I agree with Richard's and Alex's general remarks about "logicism" and what counts as "logical". It would indeed be far too quick to reject every form of logicism just because it makes the existence of an infinite number of objects a matter of "logic". Still, it is perhaps worth reiterating (as Richard indeed does) that Principia gets its infinity of objects by theft rather than honest toil: it just asserts an infinity of objects as a bald axiom rather than trying to conjure them out of some more basic logical(?) principles in a more Fregean way. So I'd still want to say that, whatever the fate of other logicisms, Russell and Whitehead 's version -- given it is based on theft! -- can't really be judged an honest implementation of the original logicist programme as e.g. described in the Principles , even prescinding from incompleteness worries. But for all that, three cheers for Principia in its centenary year!

In the Principles of Mathematics, Russell boldly asserts "All mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts, and ... all its propositions are deducible from a very small number of fundamental logical principles." Principia , a decade later, is an attempt to make good on that programmatic "logicist" claim. Now, one of the axioms of Principia is an Axiom of Infinity which in effect says that there is an infinite number of things. And you might very well wonder whether that is a truth of logic . (If someone thinks the number of things in the universe is finite, are they making a logical mistake?) Another axiom is the Axiom of Reducibility, which I won't try to explain here, but which is even less obviously a logical law -- and indeed Russell himself argued that we should accept it only because it has nice mathematical consequences in the context of the rest of Principia's system. Still, there is some room for...

When it comes to moving from the BA to beginning graduate studies, the only thing (in my experience) that grad schools really care about is just how smart you are at philosophy. So they will take note of how well you did in the BA, of what your referees write about you, and (probably most importantly) they'll make their own independent assessment of the quality of the samples of written work that they ask for. Extra-curricular activities and declarations of passion count for little!

There's no right answer. Zermelo's original set theory allowed "urelements", i.e. entities in the universe which are members of sets but not themselves sets. Some modern writers use "ZFC" to refer to a descendant of Zermelo's theory allowing urelements. George Tourlakis is an example, in his two volume Lectures in Logic and Set Theory . Some other writers (perhaps the majority) use "ZFC" to refer to the correponding theory of "pure" sets, where there are no urelements and the members of sets are themselves always other sets. Kenneth Kunen is just one example in his modern classic Set Theory . If you are interested in set theory as a tool, then the first line is arguably the more natural one to take. If you are interested in set theory for its own sake, then for most purposes you might as well take the second line (because it seems to make no big difference to the sort of questions that most set theorists are interested in: for example ZFC-with-urelements is equiconsistent with ZFC-for...

Neither. Assuming by "philosophy paper" you mean student essay, then what you need to be doing is evaluating arguments, as carefully and as honestly and as rigorously as you can. You must aim for maximum explicitness, maximum clarity, maximum organization of your thoughts. But you are writing ordered English prose, not lab notes or a mathematical proof. "Personal style" will look after itself, and shouldn't be your conscious concern. (It is always a pleasant surprise to me, e.g. when I have to mark a stack of undergraduate dissertations, how -- despite the fact that students have gone through the same teaching treadmill and drilled by weekly one-to-one essay tutorials -- distinct voices will always come through.)

I agree that philosophers should engage with relevant science. But of course, what science (if any) is relevant depends very much on what philosophical questions you are tangling with. If you are concerned with the metaphysics of time, for example, then you'll no doubt want to know something of what various kinds of physicist doing foundational work on relativity, etc., are thinking (but you needn't care at all about e.g. neuroscience). If you are concerned with the philosophy of mind then you'll probably want to know something of neuroscience and experimental psychology (but you won't care about cosmology). If you are interested in whether numbers are objects in Frege's sense, or under what circumstances abortion is permissable, or in how names latch on to the world, or whether a non-minimal state is justified, you won't care much about either neuroscienc e or cosmology, or about evolutionary biology either. So what science, if any, you need a "firm grounding" in as a philosopher will...

Perhaps it is worth taking continuing the conversation just a bit further. The idea that a proposition (statement, belief) is true if and only if it "corresponds to reality" is -- as I'm sure William would agree -- not entirely transparent. What does it commit us to, exactly? The deflationist about truth of course says that the proposition that snow is white is true if and only if reality is such that snow is white -- i.e. just if snow is white. So if the correspondence theorist is to be distinctively saying more than that, she needs to spell out what "correspondence" here comes to, over and above what the weak kind of correspondence that is already built into the deflationist view. Now, there are indeed metaphysicians who do claim to have an "industrial strength" version of the correspondence theory, who postulate the existence of facts as ingredients of the world, facts which are truth-makers whose existence is required to make propositions true (where the worldly...

Jack says "The next train to London is at 11.15"; Jill adds "That's true". Jill's remark in effect just repeats Jack's message. To say it is true that the next train to London is at 11.15 tells us no more about the world than that the next train to London is at 11.15. Dora witnesses a crime. She gives quite a long statement. "Three boys in jeans and hooded tops came into the shop just before 12. They ... etc., etc. etc., ... And finally they jumped into a red car and sped off." Dick adds "That's all true." Again, Dick is in effect just repeating Dora's statement, but saving breath. You can see why we should have use for such a very handy device in our language. Someone says something, or we read something in a book; saying "that's true" has the effect of saying the same, without all the bother of repeating what is said or written. And the same handy device is just as useful when what is said or what is written is not so common-or-garden but more theoretical. Alice says "The atomic weight...

A few comments on Hilbert's Hotel (since Charles Taliaferro has brought that up) and "actual infinities": If you want a standard presentation of the usual Hilbert's Hotel "paradox", which has nothing to do with money, then check out Wikipedia's good entry . The "paradox" just dramatizes the basic fact that an infinite set can be put in one-one correspondence with a proper subset of itself. There is nothing paradoxical about that: on the contrary, it is tantamount to a definition of what it is for a set to be (Dedekind) infinite. Can there be "actual infinities" in the sense of realizations of Dedekind infinite sets in the actual world? Well, money won't do, to be sure (but that's just a fact about money, not about the general impossibility of "actual infinities"). Suppose you think that there are space-time points, and that actual space-time is dense -- i.e. between any two points there is another one. Then the points in a space-time interval will be Dedekind infinite. [Proof: label the end...

Here's a somewhat differently slanted view -- in favour, perhaps, of being a bit "daft"! :-) No matter how many times I read three-year old Daisy her favourite book, no matter how well she knows it by heart, she hasn't read it herself. She can't read. No matter how many times the adult illiterate listens to a tape a complete reading of e.g. Bleak House , no matter how well he knows the book as a result, he hasn't actually read it. He can't read either. (The blind person who can read braille, of course, can read Bleak House. ) The audiences that heard bards sing The Iliad were fortunate indeed. But most never read it. They couldn't read. (And what about audiences before it was written down?) Similarly for the groundlings at early performances of Shakespeare. There's a difference between having read a work and knowing it well. You can read something without, as a result, remembering a word (as on long haul flights!); and you can know it very well without ever reading it. Sure,...

A formal system (of the kind to which Gödel's incompleteness theorem applies) is a consistent axiomatized theory which contains a modicum of arithmetic and is such that it is mechanically decidable whether a given sentence is or isn't an axiom. Why should we think that the "laws of the universe" can be encapsulated in a formal theory in that sense? Why suppose that all the laws can be wrapped up into a single formal system? It isn't at all obvious why that should be so: maybe the laws of the universe are so rich that they always elude being pinned down by a single formal axiomatic system (such that it is mechanically decidable what's an axiom). Indeed, we might say that Gödel's incompleteness theorem shows that, on a broad enough understanding of "laws of the universe", the laws can't be so pinned down. For any given formal theory, there will be arithmetical truths that particular formal system can't prove -- so the arithmetical laws of the universe, for a start, run always run beyond...

Isn't it just that typically not much hangs on answers about the numbers of stars, but you can e.g. ruin your clothes by getting it wrong about the paint? But if it matters (say, to win a competition with a big cash prize), I bet you would double check my statement about the number of stars, and not just fill in what I tell you on the competition entry form. And if we are idly gossiping in the pub, and I boringly mention that the new-fangled paint that they've been advertising takes for ever to dry, I don't suppose you'd leap out of your seat to check the room I've been decorating: you'd shrug and say it was my turn to buy a round of drinks. What you bother to check will depend not on the topic so much as on how much hangs on getting things right.