The Biggest Infinity

In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: 'To infinity … and beyond!' The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.

  1. Is Infinity The Biggest Number
  2. What Does Infinity Mean
  3. Biggest Number That Isn't Infinity

That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the 'reals.' (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)

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In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption. He reasons that the naturals and this zero-to-one subset of the reals having equally many members implies that the two sets can be put into a one-to-one correspondence. That is, the two sets can be paired so that every element in each set has one—and only one—'partner' in the other set.

Think of it this way: even in the absence of numerical counting, one-to-one correspondences can be used to measure relative sizes. Imagine two crates of unknown sizes, one of apples and one of oranges. Withdrawing one apple and one orange at a time thus partners the two sets into apple-orange pairs. If the contents of the two crates are emptied simultaneously, they are equally numerous; if one crate is exhausted before the other, the one with remaining fruit is more plentiful.

Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence. Every natural number n thus has a real partner rn. The reals can then be listed in order of their corresponding naturals: r1, r2, r3, and so on.

Then Cantor's wily side begins to show. He creates a real number, called p, by the following rule: make the digit n places after the decimal point in p something other than the digit in that same decimal place in rn. A simple method would be: choose 3 when the digit in question is 4; otherwise, choose 4.

For demonstration's sake, say the real number pair for the natural number 1 (r1) is Ted Williams's famed .400 batting average from 1941 (0.40570…), the pair for 2 (r2) is George W. Bush's share of the popular vote in 2000 (0.47868…) and that of 3 (r3) is the decimal component of π (0.14159…).

Is Infinity The Biggest Number

Now create p following Cantor's construction: the digit in the first decimal place should not be equal to that in the first decimal place of r1, which is 4. Therefore, choose 3, and p begins 0.3…. Then choose the digit in the second decimal place of p so that it does not equal that of the second decimal place of r2, which is 7 (choose 4; p = 0.34…). Finally, choose the digit in the third decimal place of p so that it does not equal that of the corresponding decimal place of r3, which is 1 (choose 4 again; p = 0.344…).


Continuing down the list, this mathematical method (called 'diagonalization') generates a real number p between zero and one that, by its construction, differs from every real number on the list in at least one decimal place. Ergo, it cannot be on the list.

In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are 'uncountably' numerous—making real infinity somehow larger than natural infinity.

'The idea of being 'larger than' was really a breakthrough,' says Stanley Burris, professor emeritus of mathematics at the University of Waterloo in Ontario. 'You had this basic arithmetic of infinity, but no one had thought of classifying within infinity—it was just kind of a single object before that.'

Adds mathematician Joseph Mileti of Dartmouth College: 'When I first heard the result and first saw it, it was definitely something that knocked me over. It's one of those results that's short and sweet and really, really surprising.'

For this special loudspeaker-focus issue, I asked our most senior contributors to each name the twelve loudspeakers that had the greatest impact on high-end audio. These are the speakers that introduced a new technology, changed the market, influenced future designs, or revealed some previously unheard aspect of sound quality. Although each writer worked in isolation, the individual choices exhibited remarkable unanimity. From each writer’s picks we selected, by consensus of the senior editorial staff, “The 12 Most Significant Loudspeakers of All Time.” Our final verdict is presented in ascending order of significance (#1 being the most significant). Robert Harley

12. Klipschorn

The Klipschorn is the only speaker that has been in continuous production for over 60 years. But that’s not why it makes the list. The Klipschorn is a landmark product due to its folded horn design. Paul Klipsch, inventor and entrepreneur, patented the idea of assembling chambers and passageways for a bass driver’s sound waves to gradually expand as they travel out to the opening. The sound is mechanically amplified by the expanding “folds” in the passageway of the horn. (Without this, a low-frequency horn would be the size of a full room.) In 1946, the first 20 Klipsch loudspeakers were assembled in a tin shack in Hope, Arkansas. The base horn design has never been improved—it was perfect from day one. Klipsch’s four principles of sound reproduction are: efficiency, flat frequency response, controlled directivity, and dynamic range. A Klipschorn provides a detailed wall of sound that emanates from the corner of a room. It was the first “absolute sound.” Imagine the sound of a windup Victrola being replaced overnight by the sound of unamplified live instruments in space. Most amazing is that you can buy it today. That’s 60 years of advancement in one moment. Peter Breuninger


11. MBL 101 E

In the history of high-end audio, there have been a number of fascinating and genuinely innovative drivers—Alan Hill’s plasma tweeter and Lincoln Walsh’s “transmission-line” cone, for examples. Not all of them caught on—for good reasons (the joke about the Walsh driver used to be that it took 200W to get it to make sound and 201W to blow it up, while the Hill produced enough ozone to choke a horse). Wolfgang Meletzsky’s omnidirectional “Radialstrahler”—a truly ingenious pumpkin-shaped contraption constructed of aluminum/magnesium “petals” that flex in and out in response to an audio signal (like the pleats of an accordion), producing near-equal sound pressure throughout 360 degrees (rather, dare I say it, like a pulsating, er, pumpkin)—is certainly a brilliant concept and happily it doesn’t blow up or poison the air. What it does do is produce the most enveloping soundstage this side of a surround system, absolutely thrilling large-scale dynamics, and timbres that are very true-to-life (in frequency response, the MBL is an exceedingly flat-measuring loudspeaker). Though omnis aren’t as commonplace as they once were back in the day, the sui generis 101s set a standard of excellence and sheer lifelike excitement that has kept them the foremost omnidirectional speakers for more than thirty years. Jonathan Valin

What Does Infinity Mean

10. Advent

Not very long ago, a long-time audio buddy gave me a chance to hear his Double Advent setup (and in his garage!). The experience in a sense, just about took my breath away: The speakers, even in that primitive setting, were magnificent! They remained as uncolored and neutral as ever, exceeding too many of today’s so-called “super” systems. I had, if the truth be told, forgotten (audibly) just how very special this doubling up [stacking a pair atop another pair] of Henry Kloss’ last great speaker was and remains. Wished I had had the sense to hold on to the pair I bought (back when, actually in 1972, just before I started Issue One of this rag). The Advents weren’t then entirely trouble-free thanks to mechanical problems with the original tweeters. Seen in today’s light, aside from an airy top end, the only thing missing was its ability to recreate a wide and dimensional soundstage. If you can grab a pair in good condition, and they are out there, be smarter than me. Harry Pearson

9. KLH Model Nine

More than any other electrostatic speaker of its era the KLH Model Nine was the watershed moment that proclaimed the full-range ’stat had indeed come of age. The vision of peripatetic designer and collaborator Henry Kloss (his fingerprints are also on this list’s AR3a and the Double Advents), the Model Nine and its more potent iteration known by well-heeled owners as Double Nines, is significant in that it took the transcendent qualities (midrange transparency, resolution speed, and low distortion) of the parlor room-restricted Quad ESL and added properties like 40Hz bass extension and improved output. But this six-foot dipolar had an attitude—fussy about placement and greedy for power, it blew fuses with regularity. It could be beamy in the treble, too, but when all the stars aligned few cone speakers could match this naturalistic combination of liquidity, speed, and power, making it the rare companion able to capture near symphonic playback levels. The Model Nine has been the inspiration for virtually every planar-style loudspeaker since. Neil Gader

8. Infinity IRS V

This was the last version of the original Infinity Reference System, and, by any measure, the best, standing second to none in frequency range, in a top-to-bottom coherency that had eluded designer Arnie Nudell in the earlier three versions (yes, three, there was no IV), and in an overall faithfulness to the real thing that exceeded Nudell’s best previous efforts. The EMIT tweeters had been considerably updated (so that there was less grain, less artificial brightness, and a sound just a few steps below that of Jim Winey’s Magnepan true ribbon); the EMIT and the ENIN midranges (a replacement for the Series One’s bipolar ribbons) were both now planar “ribbons”; and the non-Watkins graphite-fiber woofers, all 12 of them, were now powered by a 2000-watt amp (up from 1500 in the Series III). What this, finally, accomplished, along with a few other mods, was a seamless sonic transition between the bass and the upper drivers—a first in a Nudell product. A dream realized and a dream for this listener. Harry Pearson

Biggest Number That Isn't Infinity

7. Magnepan 1-U/1-D

Of all the loudspeakers I’ve heard in a lifetime of listening, the large, three-panel Maggie 1-Us—Magnepan’s first widely marketed planar-magnetic speaker—remain the most memorable. I’ve told the story several times before of how I originally (and unwittingly) auditioned these speakers in the early 70s and—not knowing what a Magneplanar was back then—assumed that the real grand piano ensconced behind the “screens” at the far end of the listening room was making music when, in fact (and of course), it was the Maggies that were doing same. I’ve never again been fooled that completely by a loudspeaker because nothing I’ve heard since then has sounded that much more like the real thing than the Maggie 1-Us did at the dawn of the high-end era. As HP put it in his ground-breaking TAS review: “The Magneplanars are…[a] ‘classic’…a speaker that is and will be a standard by which and to which others will be compared.” And so they were, and so they still are, in certain key respects (such as midrange realism and mid-to-upper bass resolution, scale, and slam), to me. Jonathan Valin

6. Dahlquist DQ‑10

One should always be wary of pronouncing “firsts,” but, appearing in the early seventies, Jon Dahlquist’s DQ‑10 was to my knowledge the first dynamic speaker to employ multiple drivers in an open-baffle configuration (except the acoustic‑suspension woofer, which was enclosed) staggered for proper time‑alignment and phase coherence, in an attempt to realize the openness and freedom from boxiness that Dahlquist prized in his beloved Quad ESL-57s—with the added advantages of deeper bass and dynamic extension well beyond the Quad. (The physical resemblance to the Quad was both mandated by the design and an intentional homage.) Far from flawless (including conceptually), the DQ-10 was nevertheless a ground-breaking design that preceded dozens of subsequent speakers (perhaps most prominent among them models from KEF, B&W, Spica, Thiel, Vandersteen, and Wilson) continuing up to the present day. Few large, full-range dynamic speakers before or for some time afterward equaled its openness. Paul Seydor

5. Magico Mini II

Certain speakers don’t just change the way the game is played; they change the playing field. In the twentieth century, the Quad ESL-57s (par excellence), the Maggie 1-Us, the Dahlquist DQ-10s, and the Wilson WATT/Puppy were bellwethers. In the new millennium there has been no more influential loudspeaker—no loudspeaker that has had a more profound effect on the way other loudspeaker manufacturers design and style their products—than these massive, beautiful, stand-mounted two-ways from upstarts Alon Wolf and Yair Tammam of Magico. The Mini and Mini II set new standards for neutrality, resolution, dynamic range, frequency extension, and musicality in compact speakers—and they did so not just by upping the ante on the way enclosures, drivers, and crossovers were built but also by upping the ante on the science that speaker manufacturers brought to bear on designing enclosures, drivers, and crossovers. Since the Mini II, birch-ply and aluminum boxes, high-tech composite cones, Mundorf parts, computer-assisted design have become mainstream—and other things like beryllium diaphragms, field-coil magnetics, and coincident drivers have been added to the mix. The competition in compact speakers has never been stiffer; nonetheless, the Magico Minis were the first on the ground and continue to more than hold their own against all comers. Jonathan Valin

4. Wilson Audio Tiny Tot (WATT)

David Wilson’s WATT is one of those products the world had needed without realizing it. Created in 1985 as a one-off location monitor for Wilson’s recording work, the WATT inspired desire in all who heard it. Despite an astronomical price (at the time) for a small two-way, the WATT was an instant hit and went on to become the most popular high-end loudspeaker of all time. The WATT broke new ground in several areas. First, it established that a market existed for a very high-quality small loudspeaker, paving the way for products like the Sonus faber Extrema and later, the Magico Mini. Second, the WATT was the first loudspeaker in which reducing enclosure vibration was a high design goal. The modern trend toward stiff cabinets can be traced directly to the WATT. Once music lovers (and other designers) heard a loudspeaker with the sound of the box removed, the world never looked back. Robert Harley

3. Rogers/BBC LS3/5a

The LS3/5a was a BBC design, licensable to any manufacturer. But it was the Rogers version in particular that swept the USA in the late 1970s. This small two-way (7.5' x 12' x 6.25') offered startlingly realistic vocal reproduction and a remarkably expansive and “boxless” sound picture. With its essentially neutral midband but upper bass bump and slightly projected treble, it was not entirely flat, and it had no deep bass. But for a whole generation of listeners, it redefined the possible for small speakers. Some other, larger BBC-influenced designs—the Spendor BC1 first and the Spendor SP1 and SP1/2 and Harbeth Monitor 40 later on—were better speakers overall. But none quite seized the imagination of the U.S. audio public as did the little LS3/5a. With an updated version still in production today, the LS3/5a has stood the test of time as few other speakers have. Robert E. Greene

2. Acoustic Research AR3a

Edgar Villchur invented the acoustic-suspension loudspeaker. He founded the Acoustic Research Company with Henry Kloss and began production of the AR1 in 1955. The acoustic suspension principle was elegantly simple; Villchur mounted a long-throw 12' woofer in a sealed box, using the air trapped inside the box as the spring to launch the woofer’s cone. His design so reduced the size of the cabinet that you could place it on a bookshelf, making it an instant sensation.In 1958 Villchur demonstrated a new 3-way version, the AR3, with live vs. recorded events where the musicians would stop playing the notes but continue to “pretend” to play as the ARs were switched on. Suddenly, the musicians would stop and freeze while the music continued. Jaws would drop; everyone was fooled—it made newspaper headlines! At its peak the AR3a captured 33% of the high-fidelity loudspeaker market. The Smithsonian Institution has placed the AR3 on permanent display in The National Museum of American History. Peter Breuninger

1. Quad ESL-57

There was of course never any such moniker as “ESL-57,” except in retrospect, to distinguish it from its distinguished successor the ESL-63. Designed by the legendary Peter Walker and actually introduced in 1956, it was called simply the Quad ESL, but soon became known as “Walker’s little wonder.” Little wonder: For top-to-bottom clarity, coherence, transparency, resolution, openness, naturalness, and a disappearing act that still inspires awe, the ESL established and remains to this day (even though production ceased over a quarter century ago) a reference standard among countless designers and reviewers (including the undersigned) across the globe. Despite undeniable limitations—inability to play very loud, lack of deep bass, quite directional highs—it tops virtually every list of the best, the greatest, the most significant—supply your own category—audio products ever made. Why? Because at the dawn of the stereo era this “little” wonder demonstrated what was possible in most of the essential areas of speaker performance so validly that from a certain point of view the subsequent history of speaker design has been catch-up. Paul Seydor