In relation to my earlier answer, the following article from the Economist may be of interest. It's advertised as follows: "Can the laws of physics change? Curious results from the outer reaches of the universe." The link is

www.economist.com/node/16941123?story_id=16941123&fsrc=nlw|hig|09-02-2010|editors_highlights

This is not exactly what I had in mind, but relevant nonetheless.

BTW, this question is probably best classified under "physics" rather than "mathematics."

On the standard account, given two finitely long lines, even of different lengths, their points can indeed be matched up one-to-one, e.g. by the kind of projection the teacher indicated. And the possibility of that kind of one-to-one matching is just what we mean when we say the two lines "contain the same number of points".

What makes this possible, despite the different lengths? In part, the fact that there are an infinite number of points along a finite line (the issue we are dealing with here is one of those initially puzzling matters which arises when we deal with the non-finite: intuitions tutored by finite examples can lead us astray).

And there being an infinite number of points along a finite line is related to the fact that the points on a line are dense -- that is to say, between any two points, however close together, there is another point.

Now, consider a line with end points. Between the left hand end-point a and any other point on the line there is a further point. So it follows that there is no 'next' point, immediately to the right after a. (Suppose that b is a candidate 'next' point: then by denseness, there will in fact be another point c between a and b. So b isn't 'next' after all. Nor is c, because by the same reasoning there is another point d between a and c. And so it goes.)

This means that if you remove the left-hand end-point a what remains is an 'open' line with no left-most point. (It can't be that point b is leftmost since c remains to the left of it; and it can't be point c as d remains to the left of it; and so it goes.) There are points arbitrarily close to the position that a has vacated, but no left-most one.

Hence the imagined construction where we remove the end point of a line and then remove the new end point etc. in fact makes no good sense. After we chip off the left-most point a of a line with endpoints we are left with an 'open' line with no left-most point, and the described construction falters at the very first step after the initial one!

In a finite lifetime, you won't be able fully to inspect an object with parts that are infinitely far from you, at least if we assume that you are limited by the speed of light. But there's other evidence. For example, you may be able to measure the gravitational pull of the ladder. If this pull turns out to be exactly what our theory would predict for a ladder that's like the piece of it we have before us (same material, thickness, density, etc.) and infinitely long, then this would be evidence for infinite length. (Note here that the gravitational pull exerted by any one inch of ladder declines with the square of its distance from you. So no matter how long the ladder its, its gravitational pull will not be infinite.) It's also possible that the ladder is expanding (as our universe is), or perhaps contracting. In that case you get a nice Doppler effect: a transformation of light reaching you from distant parts of the ladder -- the farther the light has traveled, the more strongly transformed it arrives. So evidence can provide reasonable assurance. And we have such reasonable assurance now that the universe is in fact finite.

But it's worth noting, as you suggest, that such evidence depends on "reasonable" assumptions. We assume that the laws of nature we've found to hold true around our space-time location also hold millions of light years away and millions of years in the past. Do we have reasonable evidence for this assumption? Well, yes, sort of, if with this assumption we can fit all our observations into one coherent account. Physics hasn't quite achieved this yet. But once physics delivers such a comprehensive theory, then it'll strike us as unreasonable -- bizarre -- to defend an infinite size of the universe by appeal to diverse laws of nature holding at different regions of space-time. But you should ask: the fact that some hypothesis strikes us as bizarre, how much reassurance is this that it's actually false?

The following story is recounted in John Aubry's Life of Thomas Hobbes:

"He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman's library Euclid's Elements lay open, and 'twas the forty-seventh proposition in the first book. He read the proposition. 'By G ,' said he, 'this is impossible!' So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. I have heard Sir Jonas Moore (and others) say that it was a great pity he had not begun the study of the mathematics sooner, for such a working head would have made great advancement in it. So had he done he would not have lain so open to his learned mathematical antagonists. But one may say of him, as one says of Jos. Scaliger, that where he errs, he errs so ingeniously, that one had rather err with him than hit the mark with Clavius. I have heard Mr Hobbes say that he was wont to draw lines on his thigh and on the sheets, abed, and also multiply and divide. He would often complain that algebra (though of great use) was too much admired, and so followed after, that it made men not contemplate and consider so much the nature and power of lines, which was a great hindrance to the growth of geometry; for that though algebra did rarely well and quickly in right lines, yet it would not bite in solid geometry."

Now Hobbes himself, in his own philosophical works, such as Leviathan, did not quite aspire to axiomatization, but he did seek proofs of the sort that can be found in geometry; his near-contemporary, Spinoza (whose views, if not his method, influenced Hobbes's own work--Hobbes even said of Spinoza, "He hath overshot me by a bar's length, for I durst not write so boldly,"--self-consciously modeled his Ethics on Euclid's Elements, following the "geometrical method," beginning each part of the Ethics with Definitions and Axioms, on which the Propositions 'proven' in each section were to be based: in the Preface to the Third Part of the Ethics, "Of the Origin and Nature of the Affects [in comtemporary language, emotions]," Spinoza claims to "treat the nature and powers of the affects...just as if it were a question of lines, planes, and bodies." However, if one opens the Ethics to any given proposition, while one will find rigorous argumentation, and 'proofs' that make reference only to the definitions, axioms, and prior propositions 'proved' in the relevant part of the Ethics, one will hardly be convinced by those proofs in the same way that Hobbes--or we--might be convinced by the proofs in Euclid's Elements. This reflects a frustrating, fascinating, feature of philosophy: the propositions it treats are not amenable to proof in the same way as propositions in mathematics. While philosophers over the centuries have sought to make philosophy into a 'rigorous science', more akin to mathematics, the propositions it treats do not seem to admit of proof like the propositions of mathematics. Indeed, it is reported that the philosopher Moritz Schlick once said, "Making lists of propositions proven by philosophers is a pastime heartily to be recommended," knowing full well that the list that one ended up with would be quite short indeed.

What's distinctive, it seems to me, of the claims in various branches of philosophy--such as ethics, epistemology, and metaphysics--is that they treat questions that do not admit of proof. One must, instead, seek to give arguments for the claims in which one is interested--just as Hobbes and Spinoza did--and those arguments will rest on reasons that may well be open to challenge by other philosophers. Consequently, there is no one method for analyzing and proving statements in metaphysics--or any other branch of philosophy, for that matter: instead, one must seek to give reasons for the claims that one seeks to advance, and to develop arguments in their support. This is what sets philosophy apart from other branches of knowledge, such as physics and mathematics.

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!

To follow on some of Richard's observations: I have never found it at all a compelling argument against logicism that it would have the existence of infinitely many natural numbers be a logical truth. That is not an argument against logicism so much as a restatement of the claim that it is incorrect.

Richard's discussion of Boolos reminds me of Gödel's own caution with regard to what his Incompleteness Theorems establish with respect to Hilbert's Program (roughly, Hilbert's attempt to show that if a basic, or finitary, proposition of mathematics can be established using the powerful, or infinitary, methods of classical mathematics, then it could already have been established using very basic, or finitary, reasoning). [For more on Hilbert's Program, you might see here or here.] I don't myself think that the phenomenon of incompleteness puts paid to Hilbert's project (as divorced from certain other beliefs that Hilbert may have held, such as the belief that all true mathematical propositions are knowable). But the second Incompleteness Theorem much more directly challenges it – indeed many have felt that it directly shows Hilbert's Program to be unachievable. But Gödel himself was more cautious in his original paper. For he noted there that (for all he has said) there may be some finitary forms of reasoning (in particular, those deployed in a proof of consistency) that are not captured within a formal system that formalizes the principles of classical mathematical reasoning.

Richard notes that the impact of the first incompleteness theorem for logicism depends on how one understands "logical". The point here (Gödel's point) is that the impact of the second incompleteness theorem for Hilbert's Program depends on how one understands "finitary".

I think it's important to distinguish the two sources of "failure", not so much as regards Principia but as regards logicism quite generally. I'll stick, as Prof Smith did, to arithmetic.

Here's a way to put Gödel's (first) incompleteness theorem: the set of truths expressible in the language of first-order arithmetic cannot be listed by any algorithmic method, i.e., it is not (as we say) "recursively enumerable". Now why is that supposed to show that logicism fails? Because the set of theorems of any first-order formal theory is recursively enumerable. This is a consequence of Gödel's first great theorem, the completeness theorem for first-order logic (and also of what we mean by a "formal" theory). So the truths aren't r.e. and the theorems are—you can list the theorems but not the truths—so the theorems can't exhaust the truths.

Now why is this a problem for logicism? Obviously, as the argument has been stated, it depends critically upon the assumption that the proposed logical basis for arithmetic is a first-order theory. In fact, of course, the argument can be generalized. Here is what George Boolos has to say about it in an unpublished (but soon to be published) paper from the early 1970s:

The argument convinces me, at any rate, that there is no reduction of arithmetical truth to logical truth, where the logic in "logical" is understood to be elementary, or first-order, logic, or indeed, any system of logic whose theses form an effectively generable set [that is, that are recursively enumerable]. (Gödel’s theorems of 1931 and the improvements and related results that followed soon afterwards meant the death of more philosophies of mathematics than just Hilbert’s formalism.)

That is Boolos summarizing a general form of the argument we just outlined: If the facts about some variety of logical deducibility are r.e. (as they are with first-order logic), then the theorems of any formal theory understood as subject to that form of logical deducibility will also be r.e. (because of what we mean by a "formal" theory), and the argument will go through. But then he goes on to say more.

The possibility of a significant reduction of arithmetic to something that might be called a system of logic is left open by our argument, however. What is excluded is that there be an effective method which identifies all and only the theses of the system: a proof procedure. It may seem that it is essential to a system of logic that it have a proof procedure, and that logics without proof-procedures are so called only laughingly or by courtesy. I think that this essentialist claim is wrong, however, and that second-order logic deserves its name. [Some ellipses omitted.]

The logical truths (what Boolos is calling "theses") of second-order logic are not r.e., so second-order logic is not subject to the argument. And there are formal second-order theories to which arithmetic can be "reduced", in the sense that their logical consequences include all arithmetical truths (and no arithmetical falsehoods). So, as Boolos says, the critical question is whether second-order "logic" is really logic, and that is what his paper is about.

Of course, none of this touches the other question asked, which in the present context will take the following form: Consider one of these formal second-order theories to which arithmetic can be reduced; is it really plausible that all of its axioms are going to be logical truths? Prof Smith gestures at one response. Since there are infinitely many numbers, any theory to which arithmetic can be reduced will have to imply the existence of infinitely many things. But surely logic doesn't imply that, so the principles can't be logical. I don't know if Prof Smith would endorse this sort of argument (Boolos did, at least sometimes), but it doesn't seem very impressive to me. At least, it doesn't really engage the logicist impulse. You can't argue against a view by pointing out that the premises assumed imply the conclusion.

Now, this isn't to say that a principle simply asserting that there are infinitely many things (like Russell's) should be regarded as an acceptable basis for a reduction of arithmetic to logic. One might well think that no such principle can be a primitive logical truth. If it's a logical truth, that has to be because it is a logical consequence of some simpler principle or principles. And I think that's correct. But there are simpler principles whose claim to logical truth (or something relevantly like it) is a good deal more plausible and that do imply the existence of infinitely many things (in fact, quite directly, of infinitely many numbers). But that is a much longer story. If you're interested, have a look at some of the papers on my web site, starting with this one.

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 argument about what principles, exactly, deserve the honoric "logic". And it would still be very interesting if Principia succeeded in showing that all elementary mathematics -- or at least, all arithmetic! -- can be reduced to a handful of laws of a very general kind.

But unfortunately, it doesn't even show that. Gödel's incompleteness theorems show that there are truths expressible in the language of elementary arithmetic which can't be proved in Principia. And the argument generalizes: well-behaved extensions of Principia will be arithmetically incomplete too.

So, in sum, Principia doesn't even seem successfully show that all propositions of arithmetic "are deducible from a very small number of fundamental logical principles" (the principles invoked arguably don't all belong to logic, and the system is certainly incomplete and incompletable).

To add a bit more, there are some interesting applications of urelements in set theory. Perhaps the most famous example is Quine's theory New Foundations. NF, as it is known, which does not permit urelements, remains something of a puzzle: It is not known if it is consistent. But NFU, which is just NF plus urelements, is known to be consistent if Peano arithmetic is. See the wikipedia entry on NF for more.

I seem to recall some interesting results due to Vann McGee about ZFU, as well, but they do not come immediately to mind.

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-pure-sets).