Gödel's incompleteness theorems and Betamax: Difference between pages

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In [[mathematical logic]], '''Gödel's incompleteness theorems''' are two celebrated theorems proved by [[Kurt Gödel]] in [[1931]].
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Somewhat simplified, the first theorem states:
| name = Betamax
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| type = [[Video recording]] media
| encoding = [[Magnetic tape]]
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| owner = [[Sony]]
| use = Video storage
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[[Sony]]'s '''Betamax''' is the 12.7 mm (0.5 inch) home [[videocassette]] tape recording format introduced on April 16, 1975 (in market on May 10) and derived from the earlier, professional 19.1 mm (0.75 inch) [[U-matic]] video cassette [[format]]. Like the video home recording system [[Video Home System|VHS]] introduced by [[JVC]] in 1976, it had no [[guard band]], and used [[azimuth recording]] to reduce [[Crosstalk (electronics)|cross-talk]]. The "Betamax" name came from a double meaning: ''beta'' being the Japanese word used to describe the way signals were recorded onto the tape, and from the fact that when the tape ran through the transport it looked like the Greek letter "[[Beta (letter)|Beta]]" (β). The suffix ''-max'' came from "maximum" to suggest greatness.<ref>Sony History: [http://www.sony.net/Fun/SH/1-13/h5.html "This is a revolution!"]</ref>
 
[[Sanyo]] marketed a version as ''Betacord'', but this was also referred casually to as "Beta." In addition to Sony and Sanyo, Betamax video recorders were also sold by [[Toshiba]], [[Pioneer Corporation|Pioneer]], [[Aiwa]] and [[NEC Corporation|NEC]], and the [[Zenith Electronics Corporation]] and [[WEGA]] Corporations contracted with [[Sony]] to produce VCRs for their product lines. Department stores like [[Sears, Roebuck and Company|Sears]], in the US and Canada, and [[Quelle (company)|Quelle]] in Germany sold Beta format VCRs under their house brands as did the [[RadioShack]] chain of electronic stores.
:''In any [[consistency proof|consistent]] formalization of mathematics that is sufficiently strong to axiomatize the [[natural numbers]] -- that is, sufficiently strong to define the operations that collectively define the natural numbers -- one can construct a true (!) statement that can be neither proved nor disproved within that system itself.''
 
Sony introduced the Betamax home video system in [[1975]] with the LV-1901 Trinitron/Betamax console. It was the most popular video format in [[1983]], gaining almost a third of the UK video recorder market, while [[Sanyo|Sanyo's]] VTC5000 was the top selling UK video recorder. By 1985, however, the market had turned sharply towards VHS.
This theorem is one of the most famous outside of mathematics, and one of the most misunderstood. It is a theorem in ''formal logic'', and as such is easy to misinterpret. There are many statements that sound similar to Gödel's first incompleteness theorem, but are in fact not true. These are discussed in [[#Misconceptions about G.F6del.27s theorems|Misconceptions about Gödel's theorems]].
 
The world's first [[camcorders]] were Sony's ''Betamovie'' Betamax recorders.
Gödel's second incompleteness theorem, which is proved by formalizing part of the proof of the first within the system itself, states:
[[Image:Sony_bmc100p.jpg|right|thumb|The world's first brand of camcorder, 1983]]
 
== The legacy of Betamax ==
:''No consistent system can be used to prove its own consistency.''
The VHS format's defeat of the Betamax format became a classic [[marketing]] case study, now identified with the verbal phrase "to Betamax", wherein a [[proprietary]] technology format is overwhelmed in the market by a competing format allowing multiple, competing, licensed manufacturers. Sony's ability to dictate an industry standard backfired when [[JVC]], and parent [[Matsushita]], made the tactical decision to forego Sony's offer of Betamax in favor of JVC's VHS technology. They felt that it would end up like [[U-Matic]] deal: Sony dominating, and they get the scraps. By 1984, forty companies utilized the VHS format in comparison with Beta's twelve. Sony finally conceded defeat in 1988 when it too began producing VHS recorders. However, Sony may be said to have had some small consolation in this saga as its [[8 mm video format|Video-8]] small-format videotape is essentially a scaled-down version of the Betamax, and Video-8 dominated the home camcorder format for the next 15 years with the rival [[VHS-C]] format, until both formats were rendered obsolete by the digital [[MiniDV]] standard.
 
The real reason for the success of VHS is [[RCA]], who asked Matsushita for a 4 hour VHS machine. RCA had earlier discussed this with Sony during Beta's development phase, but Sony's engineers felt that by slowing the tape speed from 4 to 2 cm/sec and narrowing the video track, picture quality would be too poor. Matsushita, despite protests from JVC, delivered Long Play, exactly what RCA wanted. RCA in turn would offer their 4 hour VHS decks at a suggested retail of $995. RCA's pricing and marketing of their 4 hour mode VHS machine would be crucial.
This result was devastating to a philosophical approach to mathematics known as [[Hilbert's program]]. [[David Hilbert]] proposed that the consistency of more complicated systems, such as [[real analysis]], could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger.
 
One other major consequence of the Betamax technology's introduction to the U.S. was the lawsuit ''[[Sony Corp. v. Universal City Studios]]'' (1984, the "Betamax case"), with the [[U.S. Supreme Court]] determining home videotaping to be legal in the [[United States]], wherein home videotape cassette recorders were a legal technology since they had substantial non-infringing uses. This precedent was later invoked in ''[[MGM v. Grokster]]'' (2005), where the high court agreed that the same "substantial non-infringing uses" standard applies to authors and vendors of [[peer-to-peer]] file sharing software (notably excepting those who "actively induce" copyright infringement through "purposeful, culpable expression and conduct").
== Meaning of Gödel's theorems ==
 
[[Image:Three_betamax_vcrs.jpg|right|thumb|300px|Three Sony Betamax VCRs built for the American market. Top to Bottom: (1982) SL-2000 portable with TT-2000 tuner/timer 'Base Station', (1984) SL-HF 300 Betamax HiFi unit, (1988) SL-HF 360 SuperBeta HiFi unit.]]
Gödel's theorems are theorems in [[first-order logic]], and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs (this is called [[automated reasoning]]).
 
[[Image:SL-MV1.jpg|right|thumb|300px|A rare Japanese market Betamax TV/VCR combo - Model SL-MV1.]]
To be able to perform this process, we need to know what our axioms are. We could start with a finite set of axioms, such as in [[Euclidean geometry]], or more generally we could allow an infinite list of axioms, with the requirement that we can mechanically check for any given statement if it is an axiom from that set or not. In computer science, this is known as having a [[recursive set]] of axioms. While an infinite list of axioms may sound strange, this is exactly what's used in the usual axioms for the natural numbers, the [[Peano axioms]].
 
[[Image:Betacam_betamax_tapes.jpg|thumb|300px|The early form of Betacam tapes are interchangeable with Betamax, though the recordings are not.]]
Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false.
 
In the professional and broadcast video industry, Sony's [[Betacam]], derived from Betamax as a professional format, became one of several standard formats; production houses exchange footage on Betacam videocassettes, and the Betacam system became the most widely used videotape format in the [[Electronic News Gathering|ENG]] (Electronic News Gathering) industry, replacing the 3/4" U-matic tape format (which was the first practical and cost-effective portable videotape format for broadcast television, signaling the end of 16mm film -- and the phrase "film at eleven" often heard on the six-o-clock newscast, before the film had been developed). The professional derivative of VHS, [[MII (videocassette format)|MII]] (aka Recam), faced off against Betacam and lost. Once Betacam became the ''[[de facto]]'' standard of the broadcast industry, its position in the professional market mirrored VHS's dominance in the home-video market. On a technical level, Betacam and Betamax are similar in that both share the same videocassette shape, use the same oxide tape formulation with the same [[coercivity]], and both record linear audio tracks on the same ___location of the videotape. But in the key area of video recording, Betacam and Betamax are completely different. (Unlike Betamax, Betacam uses a [[component video|component-video]] encoding system.) BetaCam tapes are mechanically interchangeable with Betamax, but not electronically. BetaCam moves the tape at 12cm/sec, with different recording/encoding techniques. Betamax is a color-under system with linear tape speeds ranging from 4cm/sec to 1.33cm/sec.
The existence of an incomplete system is in itself not particularly surprising. For example, if you take [[Euclidean geometry]] and you drop the [[parallel postulate]], you get an incomplete system. An incomplete system can mean simply that you haven't discovered all the necessary axioms.
 
Sony also offered a range of Industrial Betamax products, a Beta I only format for industrial and institutional users. Basically cheaper and smaller than U-Matic. The arrival of the Betacam system reduced the demand for both Industrial Beta and U-Matic equipment.
What Gödel showed is that in most cases, such as in [[number theory]] or [[real analysis]], you can ''never'' discover the complete list of axioms. Each time you add a statement as an axiom, there will always be another statement out of reach.
 
Betamax also had a significant part to play in the music recording industry when Sony introduced its [[Pulse-code modulation|PCM]] (Pulse Code Modulation) digital recording system as an encoding box - [[PCM adaptor]] that connected to a Betamax recorder. The Sony PCM-F1 adaptor was sold with a companion Betamax VCR SL-2000 as a portable [[Digital audio]] recording system. Many recording engineers used this system in the 1980s and 1990s to make their first digital master recordings.
You can add an infinite number of axioms; for example, you can add ''all true statements about the natural numbers'' to your list of axioms, but such a list will not be a recursive set. Given a random statement, there will be no way to know if it is an axiom of your system. If I give you a proof, in general there will be no way for you to check if that proof is valid.
 
Initially, Sony was able to tout several Betamax-only features, such as BetaScan, a high speed picture search in either direction, and BetaSkipScan, a technique that allowed the operator to see where he was on the tape by pressing the FF key (or REW, if in that mode) and the transport would switch into the BetaScan mode until the key was released. Sony believed that the M-Load transports used by VHS machines made copying these trick modes impossible. BetaScan was originally called "Videola" until the company that made the [[Moviola]] threatened legal action.
Gödel's theorem has another interpretation in the language of computer science. In first-order logic, theorems are [[recursively enumerable]]: you can write a computer program that will eventually generate any valid proof. You can ask if they satisfy the stronger property of being [[recursive set|recursive]]: can you write a computer program to definitively determine if a statement is true or false? Gödel's theorem says that in general you cannot.
 
Sony would also sell a BetaPak, a small deck designed to be used with a camera. Concerned with the need for several pieces, and cables to connect them, an integrated camera/recorder was designed, which Sony dubbed a "Camcorder". The result was Betamovie. Betamovie used the standard sized cassette, but with a modified transport. The tape was wrapped 300 degrees around a smaller, 44.671mm diameter head drum, with a single dual-azimuth head to write the video tracks. For playback, the tape would be inserted into a Beta format deck. Due to the different geometry and writing techniques employed, playback within the camcorder was not feasible. SuperBeta and Industrial Betamovie camcorders would also be sold by Sony.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to [[David Hilbert]]'s program towards a universal mathematical formalism. The generally agreed upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did.
 
Betamax introduced high fidelity audio to videotape, as Betahifi. For [[NTSC]], Betahifi worked by placing a pair of FM carriers between the chroma (C) and luminance (Y) carriers, a process known as audio frequency modulation. Each head had a specific pair of carriers, in total four individual channels were employed. Head A recorded its hifi carriers at 1.38(L) and 1.68(R) MHz, and the B head employed 1.53 and 1.83 MHz. The result was audio with an 80dB dynamic range, with less than 0.005% wow and flutter.
== Examples of undecidable statements ==
 
Prior to the introduction of Betahifi, Sony shifted the Y carrier up by 400 kHz to make room for the 4 FM carriers that would be needed for Betahifi. All Beta machines incorporated this change, plus the ability to hunt for a lower frequency pre-AFM Y carrier. Sony incorporated an "anti-hunt" circuit, to stop the machine hunting for a Y carrier that wasn't there.
The existence of an undecidable statement within a formal system is not in itself a surprising phenomenon.
 
Some Sony NTSC models were marketed as "HiFi Ready" (with a SL-HFR prefix to the model's number instead of the usual SL or SL-HF). These Betamax decks looked like a regular Betamax model, except for a special 28 pin connector on the rear. If the user desired a Betahifi model but lacked the funds at the time, he could purchase an "SL-HFRxx" and at a later date purchase the separate Hi Fi Processor. Sony offered two outboard Betahifi processors, the HFP-100 and HFP-200. They were identical except that the HFP-200 was capable of multi-channel TV sound, with the word "stereocast" printed after the Betahifi logo. This was possible because unlike a VHS HiFi deck, an NTSC Betamax didn't need an extra pair of heads. The HFP-x00 would generate the needed carriers which would be recorded by the attached deck, and during playback the AFM carriers would be passed to the HFP-x00. They also had a small "fine tracking" control on the rear panel for difficult tapes.
The subsequent combined work of Gödel and [[Paul Cohen]] has given concrete examples of undecidable statements (statements which can be neither proven nor disproven): both the [[axiom of choice]] and the [[continuum hypothesis]] are undecidable in the standard axiomatization of [[set theory]].
These results do not require the incompleteness theorem.
 
For [[PAL]], however, the bandwidth between the Chroma and Luminance carriers was not sufficient enough to allow additional FM carriers, so depth multiplexing was employed, where the audio track would be recorded in the same way that the video track was. The lower frequency audio track was written first by a dedicated head, and the video track recorded on top by the video head. The head disk had an extra pair of audio only heads with a different azimuth, positioned slightly ahead of the regular video heads, for this purpose.
In [[1936]], [[Alan Turing]] proved that the [[halting problem]]&mdash;the question of whether or not a [[Turing machine]] halts on a given program&mdash;is undecidable. This result was later generalised in the field of [[recursive function]]s to [[Rice's theorem]] which shows that all non-trivial decision problems are undecidable in a system that is [[Turing completeness|Turing-complete]].
 
Sony was confident that VHS could not achieve the same audio performance feat as Betahifi. However, to the chagrin of Sony, JVC did develop a VHS hi-fi system on the principle of depth multiplexing approximately a year after the first Betahifi VCR, the SL-5200, was introduced by Sony. Despite initial praise as providing "CD sound quality," both Beta Hi-Fi and VHS HiFi suffered from "carrier buzz," where high frequency information bled into the audio carriers, creating momentary "buzzing" and other audio flaws. Both systems also used companding noise-reduction systems, which could create "pumping" artifacts under some conditions. Both formats also suffered from interchange problems, where tapes made on one machines did not always play back well on other machines. When this happened, users were forced to revert to the old linear soundtrack.
In [[1973]], the [[Whitehead problem]] in [[group theory]] was shown to be undecidable in standard set theory.
In [[1977]], Kirby, Paris and Harrington proved that a statement in [[combinatorics]], a version of the [[Ramsey theorem]], is undecidable in the axiomatization of arithmetic given by the [[Peano axioms]] but can be proven to be true in the larger system of set theory.
[[Kruskal's tree theorem]], which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory.
[[Goodstein's theorem]] is a relatively simple statement about natural numbers that is undecidable in Peano arithmetic.
 
In 1985 Sony would introduce a new feature, High Band or SuperBeta, by again shifting the Y carrier, this time by 800 kHz. This improved the bandwidth available to the Y sideband, giving 290 to 300 lines of horizontal resolution in this mode, on a regular grade Beta cassette. A typical videocassette recorder was 240-250 lines. The heads were also narrowed to 29 microns to reduce crosstalk. Later, some models would feature further improvement, in the form of BIs, a high band version of the Beta I (BI) recording mode. There were some incompatibilities between the older Beta decks and SuperBeta, but most could play back a high band tape without major problems. SuperBeta decks had a switch to disable the SuperBeta mode for compatibility purposes.
[[Gregory Chaitin]] produced undecidable statements in [[algorithmic information theory]] and in fact proved his own incompleteness theorem in that setting.
 
JVC would counter SuperBeta with VHS HQ, or High Quality, a series of improvements to their VHS format. Originally, an HQ branded deck would have a luminance noise reduction circuit, a chroma noise reduction circuit, white clip extension, and improved sharpness circuitry. The effect was to increase the apparent resolution of a VHS recording. The major VHS [[Original Equipment Manufacturer|OEM]]s resisted HQ due to cost concerns, eventually resulting in JVC reducing the requirements for the HQ brand to ''white clip extension plus one other improvement'', either those by JVC or another circuit proposed by RCA.
One of the first problems suspected to be undecidable was the [[word problem for groups]], first posed by [[Max Dehn]] in [[1911]], which states that there is a finitely presented [[group (mathematics)|group]] that has no algorithm to state whether two words are equivalent. It was not proven to be undecidable until [[1952]].
 
JVC would introduce their SuperVHS, at 400 lines, a few years later. Due to the lower writing speeds inherent to the VHS design, Super VHS required a special high grade tape, unlike SuperBeta, which used the regular formulation Beta cassettes. Sony would also offer SuperBeta in their Industrial Beta line.
== Misconceptions about Gödel's theorems ==
 
Sony would again push the envelope with ED or "Extended Definition" Betamax, capable of up to 500 lines of resolution, using a metal formulation tape from the Betacam. They also incorporated some improvements to the transport to reduce mechanically induced aberrations in the picture. Sony introduced two ED decks and a camcorder in the late 1980s. The top end ED deck was a very capable editing deck, rivalling much more expensive U-Matic setups for its accuracy, but did not have commercial success due to lack of timecode and other pro features.
Since Gödel's first incompleteness theorem is so famous, it has given rise to many misconceptions. They are summarized here:
 
Despite the sharp decline in sales of Betamax recorders in the late 1980s and subsequent halt in production of new recorders by Sony in 2002, the format is still being used by a small number of people, most of whom are collectors or hobbyists.
# The theorem does not imply that every interesting axiom system is incomplete. For example, [[Euclidean geometry]] can be axiomatized so that it is a complete system. (In fact, [[Euclid]]'s original axioms are pretty close to being a complete axiomatization. The missing axioms express properties that seem so obvious that it took the emergence of the idea of a formal proof before their absence was noticed.)
New cassettes are still available for purchase at [[online shop]]s and used recorders are often found at [[flea market]]s, thrift stores, or on [[internet]] [[auction]] sites.
# The theorem only applies to systems that allow you to ''define'' the natural numbers as a set. It is not sufficient that the system ''contain'' the natural numbers. You must also be able to express the concept "<math>x</math> is a natural number" using your axioms and first-order logic. There are plenty of systems that contain the natural numbers and are complete. For example both the [[real numbers]] and [[complex numbers]] have complete axiomatizations.
 
== Criticism ==
== Discussion and implications ==
[[Image:Betavhs2.jpg|right|thumb|Size comparison between a Betamax cassette and a VHS cassette.]]
A multitude of technical drawbacks along with the proprietary nature of the Betamax format hurt it in its competition with VHS, in spite of the Betamax's superior video quality. Other proprietary formats from [[Sony]] such as [[Memory Stick]], [[Universal Media Disc|UMD]], [[Digital8]], and [[ATRAC]] have since been similarly criticized.
 
The main issue with the Beta format in the early days of the USA market was recording time. The original prototypes shown to Matsushita used a linear tape speed of 40 mm/sec. The technology of the day needed that speed due to the 60 micron heads employed. Sony engineers and management had decided that since one hour was acceptable to the U-Matic's buyers, it was acceptable for Beta too. So the Beta format had a smaller, one hour cassette called a K-60. (The designation would later change to L-500). The cassette was loaded with 150m of tape (close to 500 feet, which is where the "500" designation came from).
The incompleteness results affect the [[philosophy of mathematics]], particularly viewpoints like [[Symbolic Logic|formalism]], which uses formal logic to define its principles.
One can paraphrase the first theorem as saying that "we can never find an all-encompassing axiomatic system which is able to prove ''all'' mathematical truths, but no falsehoods."
 
RCA had initially planned a home video format around 1974, to be called "SelectaVision MagTape," but cancelled it after hearing rumors about Sony's Betamax format, and was considering Sony as an OEM for an RCA-branded VCR. RCA had discussions with Sony, but RCA felt the recording time was too short, insisting that they needed at least a 4-hour recording time (reportedly because that was the length of an average televised U.S. football game). Sony engineers knew that the technology available to manufacture video heads wasn't up to the task yet, but halving the tape speed and track width was a possibility. Unfortunately, the picture quality would be degraded severely, and at that time Sony engineers felt the compromise was not worthwhile.
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.
 
Soon after, RCA met with execs with the Victor Corporation of Japan (JVC), who had created their own video format, christened "VHS" (Which stood for "Video Helical Scan" and later "Video Home System"). But JVC also refused to compromise the picture quality of their format by allowing a 4-hour mode. Ironically, their parent corporation, Matsushita, later met with RCA, and agreed to manufacture a 4-hour-capable VHS machine for RCA, much to JVC's chagrin. (For NTSC markets, JVC never did make a VHS machine capable of the 4-hour "LP" mode, though they did later offer a 6-hour "EP" mode, claiming performance better than the early 4-hour machines due to improvements in video heads and video circuitry. LP machines by JVC exist for PAL markets because the basic tape speed was lower in the first place.)
The following rephrasing of the second theorem is even more unsettling to the [[foundations of mathematics]]:
 
Sony would introduce an "X2" speed of 20 mm/sec for a "2 hour Betamax." This led to confusion in the market, as some decks only recorded in X2, and others didn't even play X1 tapes. Later models would have the capability to handle a thinner tape that ran for 90 minutes in X1 mode, but many decks only recorded in X2.
:''If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.''
 
Recording time was everything, with Beta eventually managing 5 hours at BIII (13.3 mm/sec) on an ultra-thin L-830 cassette, and VHS achieving 6 hours with Extended Play on the standard T-120 cassette (and up to 9 hours on an ultra-thin T-180 cassette, later even longer with the T-200 cassette). Slower tape speeds meant a degradation in picture quality, but the consumer didn't seem to mind. Sony was never able to overcome the disadvantage of the 1 hour recording limit when Beta was introduced.
Therefore, in order to establish the consistency of a system S, one needs to utilize some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.
The consistency of the [[Peano axioms]] for [[natural numbers]] for example can be proven in [[set theory]], but not in the theory of natural numbers alone.
This provides a negative answer to problem number 2 on [[David Hilbert]]'s famous list of important open questions in mathematics (called [[Hilbert's problems]]).
 
According James Lardner's 1987 book, "Fast Forward: Hollywood, the Japanese, and the Onslaught of the VCR," Sony had met with Matsushita execs sometime in late 1974/early 1975, to discuss the forthcoming home video market. They had previously co-operated in the development and marketing of the "U" format videocassette, with Sony marketing under the U-Matic brand. At the meeting, the Matsushita exec showed them a VHS prototype, and advised them it was not too late to embrace VHS "for the good of the industry." Sony management were too close to production (and, one could argue, too proud and arrogant) to compromise, and felt their generosity had been taken advantage of. Thus, the stage was set for a battle between Sony and Matsushita in the arena of home video.
In principle, G&ouml;del's theorems still leave some hope: it might be possible to produce a general [[algorithm]] that for a given statement determines whether it is undecidable or not, thus allowing mathematicians to bypass the undecidable statements altogether. However, the negative answer to the [[Entscheidungsproblem]] shows that no such algorithm exists.
 
It should be noted that in Europe which uses the [[PAL]] television system, recording time was never such an issue. An L-750 runs for 3.25 hours with the PAL system, whereas the equivalent E180 tape would run for just 3 hours, giving Beta a longer running time for most users. The very longest L830 tape however was slightly shorter than the longest VHS tape of 4 hours at the time, but the difference of 25 minutes was not enough to be frequently cited as a reason for VHS being preferred.
There are some who hold that a statement that is unprovable within a deductive system may be quite provable in a metalanguage. And what cannot be proven in ''that'' metalanguage can likely be proven in a '''meta'''-metalanguage, recursively, ad infinitum, in principle. By invoking a sort of super Theory of Types with an axiom of Reducibility -- which by an inductive assumption applies to the entire stack of languages -- one may, for all practical purposes, overcome the obstacle of incompleteness.
 
== Popular culture ==
Note that G&ouml;del's theorems only apply to ''sufficiently strong''
Betamax was featured in a sinister context in [[David Cronenberg]]'s 1983 film ''[[Videodrome]]'', in which a video signal recorded on Betamax tape is used for [[mind control]]. By the late 1980s, however, jokes about the format's unpopularity were appearing in popular culture. In a 1989 episode of the [[situation comedy]] ''[[Married... with Children]]'', the Bundys were described as "the last family on earth with Beta"; a character had to cross state lines to get a tape from "Bob's Betas and [[Bell-bottoms]]", and was only able to rent ''[[Oh Heavenly Dog]]''.<ref>''Married... with Children'' episode 50, "The Harder They Fall", [[1989-03-26]].</ref> In the 1990 novel ''[[Good Omens]]'', a 17th-century book of prophecies includes the warning "Do Notte Buye Betamacks".<ref>{{cite book | last = Gaiman | first = Neil | authorlink = Neil Gaiman | coauthors = Terry Pratchett | title = Good Omens | publisher = Ace | date = 1996 | ___location = New York | pages = 195 | url = | doi = | id = ISBN 0-441-00325-7 }}</ref> In a 1992 episode of ''[[The Simpsons]]'', Snake steals a VCR, but on inspecting it exclaims "Oh no, Beta!"<ref>'"The Simpsons'', "Itchy and Scratchy, The Movie", [[1992-11-03]].</ref> In an episode of ''[[Futurama]]'', [[Mom (Futurama)|Mom]] says that she won't be around forever and a talking Betamax player says, "Oh shush," in response.<ref>''[[Futurama]]'', "[[Mother's Day (Futurama)|Mother's Day]]", [[2000-05-14]].</ref> In an episode of the television series [[Cowboy Bebop]], two of the main characters must go out in search of a beta player in an ancient history museum in order to view a mysterious beta cassette tape they recieved in the mail. Upon returning with a player, they woefully discover that they had found a VHS player, not betamax.
axiomatic systems.
"Sufficiently strong" means that the theory contains enough arithmetic to carry out the coding constructions needed for the proof of the first incompleteness theorem. Essentially, all that is required are some basic facts about addition and multiplication as formalized, e.g., in [[Robinson arithmetic Q]].
There are even weaker axiomatic systems that are consistent and complete, for instance [[Presburger arithmetic]] which proves every true first-order statement involving only addition.
 
As a well-known obsolete [[data storage device|storage medium]], beta videotapes are sometimes mentioned alongside other things that themselves are&mdash;or may become&mdash;outdated. For example, in a 2003 episode of ''The Simpsons'', a junkyard contains a pile of Betamax tapes, a pile of [[laserdiscs]], and next to them an empty space with a sign reading "reserved for [[DVD]]".<ref>''The Simpsons'', "[[The Fat and the Furriest]]", [[2003-11-30]].</ref> Similarly, a 2006 episode of ''[[The Grim Adventures of Billy & Mandy]]'' describes [[Fred Flintstone]] as coming from a time of "[[dinosaur]]s, Betamax tapes, [[floppy disks]], and [[acid wash]] jeans with holes in them on purpose".<ref>''The Grim Adventures of Billy and Mandy'', "Modern Primitives", [[2006-01-27]].</ref>
The axiomatic system may consist of infinitely many axioms (as first-order Peano arithmetic does), but for G&ouml;del's theorem to apply, there has to be an effective algorithm which is able to check proofs for correctness. For instance, one might take the set of all first-order sentences which are true in the standard model of the [[natural numbers]]. This system is complete; G&ouml;del's theorem does not apply because there is no effective procedure that decides if a given sentence is an axiom. In fact, that this is so is a consequence of G&ouml;del's first incompleteness theorem.
 
In [[rock climbing]], to give guidance as to how to climb a particular route is called "[[Beta (climbing)|giving beta]]". This is due to climbers sometimes videotaping other climbers on climbing routes using the Betamax format (when it was still in use), then reviewing the tapes to evaluate techniques to help them climb the same routes in the future.
Another example of a specification of a theory to which G&ouml;del's first theorem does not apply can be constructed as follows: order all possible statements about natural numbers first by length and then [[lexicographic order|lexicographically]], start with an axiomatic system initially equal to the Peano axioms, go through your list of statements one by one, and, if the current statement cannot be proven nor disproven from the current axiom system, add it to that system.
This creates a system which is complete, consistent, and sufficiently powerful, but not [[recursively enumerable set|recursively enumerable]].
 
==Notes and references==
G&ouml;del himself only proved a technically slightly weaker version of the above theorems; the first proof for the versions stated above was given by [[J. Barkley Rosser]] in 1936.
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== See also ==
In essence, the proof of the first theorem consists of constructing a statement ''p'' within a formal axiomatic system that can be given a meta-mathematical interpretation of:
* [[Videotape format war]]
* [[Peep search]] A picture search system pioneered with Betamax and available on most video formats since.
 
== External links ==
:''p'' = "This statement cannot be proven"
* [http://www.alivestudios-memories.co.uk/info3.html Running time of all lengths of Betamax tape, and how to copy the film to DVD/miniDV and AVI]
 
* [http://www.betainfoguide.com The Ultimate Betamax Info Guide - covering the Betamax format in the North American market]
As such, it can be seen as a modern variant of the [[Liar paradox]]. Unlike the Liar sentence, ''p'' does not directly refer to itself; the above interpretation can only be "seen" from outside the formal system.
* [http://www.mrbetamax.com/ Mister Betamax - extensive Beta supply site]
 
* [http://betamax.palsite.org Betamax PALsite - over 350 pages of Betamax information, running since 1997]
If the axiomatic system is consistent, G&ouml;del's proof shows that ''p'' (and its negation) cannot be proven in the system.
* [http://www.totalrewind.org The 'Total Rewind' VCR museum, covering Betamax and other vintage formats]
Therefore ''p'' is "true" (''p'' claims not to be provable, and it isn't) yet it cannot be formally proved in the system.
* [http://www.colin99.co.uk/beta.html The Betamax format in the UK, including technical information on servicing Sanyo Beta machines]
Note that adding ''p'' to the axioms of the system would not solve the problem: there would be another G&ouml;del sentence for the enlarged theory.
* [http://aolradio.podcast.aol.com/twit/DGW-030.mp3 "Daily Giz Wiz" Podcast discussing the Betamax]
 
[[Roger Penrose]] claims that this (alleged) difference between "what can be mechanically proven" and "what can be seen to be true by humans" shows that human intelligence is not mechanical in nature. This claim is also addressed by [[JR Lucas]] in [http://users.ox.ac.uk/~jrlucas/mmg.html Minds, Machines and Gödel].
 
This view is not widely accepted, because as stated by [[Marvin Minsky]], human intelligence is capable of error and of '''understanding''' statements which are in fact inconsistent or false. However, Marvin Minsky has reported that [[Kurt Gödel]] told him personally that he believed that human beings had an intuitive, not just computational, way of arriving at truth and that therefore his theorem did not limit what can be known to be true by humans.
 
The position that the theorem shows humans to have an ability that transcends formal logic can also be criticized as follows: We do not know whether the sentence ''p'' is true or not, because we do not (and can not) know whether the system is consistent.
So in fact we do not know any truth outside of the system.
All we know is the following statement:
 
:Either ''p'' is unprovable within the system, or the system is inconsistent.
 
This statement is easily proved ''within the system''.
In fact, such a proof will now be given.
 
== Proof sketch for the first theorem ==
The main problem in fleshing out the above mentioned proof idea is the following: in order to construct a statement ''p'' that is equivalent to "''p''
cannot be proved", ''p'' would have to somehow contain a reference to ''p'', which could easily give rise to an infinite regress. G&ouml;del's ingenious trick, which was later used by [[Alan Turing]] to solve the [[Entscheidungsproblem]],
will be described below.
 
To begin with, every formula or statement that can be formulated in our system gets a unique number, called its [[Goedel number|'''G&ouml;del number''']].
This is done in such a way that it is easy to mechanically convert back and forth between formulas and G&ouml;del numbers. Because our system is strong enough to reason about ''numbers'', it is now also possible to reason about ''formulas''.
 
A formula ''F''(''x'') that contains exactly one free variable ''x''
is called a ''statement form''.
As soon as ''x'' is replaced by a specific number, the statement form turns into a ''[[bona fide]]'' statement, and it then is either provable in the system, or not.
Statement forms themselves are not statements and therefore cannot be proved or disproved.
But every statement form ''F''(''x'') has a G&ouml;del number which we will denote by '''G'''(''F'').
The choice of the free variable used in the form ''F''(''x'') is not relevant to the assignment of the Gödel number '''G'''(''F'').
 
By carefully analyzing the axioms and rules of the system, one can then write down a statement form ''P''(''x'') which embodies the idea that ''x'' is the G&ouml;del number of a statement which can be proved in our system.
Formally: ''P''(''x'') can be proved if ''x'' is the G&ouml;del number of a provable statement, and its negation ''~P''(''x'') can be proved if it isn't.
(While this is good enough for this proof sketch, it is technically not completely accurate.
See G&ouml;del's paper for the problem and Rosser's paper for the resolution.
The key word is "omega-consistency".)
 
Now comes the trick: a statement form ''F''(''x'') is called ''self-unprovable'' if the form ''F'', applied to its own G&ouml;del number, is not provable.
This concept can be defined formally, and we can construct a statement form ''SU''(''z'') whose interpretation is that ''z'' is the G&ouml;del number of a self-unprovable statement form. Formally, ''SU''(''z'') is defined as: ''z''&nbsp;=&nbsp;'''G'''(''F'') for some particular form ''F''(''x''), and ''y'' is the G&ouml;del number of the statement ''F''('''G'''(''F'')), and ''~P''(''y''). Now the desired statement ''p'' that was mentioned above can be defined as:
 
:''p''&nbsp;=&nbsp;''SU''('''G'''(''SU'')).
 
Intuitively, when asking whether ''p'' is true, we ask: "Is the property of being self-unprovable itself self-unprovable?"
This is very reminiscent of the [[Barber paradox]] about the barber who shaves precisely those people who don't shave themselves: does he shave himself?
 
We will now assume that our axiomatic system is consistent.
 
If ''p'' were provable, then ''SU''('''G'''(''SU'')) would be true, and by
definition of ''SU'', ''z''&nbsp;=&nbsp;'''G'''(''SU'') would be the G&ouml;del number of a self-unprovable statement form.
Hence ''SU'' would be self-unprovable, which by definition of self-unprovable means that ''SU''('''G'''(''SU'')) is not provable, but this was our ''p'': ''p'' is not provable.
This contradiction shows that ''p'' cannot be provable.
 
If the negation of ''p''= ''SU''('''G'''(''SU'')) were provable, then by definition of ''SU'' this would mean that ''z''&nbsp;=&nbsp;'''G'''(''SU'') is not the G&ouml;del number of a self-unprovable form, which implies that ''SU'' is not self-unprovable.
By definition of self-unprovable, we conclude that ''SU''('''G'''(''SU'')) is provable, hence ''p'' is provable. Again a contradiction.
This one shows that the negation of ''p'' cannot be provable either.
 
So the statement ''p'' can neither be proved nor disproved within our system.
 
== Proof sketch for the second theorem ==
 
Let ''p'' stand for the undecidable sentence constructed above, and let's assume that the consistency of the system can be proven from within the system itself.
We have seen above that if the system is consistent, then ''p'' is not provable.
The proof of this implication can be formalized in the system itself, and therefore the statement "''p'' is not provable", or "not ''P''(''p'')" can be proven in the system.
 
But this last statement is equivalent to ''p'' itself (and this equivalence can be proven in the system), so ''p'' can be proven in the system.
This contradiction shows that the system must be inconsistent.
 
== See also ==
 
{{Home video}}
* [[Consistency]]
{{Sony Corp}}
* [[Self-reference]]
* [[Logicism]]
* [[Minds, Machines and Gödel]]
* [[Löb's Theorem]]
 
[[Category:Sony products]]
==External links and references==
[[Category:1975 introductions]]
* K. G&ouml;del: ''[http://home.ddc.net/ygg/etext/godel/ &Uuml;ber formal unentscheidbare S&auml;tze der Principia Mathematica und verwandter Systeme, I.]'' Monatshefte für Mathematik und Physik, 38 (1931), pp. 173-198. Translated in van Heijenoort: ''From Frege to G&ouml;del''. Harvard University Press, 1971.
[[Category:Commercial failures]]
* B. Rosser: ''Extensions of some theorems of Gödel and Church''. Journal of Symbolic Logic, 1 (1936), N1, pp. 87-91
[[Category:Discontinued media formats]]
* Hao Wang: ''A Logical Journey: From Gödel to Philosophy'' Bradford Books (January 10, 1997) ISBN: 0262231891
[[Category:Metaphors]]
* K&#257;rlis Podnieks: ''Around Goedel's Theorem'', http://www.ltn.lv/~podnieks/gt.html
[[Category:Video storage]]
* D. Hofstadter: ''[[G&ouml;del, Escher, Bach|G&ouml;del, Escher, Bach: An Eternal Golden Braid]]'', 1979, ISBN 0465026850. (1999 reprint: ISBN 0465026567).
*Ernest Nagel, James Roy Newman, Douglas R. Hofstadter: ''Gödel's Proof'', revised edition (2002). ISBN 0814758169.
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob2 Hilbert's second problem] (English translation)
* Norbert Domeisen, Logik der Antinomien. Bern etc.: Peter Lang. 142 S. 1990. (ISBN 3-261-04214-1), [http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0724.03003&format=complete Zentralblatt MATH]
[[Category:Theorems]]
[[Category:Mathematical logic]]
[[Category:Model theory]]
[[Category:Proof theory]]
 
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