts, that Tennenbaum’s Theorem cuts no ice against the model-theoretic sceptic (I wasn’t so clear where Paula Quinlon now stands). But I think all three other speakers in different ways wanted to squeeze something philosophical out of Tennenbaum’s Theorem. If/when published pieces emerge, I’ll say why I wasn’t so convinced. But a fun occasion (as such closely-focused workshops tend to be).