Why can no other chemical elements be used for making a light bulb filament? As far as I can see from this Wikipedia article on the incandescent light bulb, there have been only four types of light bulb filaments: those made of carbon, those made of osmium, those made of tantalum and those made of tungsten (wolfram). I wonder why it is impossible to use any other chemical element for that purpose. Is there a simple explanation of that reason?
 A: Essentially: carbon, osmium, tantalum, and tungsten, along with rhenium, are the only (known, stable) elements that have melting points high enough to remain solid at the high temperatures required to achieve the colors of standard incandescent light bulbs.
Why? Incandescent light bulbs produce light by heating a filament, which gets so hot that it emits enough radiation to light up the room. We can model the color of the light bulb well by approximating it as a blackbody at thermal equilibrium, governed by Planck's law. When electricity is passed through it, the light bulb filament heats up until it reaches, at equilibrium, the temperature referred to on the bulb label. The colder the temperature, the redder the bulb, and the higher the temperature, the bluer the bulb.
Standard incandescent light bulbs have temperatures between 2700 K and 3000 K. This is because the temperature that produces a peak at the red-most end of the visible spectrum (with $\lambda=750$ nm) is ~3800 K. Thus, as as filament temperature is reduced below 3800 K, the peak will shift further out of the visible range and the light produced will appear dimmer (due to Wien's displacement law) and so colder temperatures appear too dim to function as a light bulb.
As the filament temperature is increased, however, the operational temperature of the light bulb approaches the melting point of the filament. Since the temperature is just the mean kinetic energy of molecules, many of the molecules in the filament would have more energy individually than that mean, severely reducing the lifetime of a light bulb with a filament whose operational temperature is too close to its melting point. This is why incandescent bulbs are rarely rated above 3000 K (at least those without some kind of special coating that makes them bluer).
However, the only elements with melting points above 3000 K are the elements you mention: carbon, osmium, tantalum, and tungsten, with the exception of rhenium, one of the rarest elements on earth. See the elements in red (which have melting points at or above 3000 K) in the following periodic table (from ptable.com):

That doesn't mean there aren't other materials with high melting points that might function well as filaments for an incandescent bulb, such as tantalum carbide (which can melt at around 3900 K), for example. Here's a 1935 patent for a tantalum carbide lamp. Even rhenium has been considered. Here's a 2001 patent for a tungsten-rhenium alloy filament, although given the rarity of rhenium, it is likely not economical to use as a filament for consumer-grade light bulbs.
Non-incandescent light bulbs produce light through entirely different mechanisms, or else don't use a solid filament, which is why LED light bulbs don't require tungsten et al. at all.
A: Although the question has been excellently answered by @abeta201, my question will serve as a history lesson and how tungsten outperformed other metals.

Various physicists and scientist started to conceptualize device that will produce light on sending electricity. Joseph Swan was able to demonstrate such a working device but faltered with the reasoning for lack of vacuum and adequate supply of electricity but once better vacuum pumps were available, he started working on device which used carbonized paper filaments/carbon rods. Side-by-side, Thomas Edison also started his own research and filed his own patent. He once moved to platinum but again returned to carbon (due to cost). The first successful test was lasted 13.5 hours and his patent was approved and the device was commercialized to be used in lampposts.
Carbon filaments have a negative temperature coefficient of resistance—as they get hotter, their electrical resistance decreases. This made the lamp sensitive to fluctuations in the power supply, since a small increase of voltage would cause the filament to heat up, reducing its resistance and causing it to draw even more power and heat even further. However, it required 3 watts per candle of light. So, everybody started to  think of alternatives and they found metallic filaments.
At the wake of 20th century, various metals were tested. In 1902, Osmium was used as filament and it made a fairly satisfactory bulb requiring 1.5 watt per candle of light. A number of osmium lamps has been started to be used in Germany but the rarity and cost prevented the commercial introduction on a large scale. In 1904, bulbs made of tantalum has started to be used requiring 2 watts per candle but had a greater life expectancy only when used in direct current and not alternating current.
Following year, bulbs with tungsten filament were started to be used requiring 1.25 watt per candle of light and had a greater life expectancy in both AC and DC and it was started to be commercialized at a global level and then no one had looked for any other metal ever. Now, why tungsten worked and outperformed?

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*Highest Melting point (3410°C) and it can be operated at higher temperature and higher the operating temperature, higher the luminous efficacy. (Osmium was also near - 3050°C) but lack of thermal stability and cost precluded extensive use. Tantalum with m.p 2950°C worked better on DC)

*Tungsten's vapor pressure as a function of temperature is the lowest among all conductive metals. This reduces bulb blackening.

*Tungsten is a selective emitter; its emissivity is visible spectrum is higher than in the infrared spectrum. This contributed to the efficacy at the temperature level.

*Low specific resistance, so it was suited for heavy current

*Required only 1.25 watt per candlepower (Carbon -3, Tantalum -2)

*Higher life expectancy - 1000 hour on either current (Carbon: 400-500, Tantalum: DC -600, AC-800)

*Unlick carbo, tungsten has a positive temperature coefficient, so not it was stable but also gave a whitish light.

*It was possible to repair a tungsten filament lamp by vigorously shaking it with current on. The broken ends of filament will come in contact and weld together.

Some refractory carbides and nitrides(e.g. TaC, HfC|ZrC, ZrN) have higher melting points than tungsten and enhanced spectral selectivity, however they are thermall unstable at higher temperatures and too brittle to be fabricated into a lamp filaments.

References

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*https://en.wikipedia.org/wiki/Incandescent_light_bulb

*Power and the Engineer, Volume 34, Hill Publishing Company, 1911

*Census of Electrical Industries: 1917: Central electric light and power stations with summary of the electrical industries,  United States. Bureau of the Census U.S. Government Printing Office, 1910

*Lamps and Lighting, M.A. Cayless, Routledge, 2012
