The work of an important, though eccentric, Czech-Austrian mathematical logician, Kurt Gödel (1906-1978) dealt with completeness and consistency of logical systems. A passing acquaintance with his two theorems would have saved architect a lot of time.

Gödel's First Incompleteness Theorem states that every consistent axiomatic logical system, sufficient to express arithmetic, contains true but unprovable ("not decidable") sentences. In certain cases (when system is omega-consistent), both said sentences and their negation are unprovable. The system is consistent and true - but not "complete" because not all its sentences can be decided as true or false by either being proved or by being refuted.

The Second Incompleteness Theorem is even more earth-shattering. It says that no consistent formal logical system can prove its own consistency. The system may be complete - but then we are unable to show, using its axioms and inference laws, that it is consistent

In other words, a computational system can either be complete and inconsistent - or consistent and incomplete. By trying to construct a system both complete and consistent, a robotics engineer would run afoul of Gödel's theorem.

Note - Turing Machines

In 1936 an American (Alonzo Church) and a Briton (Alan M. Turing) published independently (as is often case in science) basics of a new branch in Mathematics (and logic): computability or recursive functions (later to be developed into Automata Theory).

The authors confined themselves to dealing with computations which involved "effective" or "mechanical" methods for finding results (which could also be expressed as solutions (values) to formulae). These methods were so called because they could, in principle, be performed by simple machines (or human-computers or human-calculators, to use Turing's unfortunate phrases). The emphasis was on finiteness: a finite number of instructions, a finite number of symbols in each instruction, a finite number of steps to result. This is why these methods were usable by humans without aid of an apparatus (with exception of pencil and paper as memory aids). Moreover: no insight or ingenuity were allowed to "interfere" or to be part of solution seeking process.

What can you say about ring tones? NOTHING…except that cell phones capable of downloading ringtones are newest “must have” item today! One of fastest growing features today is ringtones, especially downloadable ones! There are other downloadable items available on Internet too like Java Games. Various wallpapers and screen savers are hot too but by far most popular feature requested is downloadable ringtones.

The cell phone and their various ringtones have become status symbols. Ringtones show that you’re up to date and happening. You can even take a test that defines which ringtone best suits your personality. You can truly display your lifestyle and certainly your musical taste on your sleeve.

This article will describe briefly what ring tones are, what’s currently available and what future holds. It will also suggest a caveat to consider prior to downloading multitude of formats of available on market. Prices are low and sky’s limit, so read on.

• Just What Are Downloadable Ringtones?
• A Caveat Before You Buy!
• What’s Available?
• A Brief Glimpse At Ancient History.
• Ringtones – Where Do We Go From Here?
• Other Products Available On Ringtone Websites
• What Can You Say About Ringtones

Ringtones are sound that a regular or cell phone makes when it’s rings. Early ring tones were simple chimes and on later models, a combination of chimes. Limited numbers of ringtones were available and mostly consisted of different patterns of tones or chirps. An example of one of most infamous ringtones was ominous” ring ring…ring ring” that is reminiscent of Todd Beamer’s last telephone call on 9/11. You can hear that ringtone featured as an intro to Neil Young’s famous song “Let’s Roll.”

Ringtone technology developed quickly and has progressed a lot since that song was recorded. Originally, you purchased a cell phone and had a choice of monophonic ring tones that produced a one note song. The number of songs was extremely limited and downloadable ring tones were simply an Engineering student’s good idea.

Because of lightning speed technology and numerous service providers, one of latest and greatest features available today is downloadable ringtones. When you’re surfing ‘Net you’ll find plenty of sites offering tones by today’s hottest musicians; you can pick monophonic, polyphonic or true tone formats. More on that later.

A Caveat Before You Buy!

The first thing you need to know before you select, buy and download a suitable ring tone is if your cell phone is up to date enough to utilize this feature. Like everything else in technology, cell phones have evolved very quickly. As a result cell phone that you purchased just a year ago may not have ability to fully access type of ring tone currently available. So make sure you check. Re-read your manual or call your service provider to find out. It’s possible that you can upgrade your phone for a reasonably low fee.

What’s Available?

Ringtones are divided into three formats.

1. Monophonic
2. Polyphonic
3. True Tone or Real Tone

A) Monophonic Ringtones Some of first cell phones came equipped with capability of having a one tone ringer. “Monophonic” is simplest of ring tone technology that uses a midi format. This “one note wonder” changed ringtones. Gone were chirps and chimes; now you could play a song. A monophonic ringtone is single notes playing a song. You can recognize your favorite song but it sounds a little lame. Most monophonic tones are offered free of charge because of more full bodied tones now available, but if you’re trying to make a statement, however, read on.