Tropical Storm Gordon (2018)

Storm Active: September 3-8

During the final days of August, a tropical wave whose axis extended from the Caribbean south of Hispaniola northward to the adjacent Atlantic waters began to produce increased thunderstorm activity throughout the region. However, upper level winds were quite strong over the wave. This prevented further development for a few days as it moved toward the west-northwest. The system passed over the southern Bahamas on September 2 and its satellite presentation improved markedly, though it still lacked a surface circulation. Only on September 3 did Florida radar detect a well-defined surface low, prompting the naming of Tropical Storm Gordon. At the time of naming, Gordon's center was over southern Florida. It moved over the Gulf of Mexico shortly afterward but continued to bring rain to the southern half of the peninsula for the remainder of the day.

While the core intermittently showed signs of an eyewall forming on radar, the overall satellite presentation of Gordon was lackluster as it moved through the Gulf over the next day. The radius of tropical storm force winds remained very small and heavy rain did not extend much beyond it. Nevertheless, maximum winds at the center reached strong tropical storm intensity by September 4. The waters of the Gulf of Mexico were quite warm, but fortunately Gordon's forward speed remained fairly high due to consistent steering patterns, and it moved quickly northwest. Late in the day, the cyclone reached its peak intensity of 70 mph winds and a pressure of 997 mb just before making landfall along the Gulf coast near the border of Mississippi and Alabama. Mississippi was spared most of Gordon's flooding rain impacts since almost all rain was to the east of the center. Some hurricane force gusts were also reported during landfall.

The system quickly weakened inland and became a tropical depression on September 5. Its movement slowed considerably once over land and it gradually moved northward through the midwest over the following couple of days, bringing heavy rainfall to a wide swath of the central U.S. before being absorbed.

The above image shows Gordon just before landfall on the Gulf coast.

Gordon's slow movement after landfall contributed to flooding rains throughout the central U.S.

Hurricane Florence (2018)

Storm Active: August 31-September 17

On August 30, a vigorous tropical wave entered the Atlantic from the west African coastline. While the system was initially large and disorganized, its center became much better defined over the next couple of days. The low passed just south of the Cape Verde Islands on the 31st, bringing some heavy rain and winds. Around the same time, it was classified Tropical Depression Six. Ocean temperatures were only marginal for development, but wind shear was very low and the air surrounding the system quite humid (any Saharan dry air was well to the north). Because of this, the cyclone strengthened as it moved west-northwestward away from the islands, becoming Tropical Storm Florence early on September 1.

Some more intensification occurred that evening, but wind shear out of the southwest suddenly exposed the center of circulation on September 2 and the system weakened slightly. As Florence continued on its west-northwest heading over the open central Atlantic, sea surface temperatures began to warm beneath it. Overnight, the system developed an inner core and strengthened into a strong tropical storm. However, even as temperatures warmed, wind shear increased also and the satellite presentation became a bit ragged. Nevertheless, Florence held its own on the 3rd and approached hurricane strength when a central dense overcast appeared. In defiance of almost all models and forecasts, the system continued to intensify on September 4 and became a hurricane. It turned a bit toward the northwest that evening and an eye became apparent on satellite imagery. The next morning, the eye became better defined and cleared out further. Florence was then upgraded to a category 3 hurricane, the first major hurricane of the 2018 season.

That afternoon, the cyclone peaked as a category 4 hurricane with winds of 130 mph and a minimum pressure of 953 mb before wind shear out of the southwest finally began to take its toll. The southern eyewall thinned and then broke under the shear, the inner core became disrupted, and Florence lost major hurricane status on September 6. Rapid weakening continued throughout the day, and the center nearly became exposed from time to time overnight as the system weakened back to a tropical storm. Meanwhile, mid-level ridging to the north of Florence turned the shallower system back toward the west on September 7. The weakening trend halted that afternoon as shear began to abate and convection gradually worked its way back over the center. But while waters were warm and upper-level winds were more favorable, the circulation had ingested some dry air that it struggled to mix out through the day of September 8. Nevertheless, it approached hurricane strength once again by that evening.

The next day, deeper convection blossomed near the center and hints of an eye reappeared. Florence regained category 1 status that morning and began a process of rapid intensification. This process really accelerated when a symmetrical eye appearing late on September 9. By the afternoon on the 10th, the system was a dangerous category 4, surpassing its earlier peak intensity to reach a new peak of 140 mph winds and a central pressure of 939 mb. Meanwhile, a very strong subtropical ridge began building to its north and Florence turned west-northwest with a greater forward speed. Overnight, the core underwent an eyewall replacement as the small inner eyewall collapsed in favor of a broader outer one. This took some time to consolidate, leaving Florence with a lopsided appearance on infrared imagery. However, the new eyewall eventually closed off and the system maintained category 4 strength. Moreover, eyewall replacement cycles usually are accompanied by a broadening of the wind field; this held true for Florence. Its tropical storm force and hurricane force wind radii increased markedly by September 12.

By September 13, the maximum winds had decreased to category 2 strength, but the system had grown considerably. Its outer bands began to sweep across the coast of southeastern North Carolina that morning. Florence's speed slowed during the day and it stalled off the coastline. That evening, it turned toward the west and ultimately made landfall overnight as a category 1 hurricane. Even as the storm weakened, torrential rains continued to fall over much of coastal North Carolina and pushed into South Carolina on September 14. Florence became a tropical storm that day and the center crossed the border into South Carolina over land. As the circulation spun down, the core lost much of its strength, but the portion of the storm still over water to the northeast of the center was still dragging immense amounts of tropical moisture over coastal North Carolina on September 15. Along with the flooding rains came intermittent tornadoes reported in the northeast part of the circulation.

Overnight, Florence weakened to a tropical depression and finally began to pick up some speed inland toward the west and then north through Appalachia. By September 17, the system turned northeast under the influence of the mid-latitude westerlies and rains spread eastward into the mid-Atlantic, though nowhere near as severe as they had been in the Carolinas. The low pressure center of Florence became elongated later that day and the system finally transitioned to post-tropical near West Virginia. Soon, the remnants moved eastward over the Atlantic. The extratropical successor to Florence eventually spawned a new low in the subtropical Atlantic that would become Leslie. Even well after the storm had passed, river flooding continued for the Carolinas. Florence's devastating rainfall, totaling over 30 inches in parts of coastal North Carolina, made it among the costliest tropical cyclones ever recorded.

The above image shows Florence near peak intensity as a category 4 hurricane.

Florence's track was highly unusual. In a vast majority of instances, cyclones that traversed the Atlantic at the latitude that Florence did would recurve out to sea before hitting land. However, very strong subtropical ridges steered the storm into the Carolinas.

Tropical Storm Ernesto (2018)

Storm Active: August 15-18

On August 12, a low pressure area formed over the subtropical Atlantic well to the southeast off the coast of Nova Scotia. Over the following few days it moved generally toward the east and then southeast. In the meantime, very warm waters in the region allowed it to organize as atmospheric conditions improved. By August 15, the low's center had become well-defined, though the surface center was still situated under an upper-level low and the radius of maximum winds was rather broad. In light of these features, the system was classified Subtropical Depression Five that day.

The depression turned toward the north that day and strengthened into Subtropical Storm Ernesto. Marginal sea surface temperatures allowed a slight bit of intensification the next day and a transition to a fully tropical storm with convection closer to the center. At the same time, the storm began to feel the influence of the mid-latitude westerlies and accelerated northeast and then east-northeast. Ocean temperatures under the storm plummeted on August 17, but humid and unstable air in the region allowed Ernesto to maintain tropical cyclone status for somewhat longer than originally expected. Early on August 18, the system transitioned into a post-tropical storm west of Ireland. Nevertheless, it brought areas of heavy rain and gusty winds to northern Ireland and the United Kingdom as it merged with a front later that day. In both origin and track, Ernesto was very similar to its predecessor, Debby. In addition, it was the fourth system to become a subtropical storm in the 2018 season. This was the first such occurrence since 1969.

The above image shows Ernesto as a subtropical storm shortly after formation.

While Ernesto did not affect land as a subtropical or tropical cyclone, its remnants did impact the United Kingdom and Ireland.

Tropical Storm Debby (2018)

Storm Active: August 7-9

During the first few days of August, a non-tropical low meandered over the north central Atlantic. By the 4th, it was producing gale force winds, but had very little thunderstorm activity associated with it. Over the next couple days, it drifted southeastward, moving over water that was a tad warmer. The system again became stationary and then started to move back to the north, but by this time it acquired more significant convection. On August 7, the system was classified as Subtropical Storm Debby due to the spread of tropical storm force winds and outer banding from the center.

The storm slowed a bit overnight and turned toward the north-northeast on August 8. At the same time, the maximum winds increased somewhat and thunderstorms became more concentrated close to the center of circulation. As a result, Debby was reclassified as a tropical storm that morning. The cyclone also reached its peak intensity of 45 mph winds and a pressure of 1003 mb. Soon, however, the system began to weaken over the cold north Atlantic. Debby transitioned to a post-tropical storm during the afternoon of August 9 as it accelerated to the northeast, far from any land.

Debby was a small and short-lived cyclone that did not have any land impacts.

The above image shows Debby's track over the north Atlantic.

Hurricane Chris (2018)

Storm Active: July 6-12

During the first couple days of July, an disturbance formed in the subtropical Atlantic southeast of Bermuda. Over the next few days, it moved slowly northwestward and environmental conditions gradually improved for development. On July 5, the system acquired a weak low-pressure center. This became someone better defined that day, but thunderstorm activity remained quite limited. It increased on July 6, however, and the system was classified Tropical Depression Three well offshore of the Carolinas.

That night, it turned toward the north and became somewhat more organized over the warm waters. Surface pressure were still high, however, and the system's maximum winds increased only slowly. Meanwhile, steering currents collapsed and the depression moved very little on July 7. Most thunderstorm activity was displaced south and southeast of the center, keeping coastal North Carolina, which was not far to the northwest, dry. A reformation of the surface circulation to the south allowed the system to organize further and strengthen into Tropical Storm Chris by early on July 8. The next day saw gradual strengthening as the circulation tightened, but dry air intrusion prevented Chris from closing off an eyewall. By that evening, the system was on the verge of hurricane strength and finally began to move slowly toward the northeast. Though it began to move away from land, high surf continued to pound the coastline.

Since it had been stationary for days, Chris had caused cold waters to upwell underneath it (due to its strong winds). Though this decrease in temperature was mitigated somewhat by the steady flow of warm Gulf stream waters, it slowed the cyclone's intensification. However, once it started moving, the system rapidly intensified. Late on July 10, it reached its peak intensity as a category 2 hurricane with 105 mph winds and a pressure of 970 mb. As it accelerated northeast, it began to encounter cooler water, weaken, and become less symmetric. Soon, it weakened to a tropical storm and was quickly transitioning to an extratropical system. Chris became fully extratropical on July 12 as it raced northeast over cold north Atlantic waters. The post-tropical cyclone made landfall in southeastern Newfoundland that night, bringing wind gusts to near hurricane force and brief but heavy rains.

The above image shows Hurricane Chris strengthening off the U.S. east coast. Chris was the earliest second hurricane to develop in the Atlantic since 2005.

Chris did not affect land directly as a tropical cyclone, but made landfall in Newfoundland as a post-tropical cyclone.

Hurricane Beryl (2018)

Storm Active: July 5-8, 14-15

At the beginning of July, a tropical wave emerged off of the coast of Africa. This tropical wave was among the first of the season to develop significant convection over the central tropical Atlantic. Despite anomalously cool waters, conditions were still favorable enough in the deep tropics to support development. By July 4, the disturbance was quite well organized, with low shear in its environment and an evident circulation. Nevertheless, the low did not yet appear closed. The next day, organization increased further, and the system was classified as Tropical Depression Two. The depression was quite small and moving fairly quickly toward the west.

It is not uncommon for small cyclones to change rapidly in intensity, and the system strengthened quickly that evening and into July 6, becoming Tropical Storm Beryl just a few hours after formation. A minuscule pinhole eye appeared on satellite imagery and the maximum winds shot up to hurricane strength, making Beryl the first hurricane of the 2018 season by early on the 6th. The system strengthened a little bit more during that day, reaching a peak intensity of 80 mph winds and a pressure of 994 mb. Beryl turned a bit toward the west-northwest during the evening and into July 7. As it did so, it began to encounter increased wind shear. Quickly, convection was displaced toward the southeast and the center was exposed. As a result, Beryl began to quickly weaken and became a tropical storm. Convection flared up near the center from time to time over the following day but the low-level center all but disappeared as the cyclone accelerated west-northwest toward the Windward Islands. As shear continued to increase, Beryl quickly dissipated into a tropical wave.

Nevertheless, Beryl's remnants passed over the easternmost Caribbean islands on July 8, bringing heavy rains and strong winds out of the east. By the 9th, these rains had begun to move over Puerto Rico and the Virgin Islands. Overnight, the heavy rains moved over Hispaniola as the wave proceeded west-northwestward. The wave began to encounter more favorable conditions over the Bahamas, though the main impact to these areas was still locally heavy rain. After that, the disturbance turned northward and then northeastward, passing to the west of Bermuda.

Only on July 13 did the remnants of Beryl begin to show signs of reorganization, with a well-defined circulation developing. Despite marginal sea surface temperatures and atmospheric conditions, thunderstorm activity reappeared near the low-pressure center. On July 14, a full six days after dissipation, Beryl reformed. Though its windfield was concentrated near the center (typical for a tropical cyclone), the system lay under an upper-level trough. Therefore, it was deriving its energy in a manner uncharacteristic of a tropical cyclone. As a result, forecasters classified it as Subtropical Storm Beryl north of Bermuda.

The system slowed its forward motion and meandered somewhat over the next day. Vertical shear was increasing, but Beryl maintained enough convection in the southeastern quadrant to hang on to subtropical storm status. Even this was short-lived, however, and the system finally weakened into a remnant low late on July 15. This low approached Newfoundland before dissipating.

Beryl was one of the smallest hurricanes ever recorded; winds of hurricane force extended only 10 miles from the center. Note also the minuscule eye on satellite imagery.

A majority of Beryl's impacts occurred while the system was a wave moving through the Caribbean (triangle points).

Subtropical Storm Alberto (2018)

Storm Active: May 25-30

Beginning around May 20, a trough of low pressure located in the western Caribbean produced widespread thunderstorm activity as it interacted with an upper-level low. Over the next several days, the disturbance tracked generally northwestward. In the mean time, abundant moisture in the area caused sporadic rainfall from portions of Honduras to the Yucatan Peninsula to western Cuba. Even after a surface low formed, the system remained quite disorganized due to land interaction with the Yucatan and strong upper-level winds out of the west. Despite fairly hostile conditions, the low became better defined during the day of May 24. By the morning of the 25th, the surface low had emerged over water adjacent to the northeast Yucatan Peninsula with a large area of strong thunderstorms to the north and east. In addition, buoy and ship reports suggested the presence of winds to gale force. Since the low was situated under an upper-level trough, and not the upper-level high associated with traditional tropical storms, the system was designated Subtropical Storm Alberto late that morning.

During that day, the surface circulation of Alberto was far removed from the thunderstorm activity to the north and east. In fact, the overall circulation appeared to me moving northeast while the low-level swirl drifted just south of east. Nevertheless, heavy rains continued over much of Cuba and the outer bands began to affect southern Florida. Overnight, upper level winds lessened considerably, and limited convection finally appeared near the surface center. The center also turned north and accelerated early on May 26, essentially "catching up" with the rest of the circulation. As a result, Alberto's satellite presentation improved considerably. A further reformation of the center took place later that day, this time to the northeast of the previous position. This and the system's generally northward movement brought Alberto into the eastern Gulf of Mexico, not too far from the west coast of Florida. However, this coast was saved from the heavier rainfall by a dry slot in the eastern semicircle; it was now the other side that had most of the convection.

That evening and overnight, Alberto's pressure dropped considerably, its center became better defined, and it began to take on some more tropical characteristics. The storm's maximum winds increased in turn during the day of May 27. The storm also turned toward the northwest briefly under the influence of an upper-level low. Despite organization improvements, dry air was taking its toll on Alberto, invading via the western side and eroding deep convection away from the center. Situated over relatively cold eastern Gulf waters, the system also did not develop the deep warm core needed to be classified as a tropical storm. Nevertheless, Alberto reached its peak intensity of 65 mph winds and a pressure of 991 mb that evening as it approached the Florida panhandle.

Continued dry air intrusion and proximity to land decreased the storm's winds gradually as bands of heavy rain swept across the Gulf coast early on May 28. The center of Alberto made landfall that afternoon in the Florida panhandle, bringing heavy rains and localized flooding to parts of the southeast U.S. At landfall, the storm had maximum winds of 50 mph. That night, it weakened to a subtropical depression over land as it continued northward over Alabama. Curiously, the system completed its transition to a tropical cyclone (becoming a tropical depression) over Tennessee late that evening. The circulation maintained its identity and continued to cause rainfall even into May 30, when it finally became extratropical over Michigan. Alberto marked an early start to the Atlantic hurricane season for the 4th consecutive time, only the 2nd known time this has occurred (after 1951-4).

The above image shows Alberto in the eastern Gulf of Mexico on May 27.

Alberto was subtropical most of its life (square points) but transitioned over land to a tropical depression (blue circular points) and maintained this status remarkably far north.

Professor Quibb's Picks – 2018

My personal prediction for the 2018 North Atlantic hurricane season (written May 16, 2018) is as follows:

18 cyclones attaining tropical depression status,
16 cyclones attaining tropical storm status,
8 cyclones attaining hurricane status, and
4 cyclones attaining major hurricane status.

In the wake of the especially devastating 2017 season, it is difficult to predict with any certainty the outcomes for this year. Once again, models indicate that the El Niño Southern Oscillation Index (or ENSO index) will be near zero or slightly positive during this hurricane season. This index, which is a certain quantitative measure of sea surface temperature anomalies in the tropical Pacific Ocean, has some ability to predict Atlantic hurricane activity. A positive index indicates an El Niño event, which tends to correlate with higher wind shear across the Atlantic basin and less tropical cyclone development. This effect is especially pronounced in the Gulf of Mexico and Caribbean Sea. The image below shows the ENSO forecast for this season (image from the International Research Institute for Climate and Society):

However, last year's forecast was qualitatively similar, but the index ended up dipping back negative and leaving very favorable conditions for hurricane formation. Though consideration of the ENSO index alone would lead to the prediction of an average hurricane season, there is significant uncertainty. Overall, I consider the ENSO to mainly a neutral factor this year.

Present ocean temperatures in the Atlantic are slightly above average in the Gulf of Mexico and Caribbean, and significantly above average in the subtropical Atlantic and near the U.S. east coast. However, there is a large area of below average temperatures in the tropical Atlantic which is forecast by long-term models to possibly persist for a few months. The tropical Atlantic has also been dry and stable, in contrast to elevated storm activity in the Caribbean and Gulf of Mexico. These trends also show some signs of persisting into the beginning of hurricane season. I therefore expect a slow start to the season in the main development region of the tropical Atlantic (extending from Africa to the Caribbean) and a corresponding lack of Cape Verde or long-track hurricanes, though these could appear more in late September and October. There is significant potential for formation in areas closer to land, so I expect some shorter lived hurricanes in the northern Caribbean/Gulf of Mexico and U.S. east coast regions.

My estimated risks for different parts of the Atlantic basin are as follows (with 1 indicating very low risk, 5 very high, and 3 average):

U.S. East Coast: 5
The jet stream over the U.S. has been weaker than usual so far this season, and the Bermuda high stronger. However, with a weak El Nino possibly developing, long hurricane tracks westward into the Gulf still seem unlikely. The east coast, in contrast, is at a greater risk. Ocean temperatures offshore are anomalously warm and region will be very moist, suggesting a fairly high probability of tropical cyclone impacts.

Yucatan Peninsula and Central America: 3
The western Caribbean shows some signs of being a fertile area for cyclonogenesis, but with prevailing upper-level patterns as they are, it is difficult to see strong system taking due westward tracks into central America. Compared to the last few years, strong hurricanes are less of a threat, though the potential for flooding rains may be equal or greater.

Caribbean Islands: 2
As discussed above, the main development region may remain quiet for at least the first half of hurricane season. This would insulate the Caribbean islands from the approach of Cape Verde hurricanes to the west, but does not preclude development occurring locally. Nevertheless, it is somewhat more likely this year that the islands will receive a break from intense hurricane landfalls, especially the easternmost islands.

Gulf of Mexico: 3
Factors in the Gulf point in different directions. Ocean waters are warm and will likely continue to be so, particularly in eddies originating in the northern Caribbean (which also happens to be a likely source of Gulf hurricanes). On the other side, if an El Niño does develop, the Gulf of Mexico is where the suppression of hurricane activity would be most felt. Putting this together suggests a near-average risk this year.

Overall, the 2018 season is expected to be a bit above average; it should not be a repeat of the devastating 2017 season, but many areas such as the U.S. east coast may still be at high risk. Further, this is just an informal forecast and uncertainty in the outcome remains significant. Everyone in hurricane-prone areas should still take due precautions as hurricane season approaches. Dangerous storms may still occur even in overall quiet seasons.

Sources: http://www.cpc.ncep.noaa.gov/products/analysis_monitoring/lanina/enso_evolution-status-fcsts-web.pdf, https://www.tropicaltidbits.com/analysis/models/?model=cfs-avg, https://ocean.weather.gov/

Hurricane Names List – 2018

The name list for tropical cyclones forming in the North Atlantic basin for the year 2018 is as follows:

Alberto (used)
Beryl (used)
Chris (used)
Debbie (used)
Ernesto (used)
Florence (used)
Gordon (used)
Helene (used)
Isaac (used)
Joyce (used)
Kirk (used)
Leslie (used)
Michael (used)
Nadine (used)
Oscar (used)
Patty
Rafael
Sara
Tony
Valerie
William

This list is the same as the list for the 2012 season, with the exception of Sara, which replaced the retired name Sandy.

Goodstein's Theorem and Non-Standard Models of Arithmetic

This is the final post in a four-part series on logic and arithmetic, with a focus on Goodstein's Theorem. For the first post, see here.

In the previous post, Goodstein's Theorem, a statement about the properties of certain sequences of natural numbers, was proven using infinite ordinals. The use of a method "outside" arithmetic makes it reasonable that this proof cannot be encoded in the language of Peano Arithmetic (PA), the formal logical system for discussing the natural numbers. A stronger statement is also true: there is no proof of Goodstein's Theorem in PA because it cannot be deduced from the axioms of PA.

But how does one go about proving something unprovable? Certainly it is intractable to check every possible method, as the diversity of such attempts could be infinite. Mathematicians take a different approach, using tools from what is called model theory. In mathematical logic, a model of a collection of axioms is a specific structure within which the axioms (and all theorems derived from them) are interpreted to be true. Recall that the axioms of PA mentioned five specific objects, that were assumed to be given from the start: a set N, a specific member, 0, a function S from N to itself, and two binary operations on N, + and *. Of course, to actually do arithmetic we interpret N as the set of natural numbers, 0 as the number 0, S as the "successor" function taking in a number n and returning n+1, and + and * as the usual addition and multiplication. Until we provide an interpretation of to what these objects refer, namely a model, they are just symbols! We may prove statements about them, such as the fact that S(0) and S(S(0)) are distinct members of N, but this is just a mathematical sentence resulting as the end product of a series of formal deductive rules.

Any collection A = (N_A,0_A,S_A,+_A,*_A) of a set N_A, a member 0_A of the set, a function S_A:N_A→N_A, and two binary operations +_A and *_A that satisfies the axioms is a model of PA. Of course, we know fairly well what we mean by "natural numbers", namely {0,1,2,...} with 0 the first element, S sending 0 to 1, 1 to 2, etc, and the usual addition and multiplication. The entire point of selecting axioms for PA is to study ℕ = (N,0,S,+,*), the standard natural numbers. A natural question (called the question of categoricity) arises: is the standard model the only type of model for PA, or are there others? The answer is no; there are other, non-standard models A that still satisfy every axiom of PA. These were first discovered by Norwegian mathematician Thoralf Skolem in 1934. To be clear, they are not the natural numbers, at least, not as we intend them to be. Their existence exemplifies another limitation of first-order logic: axiom systems often fail to specify structures uniquely and hence fail to capture some features of the field to be studied.

Non-standard models often serve as an essential tool in independence proofs. First, we know from the previous post that the standard model ℕ of PA does satisfy Goodstein's Theorem (the standard model has the properties the natural numbers possess within the larger field of set theory, the methods of which were used in the proof). This means that the negation of Goodstein's Theorem cannot be a theorem of PA, since there is a model satisfying both the axioms and the theorem. If one could find a model of PA in which the negation of Goodstein's Theorem were true, then this would prove independence, because there would be models in which it is true and others in which it is false! Kirby and Paris used precisely this method in their 1982 proof of the result.

But what do non-standard models of natural numbers actually look like? First, we may infer what they have in common. PA axiom 1 guarantees the existence of a number 0. Axiom 2 gives it successors S(0), S(S(0)), etc. Axiom 3 says that S(n) = S(m) implies m = n. Therefore, all the successors generated from 0 are distinct from one another. This means that any model A has a set of natural numbers N_A containing the analogues of 0, 1, 2, and so on. The set of standard natural numbers N is thus contained in N_A for every A. The difference is that non-standard models have extra numbers!

At first brush, having additional "non-standard" numbers seems to contradict the Peano axioms, specifically the fifth, the axiom schema of induction. It states that if 0 has some property and that any n having the property implies that n + 1 does as well, then all natural numbers have the property. The spirit of this axiom schema, if not the letter, is that beginning at 0 and knocking down the inductive dominoes will eventually reach every natural number. If we could choose the property to be "is in the set {0,1,2,...} (the standard natural numbers N)" then this would immediately rule out nonstandard models: 0 is this set, and for any n in the set, its successor is also standard, so all of N_A is contained in {0,1,2,...} and hence we would have N_A = {0,1,2,...}. Unfortunately, it is impossible to define the set {0,1,2,...} inside of the first-order logic formulation. It is also impossible to simply add an axiom "there are no other numbers besides 0, 1, 2, etc." for the same reason. Both approaches require infinitely long logical sentences to formulate, which are forbidden in the finitary system of first-order logic.

Though the axioms of PA cannot rule out non-standard natural numbers, they are forced by the axioms to satisfy some strange conditions. Any nonstandard number c must be greater than all standard numbers. Further, PA can prove that 0 is the only number without a successor, so a "predecessor" to c, which we may call c - 1, must exist. Similarly, c - 2, c - 3, etc. must exist, as must, of course, c + 1, c + 2, etc. These must all be new non-standard numbers. Therefore, the existence of one non-standard number guarantees the existence of a whole non-standard "copy" of the integers: {...,c - 2,c - 1,c,c + 1,c + 2,...}. However, it gets much, much worse. The operation of addition is part of Peano Arithmetic, so there must be a number c + c, that may be proven to be greater than all numbers c + 1, c + 2, and so on. From here, we get another new infinite collection of non-standards {...,c + c - 2,c + c - 1,c + c,c + c + 1,c + c + 2,...}. A similar story occurs for c + c + c = c*3 and larger numbers as well, but we can also go in reverse. One can prove in PA that every number is either even or odd; that is, for any n, there is an m satisfying either m + m = n (if n is even), or m + m + 1 = n (if n is odd). This theorem means that c is even or odd, so there must be a smaller non-standard d with d + d = c or d + d + 1 = c. This d has its own infinite set of non-standard neighbors. The reader may continue this type of exercise and eventually derive the type of picture illustrated above: any non-standard model of natural numbers must contain the standard numbers plus (at least) an infinite number of copies of the integers, ℤ, one for each member of the set of rational numbers, ℚ.

As strange as these models are, they cannot be ruled out in PA, nor is there a natural addition to the axioms that may do so. Rather than being just a defect of first-order logic however, non-standard models are a useful tool for examining the structure of different theories. Now that we have a non-standard model at our disposal, it seems reasonable that Goodstein's Theorem should fail for some non-standard models: "Goodstein sequences" beginning at non-standard natural numbers do not seem likely to terminate at zero. After all, they have infinitely many copies of the integers to move around in! These sequences often cannot be computed explicitly, but using other logical machinery, one can prove the fact that they do not necessarily terminate. This establishes the independence of the theorem from PA.

Goodstein sequences, interesting in their own right for their rapid growth, allow an interesting perspective on Peano Arithmetic and its limitations. The questions of independence and non-standard models arise frequently in the foundations of mathematics, as we seek to define precisely the scope of our mathematical theories.

Sources: http://www.cs.tau.ac.il/~nachumd/term/Kirbyparis.pdf, http://blog.kleinproject.org/?p=674, http://www.ams.org/journals/proc/1983-087-04/S0002-9939-1983-0687646-0/S0002-9939-1983-0687646-0.pdf, http://settheory.net/model-theory/non-standard-arithmetic, http://www.columbia.edu/~hg17/nonstandard-02-16-04-cls.pdf, http://boolesrings.org/victoriagitman/files/2015/04/introToPAModels.pdf, http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers/