Lake Vostok

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https://en.wikipedia.org/wiki/Lake_Vostok

 

Lake Vostok
Radar satellite image of Lake Vostok
Coordinates	77°30′S 106°00′ECoordinates: 77°30′S 106°00′E
Lake type	Ancient lake, Subglacial rift lake
Basin countries	Antarctica
Max. length	250 km (160 mi)
Max. width	50 km (30 mi)
Surface area	12,500 km2 (4,830 sq mi)
Average depth	432 m (1,417 ft)
Max. depth	510 m (1,700 ft) to 900 m (3,000 ft)
Water volume	5,400 km3 (1,300 cu mi) ± 1,600 km3 (400 cu mi)
Residence time	13,300 yrs
Surface elevation	c. −500 m (−1,600 ft)
Islands	1
Settlements	Vostok Station

Lake Vostok (Russian: Озеро Восток, Ozero Vostok, lit. "Lake East") is the largest of Antarctica's almost 400 known subglacial lakes. Lake Vostok is located at the southern Pole of Cold, beneath Russia's Vostok Station under the surface of the central East Antarctic Ice Sheet, which is at 3,488 m (11,444 ft) above mean sea level. The surface of this fresh water lake is approximately 4,000 m (13,100 ft) under the surface of the ice, which places it at approximately 500 m (1,600 ft) below sea level.

Measuring 250 km (160 mi) long by 50 km (30 mi) wide at its widest point, it covers an area of 12,500 km² (4,830 sq mi) making it the 16th largest lake by surface area. With an average depth of 432 m (1,417 ft), it has an estimated volume of 5,400 km³ (1,300 cu mi), making it the 6th largest lake by volume.

The lake is divided into two deep basins by a ridge. The liquid water depth over the ridge is about 200 m (700 ft), compared to roughly 400 m (1,300 ft) deep in the northern basin and 800 m (2,600 ft) deep in the southern.

The lake is named after Vostok Station, which in turn is named after the Vostok (Восток), a sloop-of-war ship, which means "East" in Russian. The existence of a subglacial lake in the Vostok region was first suggested by Russian geographer Andrey Kapitsa based on seismic soundings made during the Soviet Antarctic Expeditions in 1959 and 1964 to measure the thickness of the ice sheet. The continued research by Russian and British scientists led by 1993 to the final confirmation of the existence of the lake by J. P. Ridley using ERS-1 laser altimetry.

The overlying ice provides a continuous paleoclimatic record of 400,000 years, although the lake water itself may have been isolated for 15 to 25 million years. On 5 February 2012, a team of Russian scientists completed the longest ever ice core of 3,768 m (12,400 ft) and pierced the ice shield to the surface of the lake.

The first core of freshly frozen lake ice was obtained on 10 January 2013 at a depth of 3,406 m (11,175 ft). However, as soon as the ice was pierced, water from the underlying lake gushed up the borehole, mixing it with the Freon and kerosene used to keep the borehole from freezing. A new borehole was drilled and an allegedly pristine water sample was obtained in January 2015. The Russian team plans to eventually lower a probe into the lake to collect water samples and sediments from the bottom. It is hypothesized that unusual forms of life could be found in the lake's liquid layer, a fossil water reserve. Because Lake Vostok may contain an environment sealed off below the ice for millions of years, the conditions could resemble those of ice-covered oceans hypothesized to exist on Jupiter's moon Europa, and Saturn's moon Enceladus.

Discovery

Location of Lake Vostok in East Antarctica

Russian scientist Peter Kropotkin first proposed the idea of fresh water under Antarctic ice sheets at the end of the 19th century. He theorized that the tremendous pressure exerted by the cumulative mass of thousands of vertical meters of ice could decrease the melting point at the lowest portions of the ice sheet to the point where the ice would become liquid water. Kropotkin's theory was further developed by Russian glaciologist I. A. Zotikov, who wrote his Ph.D. thesis on this subject in 1967.

Russian geographer Andrey Kapitsa used seismic soundings in the region of Vostok Station made during the Soviet Antarctic Expedition in 1959 and 1964 to measure the thickness of the ice sheet. Kapitsa was the first to suggest the existence of a subglacial lake in the region, and the subsequent research confirmed his hypothesis.

When British scientists in Antarctica performed airborne ice-penetrating radar surveys in the early 1970s, they detected unusual radar readings at the site which suggested the presence of a liquid freshwater lake below the ice. In 1991, Jeff Ridley, a remote sensing specialist with the Mullard Space Science Laboratory at University College London, directed the ERS-1 satellite to turn its high-frequency array toward the center of the Antarctic ice cap. The data from ERS-1 confirmed the findings from the 1973 British surveys, but these new data were not published in the Journal of Glaciology until 1993. Space-based radar revealed that this subglacial body of fresh water is one of the largest lakes in the world, and one of some 140 subglacial lakes in Antarctica. Russian and British scientists delineated the lake in 1996 by integrating a variety of data, including airborne ice-penetrating radar imaging observations and space-based radar altimetry. It has been confirmed that the lake contains large amounts of liquid water under the more than 3-kilometer-thick (1.9 mi) ice cap. The lake has at least 22 cavities of liquid water, averaging 10 kilometers (6 mi) each.

The station after which the lake is named commemorates the Vostok (Восток), the 900-ton sloop-of-war ship sailed by one of the discoverers of Antarctica, Russian explorer Admiral Fabian von Bellingshausen. Because the word Vostok means "East" in Russian, the names of the station and lake also reflect the fact that they are located in East Antarctica.

In 2005 an island was found in the central part of the lake. Then, in January 2006, the discovery of two nearby smaller lakes under the ice cap was published; they are named 90 Degrees East and Sovetskaya. It is suspected that these Antarctic subglacial lakes may be connected by a network of subglacial rivers. Centre for Polar Observation & Modelling glaciologists propose that many of the subglacial lakes of Antarctica are at least temporarily interconnected. Because of varying water pressure in individual lakes, large subsurface rivers may suddenly form and then force large amounts of water through the solid ice.

Geological history

Africa separated from Antarctica around 160 million years ago, followed by the Indian subcontinent, in the early Cretaceous (about 125 million years ago). About 66 million years ago, Antarctica (then connected to Australia) still had a tropical to subtropical climate, complete with marsupial fauna and an extensive temperate rainforest.

The Lake Vostok basin is a small (50-kilometer-wide (31 mi)) tectonic feature within the overall setting of a several-hundred-kilometer-wide continental collision zone between the Gamburtsev Mountain Range, a subglacial mountain range and the Dome C region. The lake water is cradled on a bed of sediments 70 meters (230 ft) thick, offering the possibility that they contain a unique record of the climate and life in Antarctica before the ice cap formed.

Traits

Vostok Station

 

The lake water is estimated to have been sealed off under the thick ice sheet about 15 million years ago. Initially, it was thought that the same water had made up the lake since the time of its formation, giving a residence time in the order of one million years. Later research by Robin Bell and Michael Studinger from the Lamont–Doherty Earth Observatory of Columbia University suggested that the water of the lake is continually freezing and being carried away by the motion of the Antarctic ice sheet, while being replaced by water melting from other parts of the ice sheet in these high pressure conditions. This resulted in an estimate that the entire volume of the lake is replaced every 13,300 years — its effective mean residence time.

The coldest naturally occurring temperature ever observed on Earth, −89 °C (−128 °F), was recorded at Vostok Station on 21 July 1983. The average water temperature is calculated to be around −3 °C (27 °F); it remains liquid below the normal freezing point because of high pressure from the weight of the ice above it. Geothermal heat from the Earth's interior may warm the bottom of the lake, while the ice sheet itself insulates the lake from cold temperatures on the surface. 

Lake Vostok is an oligotrophic extreme environment, one that is expected to be supersaturated with nitrogen and oxygen, measuring 2.5 litres (0.088 cu ft) of nitrogen and oxygen per 1 kg (2.2 lb) of water, that is 50 times higher than those typically found in ordinary freshwater lakes on Earth's surface. The sheer weight and pressure around 345 bars (5,000 psi) of the continental ice cap on top of Lake Vostok is estimated to contribute to the high gas concentration.

Besides dissolving in the water, oxygen and other gases are trapped in a type of structure called a clathrate. In clathrate structures, gases are enclosed in an icy cage and look like packed snow. These structures form at the high-pressure depths of Lake Vostok and would become unstable if brought to the surface.

In April 2005, German, Russian, and Japanese researchers found that the lake has tides. Depending on the position of the Sun and the Moon, the surface of the lake rises about 12 mm (0.47 in). The lake is under complete darkness, under 355 bar (5,150 psi) of pressure, and expected to be rich in oxygen, so there is speculation that any organisms inhabiting the lake could have evolved in a manner unique to this environment. There is a 1 microtesla magnetic anomaly on the east coast of the lake, spanning 105 by 75 km (65 by 47 mi). Researchers hypothesize that the anomaly may be caused by a thinning of the Earth's crust in that location.

Living Hydrogenophilus thermoluteolus micro-organisms have been found in Lake Vostok's deep ice core drillings; they are an extant surface-dwelling species. This suggests the presence of a deep biosphere utilizing a geothermal system of the bedrock encircling the subglacial lake. There is optimism that microbial life in the lake may be possible despite high pressure, constant cold, low nutrient input, potentially high oxygen concentration and an absence of sunlight. Jupiter's moon Europa and Saturn's moon Enceladus may also harbor lakes or oceans below a thick crust of ice. Any confirmation of life in Lake Vostok could strengthen the prospect for the presence of life on icy moons.

Research

Ice cores drilled at Vostok Station, which is seen in the background

Researchers working at Vostok Station produced one of the world's longest ice cores in 1998. A joint Russian, French, and United States team drilled and analyzed the core, which is 3,623 m (11,886 ft) long. Ice samples from cores drilled close to the top of the lake have been assessed to be as old as 420,000 years. The assumption is that the lake has been sealed from the surface since the ice sheet was formed 15 million years ago. Drilling of the core was deliberately halted roughly 100 m (300 ft) above the suspected boundary between the ice sheet and the liquid waters of the lake. This was to prevent contamination of the lake with the 60-ton column of Freon and kerosene used to prevent the borehole from collapsing and freezing over.

From this core, specifically from ice that is thought to have formed from lake water freezing onto the base of the ice sheet, extremophile microbes were found, suggesting that the lake water supports life. Scientists suggested that the lake could possess a unique habitat for ancient bacteria with an isolated microbial gene pool containing characteristics developed perhaps 500,000 years ago.

An artist's cross-section of Lake Vostok's drilling

 

In January 2011, the head of the Russian Antarctic Expedition, Valery Lukin, announced that his team had only 50 m (200 ft) of ice left to drill in order to reach the water. The researchers then switched to a new thermal drill head with a "clean" silicone oil fluid to drill the rest of the way. Instead of drilling all the way into the water, they said they would stop just above it when a sensor on the thermal drill detected free water. At that point, the drill was to be stopped and extracted from the bore hole. Removal of the drill would lower the pressure beneath it, drawing water into the hole to be left to freeze, creating a plug of ice in the bottom of the hole. Drilling stopped on 5 February 2011 at a depth of 3,720 m (12,200 ft) so that the research team could make it off the ice before the beginning of the Antarctic winter season. The drilling team left by aircraft on 6 February 2011.

By plan, the following summer, the team was to drill down again to take a sample of that ice and analyze it. The Russians resumed drilling into the lake in January 2012 and reached the upper surface of the water on 6 February 2012. The researchers allowed the rushing lake water to freeze within the bore hole and months later, they collected ice core samples of this newly formed ice and sent to the Laboratory for Glaciology and Environmental Geophysics in Grenoble, France, for analysis.

Biology results

United Kingdom and United States

Scientists first reported evidence of microbes in the accretion ice in 1999. Since then, a different team led by Scott O. Rogers has been identifying a variety of bacteria and fungi from accretion ice (not from the subglacial water layer) collected during U.S. drilling projects in the 1990s. According to him, this indicates that the lake below the ice is not sterile but contains a unique ecosystem. Then Scott Rogers published in July 2013 that his team performed nucleic acid (DNA and RNA) sequencing and the results allowed deduction of the metabolic pathways represented in the accretion ice and, by extension, in the lake. The team found 3,507 unique gene sequences, and approximately 94% of the sequences were from bacteria and 6% were from Eukarya. Taxonomic classifications (to genus and/or species) or identification were possible for 1,623 of the sequences. In general, the taxa were similar to organisms previously described from lakes, brackish water, marine environments, soil, glaciers, ice, lake sediments, deep-sea sediments, deep-sea thermal vents, animals and plants. Sequences from aerobic, anaerobic, psychrophilic, thermophilic, halophilic, alkaliphilic, acidophilic, desiccation-resistant, autotrophic, and heterotrophic organisms were present, including a number from multicellular eukaryotes.

However, microbiologist David Pearce of the University of Northumbria in Newcastle, UK, stated that the DNA could simply be contamination from the drilling process, and not representative of Lake Vostok itself. The old ice cores were drilled in the 1990s to look for evidence of past climates buried in the ice, rather than for life, so the drilling equipment was not sterilized. Also Sergey Bulat, a Lake Vostok expert at the Petersburg Nuclear Physics Institute in Gatchina, Russia, doubts that any of the cells or DNA fragments in the samples would belong to organisms that might actually exist in the lake. He says that it is very probable that the samples are heavily contaminated with tissue and microbes from the outside world.

Russia and France

Russian and French scientists have been carrying out molecular DNA studies of the water from Lake Vostok that was frozen in the borehole, by constructing numerous DNA libraries, which are collections of fragments of DNA that allow scientists to identify which species of bacteria may belong to. Samples taken from the lake so far contain about one part of kerosene per 1000 of water, and they are contaminated with bacteria previously present in the drill bit and the kerosene drilling fluid. So far, the scientists have been able to identify 255 contaminant species, but also have found an unknown bacterium when they initially drilled down to the lake's surface in 2012, with no matches in any international databases, and they hope it may be a unique inhabitant of Lake Vostok. However, Vladimar Korolev, the laboratory head of the study at the same institution, said that the bacteria could in principle be a contaminant that uses kerosene—the antifreeze used during drilling—as an energy source.

Critics from the scientific community state that no valuable information can be obtained until they can test clean samples of lake water, uncontaminated by drilling fluid. Regardless of the contamination issues, in May 2013 the drilling facility at the Russian Vostok Antarctic station was declared a historic monument as "the result of the recognition of the achievements of the Russian research of Antarctica by the international scientific community, and of the unique operations on opening the subglacial Lake Vostok performed by Russian scientists on February 5, 2012."

In January 2015, the Russian press stated that Russian scientists have made a new "clean" borehole into Lake Vostok using a special 50-kilogram probe that collected about 1 liter of water not adulterated by the antifreezing fluid. It was predicted that the water would rise 30–40 m in the bottom part of the borehole, but in fact the water rose from the lake to a height of more than 500 m. In October of that same year, the work was suspended for that southern summer because of insufficient funding by the federal Russian government.

Contamination due to drilling fluids

The drilling project has been opposed by some environmental groups and scientists who have argued that hot-water drilling would have a more limited environmental impact. The main concern is that the lake could become contaminated with the antifreeze that the Russians used to keep the bore hole from refreezing. Scientists of the United States National Research Council have taken the position that it should be assumed that microbial life exists in Lake Vostok and that after such a long isolation, any life forms in the lake require strict protection from contamination.

The original drilling technique employed by the Russians involved the use of Freon and kerosene to lubricate the borehole and prevent it from collapsing and freezing over; 60 short tons (54 t) of these chemicals have been used thus far on the ice above Lake Vostok. Other countries, particularly the United States and Britain, have failed to persuade the Russians not to pierce to the lake until cleaner technologies such as hot-water drilling are available. Though the Russians claim to have improved their operations, they continue to use the same borehole, which has already been contaminated with kerosene. According to the head of Russian Antarctic Expeditions, Valery Lukin, new equipment was developed by researchers at the Petersburg Nuclear Physics Institute that would ensure the lake remains uncontaminated upon intrusion. Lukin has repeatedly reassured other signatory nations to the Antarctic Treaty System that the drilling will not affect the lake, arguing that on breakthrough, water will rush up the borehole, freeze, and seal the other fluids out.

Some environmentalist groups remain unconvinced by these arguments. The Antarctic and Southern Ocean Coalition has argued that this manner of drilling is a profoundly misguided step which endangers Lake Vostok and other subglacial lakes in Antarctica (which some scientists are convinced are inter-linked with Lake Vostok). The coalition has asserted that "it would be far preferable to join with other countries to penetrate a smaller and more isolated lake before re-examining whether penetration of Lake Vostok is environmentally defensible. If we are wise, the Lake will be allowed to reveal its secrets in due course."

Lukin claims that hot-water drilling is much more dangerous for the microbiotic fauna, as it would boil the living species, plus disturb the entire structure of water layers of the lake. Additionally, hot-water drilling would have required more power than the Russian expedition could have generated at their remote camp. However, the water samples obtained by the Russian team were heavily contaminated with drilling fluid, so they reported in May 2017 that it was impossible at this time to obtain reliable data on the real chemical and biological composition of the lake water.

Obduction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Obduction

Obduction is the overthrusting of continental crust by oceanic crust or mantle rocks at a convergent plate boundary, such as closing of an ocean or a mountain building episode. This process is uncommon because the denser oceanic lithosphere usually subducts underneath the less dense continental plate.

Obduction occurs where a fragment of continental crust is caught in a subduction zone with resulting overthrusting of oceanic mafic and ultramafic rocks from the mantle onto the continental crust. Obduction often occurs where a small tectonic plate is caught between two larger plates, with the crust (both island arc and oceanic) welding onto an adjacent continent as a new terrane. When two continental plates collide, obduction of the oceanic crust between them is often a part of the resulting orogeny.

Most obductions appear to have initiated at back-arc basins above the subduction zones during the closing of an ocean or an orogeny.

Characteristic rocks

The characteristic rocks of obducted oceanic lithosphere are the ophiolites. Ophiolites are an assemblage of oceanic lithosphere rocks that have been emplaced onto a continent. This assemblage consists of deep-marine sedimentary rock (chert, limestone, clastic sediments), volcanic rocks (pillow lavas, glass, ash, sheeted dykes and gabbros) and peridotite (mantle rock).

Types of obductions

Upwedging in subduction zones

This process is operative beneath and behind the inner walls of oceanic trenches (subduction zone) where slices of oceanic crust and mantle are ripped from the upper part of the descending plate and wedged and packed in high pressure assemblages against the leading edge of the other plate.

Weakening and cracking of oceanic crust and upper mantle is likely to occur in the tensional regime. This results in the incorporation of ophiolite slabs into the overriding plate.

Progressive packing of ophiolite slices and arc fragments against the leading edge of a continent may continue over a long period of time and lead to a form of continental accretion.

Compressional telescoping onto Atlantic-type continental margins

The simplest form of this type of obduction may follow from the development of a subduction zone near the continental margin. Above and behind the subduction zone, a welt of oceanic crust and mantle rides up over the descending plate. The ocean, intervening between the continental margin and the subduction zone is progressively swallowed until the continental margin arrives at the subduction zone and a giant wedge or slice (nappe) of oceanic crust and mantle is pushed across the continental margin. Because the buoyancy of the relatively light continental crust is likely to prohibit its extensive subduction, a flip in subduction polarity will occur yielding an ophiolite sheet lying above a descending plate.

If however, a large tract of ocean intervenes between the continental margin the subduction zone, a fully developed arc and back arc basin may eventually arrive and collide with the continental margin. Further convergence may lead to overthrusting of the volcanic arc assemblage and may be followed by flipping the subduction polarity.

According to the rock assemblage as well as the complexly deformed ophiolite basement and arc intrusions, the Coastal Complex of western Newfoundland may well have been formed by this mechanism.

Gravity sliding onto Atlantic-type continental margins

This concept involves the progressive uplift of an actively spreading oceanic ridge, the detachment of slices from the upper part of the lithosphere and the subsequent gravity sliding of these slices onto the continental margin as ophiolites. This concept was advocated by Reinhardt  for the emplacement of the Semail Ophiolite complex in Oman and argued by Church  and Church and Stevens  for the emplacement of the Bay of Islands sheet in western Newfoundland. This concept has subsequently been replaced by hypotheses that advocate subduction of the continental margin beneath oceanic lithosphere. 

Transformation of a spreading ridge to a subduction zone

Many ophiolite complexes were emplaced as thin hot obducted sheets of oceanic lithosphere shortly after their generation by plate accretion. The change from a spreading plate boundary to a subduction plate boundary may result from rapid rearrangement of relative plate motion. A transform fault may also become a subduction zone, with the side with the higher, hotter, thinner lithosphere riding over the lower, colder lithosphere. This mechanism would lead to obduction of ophiolite complex if it occurred near a continental margin.

Interference of a spreading ridge and a subduction zone

In the situation where a spreading ridge approaches a subduction zone, the ridge collides with the subduction zone, at which time there will develop a complex interaction of subduction-related tectonic sedimentary, and spreading-related tectonic igneous activity. The left-over ridge may either subduct or ride upward across the trench onto arc trench gap and arc terranes as a hot ophiolite slice. These two mechanisms are shown in figure 2 B and C. Two examples of this interaction of a ridge colliding into a trench are well documented. The first one is the progressive diminution of the Farallon plate off California. Ophiolite obduction by the above proposed mechanism would not be expected as the two plates share a dextral transform boundary. However, the major collision of the Kula/Pacific plate with the Alaskan/Aleutian resulted in the initiation of subduction of the Pacific plate beneath Alaska, with no sign of either obduction or indeed any major manifestation of a ridge being “swallowed”.

Obduction from rear-arc basin

Dewey and Bird suggested that a common form of ophiolite obduction is related to the closure of rear-arc marginal basins and that, during such closure by subduction, slices of oceanic crust and mantle may be expelled onto adjacent continental forelands and emplaced as ophiolite sheets. In the high heat-flow region of a volcanic arc and rear-arc basin the lithosphere is particularly thin. This thin lithosphere may preferentially fail along gently dipping thrust surface if a compressional stress is applied to the region. Under these circumstances a thin sheet of lithosphere may become detached and begin to ride over adjacent lithosphere to finally become emplaced as a thin ophiolite sheet on the adjacent continental foreland. This mechanism is a form of plate convergence where a thin, hot layer of oceanic lithosphere is obducted over cooler and thicker lithosphere.

Obduction during continental collision

As an ocean is progressively trapped in between two colliding continental lithospheres, the rising wedges of oceanic crust and mantle rise are caught in the jaws of the continent/continent vise and detach and begin to move up the advancing continental rise. Continued convergence may lead to the overthrusting of the arc-trench gap and eventually overthrusting of the metamorphic plutonic and volcanic rocks of the volcanic arc.

Following total subduction of an oceanic tract, continuing convergence may lead to a further sequence of intra-continental mechanisms of crustal shortening. This mechanism is thought to be responsible for the various ocean basins of the Mediterranean region. The Alpine belt is believed to register a complex history of plate interactions during the general convergence of the Eurasian plate and African plates.

Examples

There are many examples of oceanic crustal rocks and deeper mantle rocks that have been obducted and exposed at the surface worldwide. New Caledonia is one example of recent obduction. The Klamath Mountains of northern California contain several obducted oceanic slabs. Obducted fragments also are found in Oman, the Troodos Mountains of Cyprus, Newfoundland, New Zealand, the Alps of Europe, the Shetland islands of Unst and Fetlar, and the Appalachians of eastern North America.

Subduction

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Subduction

 

Diagram of the geological process of subduction

Subduction is a geological process that takes place at convergent boundaries of tectonic plates where one plate moves under another and is forced to sink due to high gravitational potential energy into the mantle. Regions where this process occurs are known as subduction zones. Rates of subduction are typically measured in centimeters per year, with the average rate of convergence being approximately two to eight centimeters per year along most plate boundaries.

Plates include both oceanic crust and continental crust. Stable subduction zones involve the oceanic lithosphere of one plate sliding beneath the continental or oceanic lithosphere of another plate due to the higher density of the oceanic lithosphere. This means that the subducted lithosphere is always oceanic while the overriding lithosphere may or may not be oceanic. Subduction zones are sites that usually have a high rate of volcanism and earthquakes. Furthermore, subduction zones develop belts of deformation and metamorphism in the subducting crust, whose exhumation is part of orogeny and also leads to mountain building in addition to collisional thickening.

General description

Subduction zones are sites of gravitational sinking of Earth's lithosphere (the crust plus the top non-convecting portion of the upper mantle). Subduction zones exist at convergent plate boundaries where one plate of oceanic lithosphere converges with another plate. The descending slab, the subducting plate, is over-ridden by the leading edge of the other plate. The slab sinks at an angle of approximately twenty-five to forty-five degrees to Earth's surface. This sinking is driven by the temperature difference between the subducting oceanic lithosphere and the surrounding mantle asthenosphere, as the colder oceanic lithosphere has, on average, a greater density. At a depth of greater than 60 kilometers, the basalt of the oceanic crust is converted to a metamorphic rock called eclogite. At that point, the density of the oceanic crust increases and provides additional negative buoyancy (downwards force). It is at subduction zones that Earth's lithosphere, oceanic crust and continental crust, sedimentary layers and some trapped water are recycled into the deep mantle.

Earth is so far the only planet where subduction is known to occur. Subduction is the driving force behind plate tectonics, and without it, plate tectonics could not occur.

Oceanic subduction zones dive down into the mantle beneath 55,000 km (34,000 mi) of convergent plate margins (Lallemand, 1999), almost equal to the cumulative 60,000 km (37,000 mi) of mid-ocean ridges. Subduction zones burrow deeply but are imperfectly camouflaged, and geophysics and geochemistry can be used to study them. Not surprisingly, the shallowest portions of subduction zones are known best. Subduction zones are strongly asymmetric for the first several hundred kilometers of their descent. They start to go down at oceanic trenches. Their descents are marked by inclined zones of earthquakes that dip away from the trench beneath the volcanoes and extend down to the 660-kilometer discontinuity. Subduction zones are defined by the inclined array of earthquakes known as the Wadati–Benioff zone after the two scientists who first identified this distinctive aspect. Subduction zone earthquakes occur at greater depths (up to 600 km (370 mi)) than elsewhere on Earth (typically less than 20 km (12 mi) depth); such deep earthquakes may be driven by deep phase transformations, thermal runaway, or dehydration embrittlement.

The subducting basalt and sediment are normally rich in hydrous minerals and clays. Additionally, large quantities of water are introduced into cracks and fractures created as the subducting slab bends downward. During the transition from basalt to eclogite, these hydrous materials break down, producing copious quantities of water, which at such great pressure and temperature exists as a supercritical fluid. The supercritical water, which is hot and more buoyant than the surrounding rock, rises into the overlying mantle where it lowers the pressure in (and thus the melting temperature of) the mantle rock to the point of actual melting, generating magma. The magmas, in turn, rise (and become labeled diapirs) because they are less dense than the rocks of the mantle. The mantle-derived magmas (which are basaltic in composition) can continue to rise, ultimately to Earth's surface, resulting in a volcanic eruption. The chemical composition of the erupting lava depends upon the degree to which the mantle-derived basalt interacts with (melts) Earth's crust and/or undergoes fractional crystallization. 

Above subduction zones, volcanoes exist in long chains called volcanic arcs. Volcanoes that exist along arcs tend to produce dangerous eruptions because they are rich in water (from the slab and sediments) and tend to be extremely explosive. Krakatoa, Nevado del Ruiz, and Mount Vesuvius are all examples of arc volcanoes. Arcs are also known to be associated with precious metals such as gold, silver and copper believed to be carried by water and concentrated in and around their host volcanoes in rock called "ore". 

Theory on origin

Initiation

Although the process of subduction as it occurs today is fairly well understood, its origin remains a matter of discussion and continuing study. Subduction initiation can occur spontaneously if denser oceanic lithosphere is able to founder and sink beneath adjacent oceanic or continental lithosphere; alternatively, existing plate motions can induce new subduction zones by forcing oceanic lithosphere to rupture and sink into the asthenosphere. Both models can eventually yield self-sustaining subduction zones, as oceanic crust is metamorphosed at great depth and becomes denser than the surrounding mantle rocks. Results from numerical models generally favor induced subduction initiation for most modern subduction zones, which is supported by geologic studies, but other analogue modeling shows the possibility of spontaneous subduction from inherent density differences between two plates at passive margins, and observations from the Izu-Bonin-Mariana subduction system are compatible with spontaneous subduction nucleation. Furthermore, subduction is likely to have spontaneously initiated at some point in Earth's history, as induced subduction nucleation requires existing plate motions, though an unorthodox proposal by A. Yin suggests that meteorite impacts may have contributed to subduction initiation on early Earth.

Geophysicist Don L. Anderson has hypothesized that plate tectonics could not happen without the calcium carbonate laid down by bioforms at the edges of subduction zones. The massive weight of these sediments could be softening the underlying rocks, making them pliable enough to plunge.

Modern-style subduction

Modern-style subduction is characterized by low geothermal gradients and the associated formation of high-pressure low temperature rocks such as eclogite and blueschist. Likewise, rock assemblages called ophiolites, associated to modern-style subduction, also indicate such conditions. Eclogite xenoliths found in the North China Craton provide evidence that modern-style subduction occurred at least as early as 1.8 Ga ago in the Paleoproterozoic Era. Nevertheless, the eclogite itself was produced by oceanic subdcution during the assembly of supercontinents at about 1.9–2.0 Ga.

Blueschist is a rock typical for present-day subduction settings. Absence of blueschist older than Neoproterozoic reflect more magnesium-rich compositions of Earth's oceanic crust during that period. These more magnesium-rich rocks metamorphose into greenschist at conditions when modern oceanic crust rocks metamorphose into blueschist. The ancient magnesium-rich rocks means that Earth's mantle was once hotter, but not that subduction conditions were hotter. Previously, lack of pre-Neoproterozoic blueschist was thought to indicate a different type of subduction. Both lines of evidence refutes previous conceptions of modern-style subduction having been initiated in the Neoproterozoic Era 1.0 Ga ago.

Effects

Volcanic activity

Oceanic plates are subducted creating oceanic trenches.

 

Volcanoes that occur above subduction zones, such as Mount St. Helens, Mount Etna and Mount Fuji, lie at approximately one hundred kilometers from the trench in arcuate chains, hence the term volcanic arc. Two kinds of arcs are generally observed on Earth: island arcs that form on oceanic lithosphere (for example, the Mariana and the Tonga island arcs), and continental arcs such as the Cascade Volcanic Arc, that form along the coast of continents. Island arcs are produced by the subduction of oceanic lithosphere beneath another oceanic lithosphere (ocean-ocean subduction) while continental arcs formed during subduction of oceanic lithosphere beneath a continental lithosphere (ocean-continent subduction). An example of a volcanic arc having both island and continental arc sections is found behind the Aleutian Trench subduction zone in Alaska.

The arc magmatism occurs one hundred to two hundred kilometers from the trench and approximately one hundred kilometers above the subducting slab. This depth of arc magma generation is the consequence of the interaction between hydrous fluids, released from the subducting slab, and the arc mantle wedge that is hot enough to melt with the addition of water. It has also been suggested that the mixing of fluids from a subducted tectonic plate and melted sediment is already occurring at the top of the slab before any mixing with the mantle takes place.

Arcs produce about 25% of the total volume of magma produced each year on Earth (approximately thirty to thirty-five cubic kilometers), much less than the volume produced at mid-ocean ridges, and they contribute to the formation of new continental crust. Arc volcanism has the greatest impact on humans because many arc volcanoes lie above sea level and erupt violently. Aerosols injected into the stratosphere during violent eruptions can cause rapid cooling of Earth's climate and affect air travel. 

Earthquakes and tsunamis

The strains caused by plate convergence in subduction zones cause at least three types of earthquakes. Earthquakes mainly propagate in the cold subducting slab and define the Wadati–Benioff zone. Seismicity shows that the slab can be tracked down to the upper mantle/lower mantle boundary (approximately six hundred kilometer depth).

Nine of the ten largest earthquakes of the last 100 years were subduction zone events, which included the 1960 Great Chilean earthquake, which, at M 9.5, was the largest earthquake ever recorded; the 2004 Indian Ocean earthquake and tsunami; and the 2011 Tōhoku earthquake and tsunami. The subduction of cold oceanic crust into the mantle depresses the local geothermal gradient and causes a larger portion of Earth to deform in a more brittle fashion than it would in a normal geothermal gradient setting. Because earthquakes can occur only when a rock is deforming in a brittle fashion, subduction zones can cause large earthquakes. If such a quake causes rapid deformation of the sea floor, there is potential for tsunamis, such as the earthquake caused by subduction of the Indo-Australian Plate under the Euro-Asian Plate on December 26, 2004 that devastated the areas around the Indian Ocean. Small tremors which cause small, nondamaging tsunamis, also occur frequently.

A study published in 2016 suggested a new parameter to determine a subduction zone's ability to generate mega-earthquakes. By examining subduction zone geometry and comparing the degree of curvature of the subducting plates in great historical earthquakes such as the 2004 Sumatra-Andaman and the 2011 Tōhoku earthquake, it was determined that the magnitude of earthquakes in subduction zones is inversely proportional to the degree of the fault's curvature, meaning that "the flatter the contact between the two plates, the more likely it is that mega-earthquakes will occur."

Outer rise earthquakes occur when normal faults oceanward of the subduction zone are activated by flexure of the plate as it bends into the subduction zone. The 2009 Samoa earthquake is an example of this type of event. Displacement of the sea floor caused by this event generated a six-meter tsunami in nearby Samoa.

Anomalously deep events are a characteristic of subduction zones, which produce the deepest quakes on the planet. Earthquakes are generally restricted to the shallow, brittle parts of the crust, generally at depths of less than twenty kilometers. However, in subduction zones, quakes occur at depths as great as 700 km (430 mi). These quakes define inclined zones of seismicity known as Wadati–Benioff zones which trace the descending lithosphere. 

Seismic tomography has helped detect subducted lithosphere, slabs, deep in the mantle where there are no earthquakes. About one hundred slabs have been described in terms of depth and their timing and location of subduction. The great seismic discontinuities in the mantle, at 410 km (250 mi) depth and 670 km (420 mi), are disrupted by the descent of cold slabs in deep subduction zones. Some subducted slabs seem to have difficulty penetrating the major discontinuity that marks the boundary between upper mantle and lower mantle at a depth of about 670 kilometers. Other subducted oceanic plates have sunk all the way to the core-mantle boundary at 2890 km depth. Generally slabs decelerate during their descent into the mantle, from typically several cm/yr (up to ~10 cm/yr in some cases) at the subduction zone and in the uppermost mantle, to ~1 cm/yr in the lower mantle. This leads to either folding or stacking of slabs at those depths, visible as thickened slabs in Seismic tomography. Below ~1700 km, there might be a limited acceleration of slabs due to lower viscosity as a result of inferred mineral phase changes until they approach and finally stall at the core-mantle boundary. Here the slabs are heated up by the ambient heat and are not detected anymore ~300 Myr after subduction.

Orogeny

Orogeny is the process of mountain building. Subducting plates can lead to orogeny by bringing oceanic islands, oceanic plateaus, and sediments to convergent margins. The material often does not subduct with the rest of the plate but instead is accreted (scraped off) to the continent, resulting in exotic terranes. The collision of this oceanic material causes crustal thickening and mountain-building. The accreted material is often referred to as an accretionary wedge, or prism. These accretionary wedges can be identified by ophiolites (uplifted ocean crust consisting of sediments, pillow basalts, sheeted dykes, gabbro, and peridotite).

Subduction may also cause orogeny without bringing in oceanic material that collides with the overriding continent. When the subducting plate subducts at a shallow angle underneath a continent (something called "flat-slab subduction"), the subducting plate may have enough traction on the bottom of the continental plate to cause the upper plate to contract leading to folding, faulting, crustal thickening and mountain building. Flat-slab subduction causes mountain building and volcanism moving into the continent, away from the trench, and has been described in North America (i.e. Laramide orogeny), South America and East Asia.

The processes described above allow subduction to continue while mountain building happens progressively, which is in contrast to continent-continent collision orogeny, which often leads to the termination of subduction.

Subduction angle

Subduction typically occurs at a moderately steep angle right at the point of the convergent plate boundary. However, anomalous shallower angles of subduction are known to exist as well some that are extremely steep.

• Flat-slab subduction (subducting angle less than 30°) occurs when subducting lithosphere, called a slab, subducts nearly horizontally. The relatively flat slab can extend for hundreds of kilometers. That is abnormal, as the dense slab typically sinks at a much steeper angle directly at the subduction zone. Because subduction of slabs to depth is necessary to drive subduction zone volcanism (through the destabilization and dewatering of minerals and the resultant flux melting of the mantle wedge), flat-slab subduction can be invoked to explain volcanic gaps. Flat-slab subduction is ongoing beneath part of the Andes causing segmentation of the Andean Volcanic Belt into four zones. The flat-slab subduction in northern Peru and the Norte Chico region of Chile is believed to be the result of the subduction of two buoyant aseismic ridges, the Nazca Ridge and the Juan Fernández Ridge, respectively. Around Taitao Peninsula flat-slab subduction is attributed to the subduction of the Chile Rise, a spreading ridge. The Laramide Orogeny in the Rocky Mountains of United States is attributed to flat-slab subduction. Then, a broad volcanic gap appeared at the southwestern margin of North America, and deformation occurred much farther inland; it was during this time that the basement-cored mountain ranges of Colorado, Utah, Wyoming, South Dakota, and New Mexico came into being. The most massive subduction zone earthquakes, so-called "megaquakes", have been found to occur in flat-slab subduction zones.
• Steep-angle subduction (subducting angle greater than 70°) occurs in subduction zones where Earth's oceanic crust and lithosphere are old and thick and have, therefore, lost buoyancy. The steepest dipping subduction zone lies in the Mariana Trench, which is also where the oceanic crust, of Jurassic age, is the oldest on Earth exempting ophiolites. Steep-angle subduction is, in contrast to flat-slab subduction, associated with back-arc extension of crust making volcanic arcs and fragments of continental crust wander away from continents over geological times leaving behind a marginal sea.

Importance

Subduction zones are important for several reasons:

1. Subduction Zone Physics: Sinking of the oceanic lithosphere (sediments, crust, mantle), by contrast of density between the cold and old lithosphere and the hot asthenospheric mantle wedge, is the strongest force (but not the only one) needed to drive plate motion and is the dominant mode of mantle convection.
2. Subduction Zone Chemistry: The subducted sediments and crust dehydrate and release water-rich (aqueous) fluids into the overlying mantle, causing mantle melting and fractionation of elements between surface and deep mantle reservoirs, producing island arcs and continental crust. Hot fluids in subduction zones also alter the mineral compositions of the subducting sediments and potentially the habitability of the sediments for microorganisms.
3. Subduction zones drag down subducted oceanic sediments, oceanic crust, and mantle lithosphere that interact with the hot asthenospheric mantle from the over-riding plate to produce calc-alkaline series melts, ore deposits, and continental crust.
4. Subduction zones pose significant threats to lives, property, economic vitality, cultural and natural resources, and quality of life. The tremendous magnitudes of earthquakes or volcanic eruptions can also have knock-on effects with global impact.

Subduction zones have also been considered as possible disposal sites for nuclear waste in which the action of subduction itself would carry the material into the planetary mantle, safely away from any possible influence on humanity or the surface environment. However, that method of disposal is currently banned by international agreement. Furthermore, plate subduction zones are associated with very large megathrust earthquakes, making the effects on using any specific site for disposal unpredictable and possibly adverse to the safety of longterm disposal.

Object theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Object_theory

Object theory is a theory in philosophy and mathematical logic concerning objects and the statements that can be made about objects.

In some cases "objects" can be concretely thought of as symbols and strings of symbols, here illustrated by a string of four symbols " ←←↑↓←→←↓" as composed from the 4-symbol alphabet { ←, ↑, →, ↓ } . When they are "known only through the relationships of the system [in which they appear], the system is [said to be] abstract ... what the objects are, in any respect other than how they fit into the structure, is left unspecified." (Kleene 1952:25) A further specification of the objects results in a model or representation of the abstract system, "i.e. a system of objects which satisfy the relationships of the abstract system and have some further status as well" (ibid).

A system, in its general sense, is a collection of objects O = {o₁, o₂, ... o_n, ... } and (a specification of) the relationship r or relationships r₁, r₂, ... r_n between the objects:

Example: Given a simple system = { { ←, ↑, →, ↓ }, ∫ } for a very simple relationship between the objects as signified by the symbol ∫ :

∫→ => ↑, ∫↑ => ←, ∫← => ↓, ∫↓ => →

A model of this system would occur when we assign, for example the familiar natural numbers { 0, 1, 2, 3 }, to the symbols { ←, ↑, →, ↓ }, i.e. in this manner: → = 0, ↑ = 1, ← = 2, ↓ = 3 . Here, the symbol ∫ indicates the "successor function" (often written as an apostrophe ' to distinguish it from +) operating on a collection of only 4 objects, thus 0' = 1, 1' = 2, 2' = 3, 3' = 0.

Or, we might specify that ∫ represents 90-degree counter-clockwise rotations of a simple object → .

 

The genetic versus axiomatic method

The following is an example of the genetic or constructive method of making objects in a system, the other being the axiomatic or postulational method. Kleene states that a genetic method is intended to "generate" all the objects of the system and thereby "determine the abstract structure of the system completely" and uniquely (and thus define the system categorically). If axioms rather than a genetic method is used, such axiom-sets are said to be categorical.

Unlike the ∫ example above, the following creates an unbounded number of objects. The fact that O is a set, and □ is an element of O, and ■ is an operation, must be specified at the outset; this is being done in the language of the metatheory (see below):

Given the system ( O, □, ■ ): O = { □, ■□, ■■□, ■■■□, ■■■■□, ■■■■■□, ..., ■ⁿ□, etc. }

 

Abbreviations

The object ■ⁿ□ demonstrates the use of "abbreviation", a way to simplify the denoting of objects, and consequently discussions about them, once they have been created "officially". Done correctly the definition would proceed as follows:

■□ ≡ ■¹□, ■■□ ≡ ■²□, ■■■□ ≡ ■³□, etc, where the notions of ≡ ("defined as") and "number" are presupposed to be understood intuitively in the metatheory.

Kurt Gödel 1931 virtually constructed the entire proof of his incompleteness theorems (actually he proved Theorem IV and sketched a proof of Theorem XI) by use of this tactic, proceeding from his axioms using substitution, concatenation and deduction of modus ponens to produce a collection of 45 "definitions" (derivations or theorems more accurately) from the axioms. 

A more familiar tactic is perhaps the design of subroutines that are given names, e.g. in Excel the subroutine " =INT(A1)" that returns to the cell where it is typed (e.g. cell B1) the integer it finds in cell A1.

Models

A model of the above example is a left-ended Post–Turing machine tape with its fixed "head" located on the left-end square; the system's relation is equivalent to: "To the left end, tack on a new square □, right-shift the tape, then print ■ on the new square". Another model is the natural numbers as created by the "successor" function. Because the objects in the two systems e.g. ( □, ■□, ■■□, ■■■□ ... ) and (0, 0′, 0′′, 0′′′, ...) can be put into a 1-1 correspondence, the systems are said to be (simply) isomorphic (meaning "same shape"). Yet another isomorphic model is the little sequence of instructions for a counter machine e.g. "Do the following in sequence: (1) Dig a hole. (2) Into the hole, throw a pebble. (3) Go to step 2."

As long as their objects can be placed in one-to-one correspondence ("while preserving the relationships") models can be considered "equivalent" no matter how their objects are generated (e.g. genetically or axiomatically):

"Any two simply isomorphic systems constitute representations [models] of the same abstract system, which is obtained by abstracting from either of them, i.e. by leaving out of account all relationships and properties except the ones to be considered for the abstract system." (Kleene 1935:25)

 

Tacit assumptions, tacit knowledge

An alert reader may have noticed that writing symbols □, ■□, ■■□, ■■■□, etc. by concatenating a marked square, i.e. ■, to an existing string is different from writing the completed symbols one after another on a Turing-machine tape. Another entirely possible scenario would be to generate the symbol-strings one after another on different sections of tape e.g. after three symbols: ■■■□■■□■□□. The proof that these two possibilities are different is easy: they require different "programs". But in a sense both versions create the same objects; in the second case the objects are preserved on the tape. In the same way, if a person were to write 0, then erase it, write 1 in the same place, then erase it, write 2, erase it, ad infinitum, the person is generating the same objects as if they were writing down 0 1 2 3 ... writing one symbol after another to the right on the paper. 

Once the step has been taken to write down the symbols 3 2 1 0 one after another on a piece of paper (writing the new symbol on the left this time), or writing ∫∫∫※∫∫※∫※※ in a similar manner, then putting them in 1-1 correspondence with the Turing-tape symbols seems obvious. Digging holes one after the other, starting with a hole at "the origin", then a hole to its left with one pebble in it, then a hole to its left with two pebbles in it, ad infinitum, raises practical questions, but in the abstract it too can be seen to be conducive to the same 1-1 correspondence. 

However, nothing in particular in the definition of genetic versus axiomatic methods clears this up—these are issues to be discussed in the metatheory. The mathematician or scientist is to be held responsible for sloppy specifications. Breger cautions that axiomatic methods are susceptible to tacit knowledge, in particular, the sort that involves "know-how of a human being" (Breger 2000:227).

A formal system

In general, in mathematics a formal system or "formal theory" consists of "objects" in a structure:

• The symbols to be concatenated (adjoined),
• The formation-rules (completely specified, i.e. formal rules of syntax) that dictate how the symbols and the assemblies of symbols are to be formed into assemblies (e.g. sequences) of symbols (called terms, formulas, sentences, propositions, theorems, etc.) so that they are in "well-formed" patterns (e.g. can a symbol be concatenated at its left end only, at its right end only, or both ends simultaneously? Can a collection of symbols be substituted for (put in place of) one or more symbols that may appear anywhere in the target symbol-string?),
• Well-formed "propositions" (called "theorems" or assertions or sentences) assembled per the formation rules,
• A few axioms that are stated up front and may include "undefinable notions" (examples: "set", "element", "belonging" in set theory; "0" and " ' " (successor) in number theory),
• At least one rule of deductive inference (e.g. modus ponens) that allow one to pass from one or more of the axioms and/or propositions to another proposition.

Informal theory, object theory, and metatheory

A metatheory exists outside the formalized object theory—the meaningless symbols and relations and (well-formed-) strings of symbols. The metatheory comments on (describes, interprets, illustrates) these meaningless objects using "intuitive" notions and "ordinary language". Like the object theory, the metatheory should be disciplined, perhaps even quasi-formal itself, but in general the interpretations of objects and rules are intuitive rather than formal. Kleene requires that the methods of a metatheory (at least for the purposes of metamathematics) be finite, conceivable, and performable; these methods cannot appeal to the completed infinite. "Proofs of existence shall give, at least implicitly, a method for constructing the object which is being proved to exist."

Kleene summarizes this as follows: "In the full picture there will be three separate and distinct "theories""

"(a) the informal theory of which the formal system constitutes a formalization

"(b) the formal system or object theory, and

"(c) the metatheory, in which the formal system is described and studied" (p. 65)

He goes on to say that object theory (b) is not a "theory" in the conventional sense, but rather is "a system of symbols and of objects built from symbols (described from (c))". 

Expansion of the notion of formal system

Well-formed objects

If a collection of objects (symbols and symbol-sequences) is to be considered "well-formed", an algorithm must exist to determine, by halting with a "yes" or "no" answer, whether or not the object is well-formed (in mathematics a wff abbreviates well-formed formula). This algorithm, in the extreme, might require (or be) a Turing machine or Turing-equivalent machine that "parses" the symbol-string as presented as "data" on its tape; before a universal Turing machine can execute an instruction on its tape, it must parse the symbols to determine the exact nature of the instruction and/or datum encoded there. In simpler cases a finite state machine or a pushdown automaton can do the job. Enderton describes the use of "trees" to determine whether or not a logic formula (in particular a string of symbols with parentheses) is well formed. Alonzo Church 1934 describes the construction of "formulas" (again: sequences of symbols) as written in his λ-calculus by use of a recursive description of how to start a formula and then build on the starting-symbol using concatenation and substitution.

Example: Church specified his λ-calculus as follows (the following is simplified version leaving out notions of free- and bound-variable). This example shows how an object theory begins with a specification of an object system of symbols and relations (in particular by use of concatenation of symbols):

(1) Declare the symbols: {, }, (, ), λ, [, ] plus an infinite number of variables a, b, c, ..., x, ...

(2) Define formula: a sequence of symbols

(3) Define the notion of "well-formed formula" (wff) recursively starting with the "basis" (3.i):

• (3.1) (basis) A variable x is a wff
• (3.2) If F and X are wffs, then {F}(X) is a wff; if x occurs in F or X then it is said to be a variable in {F}(X).
• (3.3) If M is well-formed and x occurs in M then λx[M] is a wff.

(4) Define various abbreviations:

• {F}[X] abbreviates to F(X) if F is a single symbol
• ${\displaystyle {{F}[X]}[Y]}$ abbreviates to {F}(X,Y) or F(X,Y) if F is a single symbol
• λx₁λx₂[...λx_n[M]...] abbreviates to λx₁x₂...x_n•M
• λab•a(b) abbreviates to 1
• λab•a(a(b)) abbreviates to 2, etc.

(5) Define the notion of "substitution" of formula N for variable x throughout M (Church 1936)

Undefined (primitive) objects

Certain objects may be "undefined" or "primitive" and receive definition (in the terms of their behaviors) by the introduction of the axioms. 

In the next example, the undefined symbols will be { ※, ↀ, ∫ }. The axioms will describe their behaviors. 

Axioms

Kleene observes that the axioms are made up of two sets of symbols: (i) the undefined or primitive objects and those that are previously known. In the following example, it is previously known in the following system ( O, ※, ↀ, ∫ ) that O constitutes a set of objects (the "domain"), ※ is an object in the domain, ↀ and ∫ are symbols for relations between the objects, => indicates the "IF THEN" logical operator, ε is the symbol that indicates "is an element of the set O", and "n" will be used to indicate an arbitrary element of set-of-objects O.

After (i) a definition of "string S"—an object that is a symbol ※ or concatenated symbols ※, ↀ or ∫, and (ii) a definition of "well-formed" strings -- (basis) ※ and ↀS, ∫S where S is any string, come the axioms:

• ↀ※ => ※, in words: "IF ↀ is applied to object ※ THEN object ※ results."
• ∫n ε O, in words "IF ∫ is applied to arbitrary object "n" in O THEN this object ∫n is an element of O".
• ↀn ε O, "IF ↀ is applied to arbitrary object "n" in O THEN this object ↀn is an element of O".
• ↀ∫n => n, "IF ↀ is applied to object ∫n THEN object n results."
• ∫ↀn => n, "IF ∫ is applied to object ↀn THEN object n results."

So what might be the (intended) interpretation of these symbols, definitions, and axioms?

If we define ※ as "0", ∫ as "successor", and ↀ as "predecessor" then ↀ※ => ※ indicates "proper subtraction" (sometimes designated by the symbol ∸, where "predecessor" subtracts a unit from a number, thus 0 ∸1 = 0). The string " ↀ∫n => n " indicates that if first the successor is applied to an arbitrary object n and then the predecessor ↀ is applied to ∫n, the original n results." 

Is this set of axioms "adequate"? The proper answer would be a question: "Adequate to describe what, in particular?" "The axioms determine to which systems, defined from outside the theory, the theory applies." (Kleene 1952:27). In other words, the axioms may be sufficient for one system but not for another.

In fact, it is easy to see that this axiom set is not a very good one—in fact, it is inconsistent (that is, it yields inconsistent outcomes, no matter what its interpretation):

Example: Define ※ as 0, ∫※ as 1, and ↀ1 = 0. From the first axiom, ↀ※ = 0, so ∫ↀ※ = ∫0 = 1. But the last axiom specifies that for any arbitrary n including ※ = 0, ∫ↀn => n, so this axiom stipulates that ∫ↀ0 => 0, not 1.

Observe also that the axiom set does not specify that ∫n ≠ n. Or, excepting the case n = ※, ↀn ≠ n. If we were to include these two axioms we would need to describe the intuitive notions "equals" symbolized by = and not-equals symbolized by ≠.