If you've ever thought much about numbers or talked with a preschooler learning to count, you've probably encountered some of the questions that led to Cantor's discovery a century ago. How many natural numbers are there? (Naturals are just the numbers we count with: 1, 2, 3, 4 and so on up forever.) And what about the even naturals: 2, 4, 6, 8 and so on? Infinitely many in both cases, right? OK, but are there more naturals than evens? Clearly every even natural number is a natural number, but there are plenty of naturals that aren't even -- namely the odds: 3, 5, 7, 9 and so on. Does that mean that the set of naturals is bigger than the set of even naturals?

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"The Amber Spyglass" is the final book in the most ambitious allegory being published today, Philip Pullman's "His Dark Materials" trilogy. With epigraphs from William Blake, Rainer Rilke and John Ashbery, and tributes to John Milton and Henrich von Kleist in the acknowledgments, Pullman places himself in a tradition of serious symbol makers, which might be expected to intimidate the children to whom the series is directed (or, at least, to whom it was directed when he began it). But while Pullman may have become caught up in adult theology -- and while he has won more grown-up readers with each "Dark Materials" book -- he keeps the swooping plots and passionate characters that make his earlier books so appealing to young readers.

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Suzy McKee Charnas' Holdfast Chronicles fall squarely in the tradition of feminist Utopias/dystopias that produced Joanna Russ' "The Female Man" or Margaret Atwood's "The Handmaid's Tale," nourishing writers like Ursula K. Le Guin and Sherri S. Tepper. The latest book in the series, "The Conqueror's Child" (1999), recently won the James Tiptree Jr. Award, an annual literary prize for science fiction or fantasy that "explores and expands the roles of women and men." Produced over the course of 30 years, the Holdfast tetralogy offers a fascinating look back at the permutations of the feminist imagination in recent years, and it underlines the ideals and challenges faced by feminists -- sometimes on purpose and sometimes in spite of itself.

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When fantasists and science fiction writers want to create a self-contained world, they often pick a planet -- an island surrounded by a vacuum. Stewart, however, stays Earth-bound off the Texas coast, rooting his story in our planet's current history. In 1900 a flood changed the face of Galveston; 104 years later another flood -- of magic, not water -- disrupted the island utterly and sent the survivors into a state of siege.

Ever since Mardi Gras 2004, the island has been divided in two. Downtown, in Carnival, eternal night reigns. There, time stands still for a perpetual party hosted by the master of ceremonies, the hunchbacked god Momus. Meanwhile, beyond Carnival's gates, Galveston's leading citizens have been struggling for decades to hold their world together. The four Mardi Gras Krewes, clubs of townspeople who used to organize the Mardi Gras parade before the Flood, have grown in political power.

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Well, maybe. Joyce walks with the grace of a circus star, or a Henry James, on that narrow line between seeming and being. Can Maggie really transform herself into a bird, or is she just high on the herbs that go into her potions? Did Sam invent the tooth fairy, as his parents and his psychiatrist believe, or did the fairy invent Sam, as the creature insists? Do the indigo seekers see true visions or psychotic hallucinations? Joyce builds suspense by keeping his readers guessing up to the very end -- and sometimes beyond.

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Take Flip, the pierced, tattooed mailroom clerk at HiTek, a corporate think tank, where Willis' 1996 novel "Bellwether" takes place. Grudging and aggressively incompetent, she tosses a monkey wrench into the research of every scientist in the company. At least, that's how it seems to Sandra Foster, the narrator. Sandra, a sociologist, is trying to discover how fads start (in particular, hair-bobbing in the 1920s). Flip foils her at every turn, trashing her carefully arranged clippings or mis-delivering important packages. Yet if not for Flip, Sandra might never have met chaos researcher Bennett O'Reilly, and the two might never have started on their singularly fertile collaboration.

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Although I love a good spaceship or wizard (or marzipan or okra, for that matter), I sympathize. Bad science fiction and fantasy can get pretty stratospherically bad -- and at spectacular length. Nevertheless, when readers scorn the good stuff, they're closing their minds to treasures that have grown rare elsewhere in fiction. Adventure, allegory, invention and myth have taken refuge in these genres.

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Dental advocates like Herself who become tempted to slip a book
into the trick-or-treat bag instead of candy will find plenty of goodies
to choose from. Anthologies of ghost stories, chapter books about
witches, and gothics for tots crowd the shelves this time of year.
They'll keep the Halloween spirit alive well past Thanksgiving, when
long-hoarded sugar pumpkins have turned to rock.

Angela Barrett -- whose heroic "Joan of Arc" and "Snow White" have become classics -- provides delicate illustrations for "The Random House Book of Ghost Stories," which features tales that are, on the whole, more melancholy or comic than frightening. Originally published in Great Britain, it brings together such British masters as Leon Garfield, Penelope Lively and Philippa Pearce.

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When I was 7, I used to lie awake at night wondering about infinity. I would imagine myself swimming up from my bed into the darkness of space, on and on, out and out. It had to stop somewhere, I thought, so I imagined a brick wall marking the end. But what was beyond the brick wall? Sometimes I broke through into more darkness, miles of it, hours of it, until the emptiness made me panic and I added another wall -- only to break through it into more space. Sometimes, to stave off the darkness, I would imagine the wall extending in its red-brown solidity, filling the space beyond it with bricks and mortar, until that, too, became oppressive. Eventually, unnerved, I'd crawl into bed with my parents.

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Parents eager to keep the family entertainment bills under control (those $9 tickets really add up) will be tempted to substitute the print version of "The Iron Giant" for the movie of the same name. So will those who hope to guide their children's enthusiasm into literary channels. After all, a mere $16 buys the 30th anniversary edition of the classic chapter book by the late poet laureate of England, its jacket spangled with gushing blurbs from juvenile lit greats, such as Madeline L'Engle and the author of "The Phantom Tollbooth."

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Amid the endless parade of top-10 lists and published forebodings that mark the millennium, Kathleen Krull's "They Saw the Future" stands out. This large-format book for 10-to-14-year-olds is a uniquely forward-looking history, illustrated with handsome collage paintings. In 12 chapters devoted to "oracles, psychics, scientists, great thinkers, and pretty good guessers," it explores the human longing to understand and thereby control the future.

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Oh, yeah? Amelia Atwater-Rhodes is the latest counterexample. She finished her first novel, "In the Forests of the Night," last year when she was only 13. An atmospheric revenge tale about a teen vampire, it's as suspenseful and well-constructed as many novels by authors several times her age. "As a teen, I bring a different perspective to writing," Atwater-Rhodes told Teen People. "I can offer immediate emotions, experiences and insight that adult writers often have to reach back and find in order to write about them."

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But how many kids do you know who can tell an azalea from a zinnia? In this season when tiny, perfect leaves poke through the ends of rural twigs like the fingers of newborns, when suburban lawns start to bloom bright, and shaggy weeds assert themselves even along the edges of city cement, shouldn't those ravenous young minds be scarfing up plant facts?

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"In Celtic societies, women were given fairly equal status with their men. Some became rulers of tribes and even fought in battle," writes Robert Byrd in the afterword to "Finn MacCoul and his Fearless Wife: A Giant of a Tale from Ireland,"one of a pair of newly published picture books that celebrates Ireland's
mythological heroines. Byrd draws on several
sources, including Yeats, for his story of Oonagh, the
wife of the timorous, goofy giant Finn, who
outsmarts Cucullin, a big -- and I mean BIG -- bully
from Scotland. Finn attracts Cucullin's attention by
building the Giant's Causeway, a stone bridge
between Ireland and Scotland. (Geologists call it a
basalt formation, but storytellers know better.) When
Cucullin comes looking for him, Finn runs home to
Oonagh. Thinking fast, she hides him in a cradle,
then passes him off to Cucullin as Finn's own infant.
Finn and Oonagh pull the cheese trick, a folktale
staple: She challenges the great Scot to crusha stone
in his hand; when he can't, she gives Finn a
look-alike cheese,which he shatters easily. If Finn's
baby son is capable of such feats, thinks Cucullin,
what must the father be like? He doesn't stick
around to find out.

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Closest to alphabet books are letter-play books, such as William Steig's ingenious "CDB!" and its sequel, "CDC?" Each page contains a string of letters that yield their meaning when pronounced sequentially. Steig helps out with his goofy cartoon illustrations. The title spread of "CDB!" for example, shows a boy and girl enthusiastically inspecting an insect that's hovering near some flowers. (See the bee -- get it?) "D B," it turns out, "S A B Z B." Girls named MLE and KT and boys named PT will find these books particularly XLN.

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ABCs usher children into the prestigious world of the written word. Remember your first one? Remember tracing the letters with your finger, as if its meaning would somehow flow in through your hands? Perhaps you were an Eric Carle child, like me -- his colors seemed so vivid I thought I could feel them with my eyes closed. Like the old favorites, the recent crop of ABCs introduces aspiring readers to the uses of those funny squiggles adults set such store by, then elaborates on their subtleties.

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To honor the pleasure we both took in transformations, I'm planning to
give Mom Joan Steiner's "Look-Alikes." This volume of photographic still
lifes is marketed as a puzzle book for children, but it's really
something more profound: a meditation on nostalgia and utopia, or a
series of visual lyrics, or maybe a vivid dream.

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One way to borrow a musical voice is to illustrate a song, as Kathy
Jakobsen does in "This Land Is Your Land." A small, upright Woody
Guthrie strides through Jakobsen's folk-arty paintings while his song
plays itself out in the reader's inner ear and perhaps even aloud if
there's a parent around who can carry a tune.

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