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To infinity and beyond! Understanding more about the concept of infinity

Elena, Year 13, explores the concept of infinity, challenging our understanding.

Most people nowadays are comfortable with the idea that numbers can go on forever. Theoretically, you could keep counting on and on.

If I asked: Which is bigger? The set of all counting numbers, or the set of all even numbers?

Above: Infinity symbol (Wikipedia)

The natural inclination would be to say that the set of all numbers is larger than the set of even numbers. However, these two sets of infinities are actually equal.

Hilbert’s Hotel Paradox

In the 1920s, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds around the concept of infinity.

Imagine a grand hotel with an infinite number of rooms and an infinite number of guests in those rooms.

To challenge the common idea about infinity, he asked what would happen if a new guest came along and asked to stay in a room. At first, it seems impossible, as there is already an infinite number of guests.

However, the trick is to ask each guest to move along one room:

The guest in room 1 moves to room 2, the one in room 2 moves to room 3 and so on. Since there is an infinite number of rooms, there will still be room for all the previous guests.

So the new guest would have a space in room number 1. This shows that infinity+1 is still equal to infinity.

This solution is only possible because the infinite hotel only deals with the lowest levels of infinity, mainly the countable infinity of the natural numbers.

Cardinality

If we were dealing with higher orders of infinity, such as the real numbers, these strategies would not be possible as there is no way of systematically including every number.

The infinite hotel on the real number line would have an infinite number of rooms in the basement (negative numbers), fractional rooms, square root rooms and other irrational number rooms.

Even though this example shows how adding infinities equals another infinity, some infinities are known to be bigger than others.

This concept was first introduced in 1891 by a German mathematician called Georg Cantor, who tackled the idea of the existence of transfinite numbers and their cardinality.

Transfinite numbers are numbers which are used to describe the size of an infinite set of numbers, while cardinality is the size of a set.

Having the same cardinality means that there is a one-to-one correspondence between sets of numbers – this is called a bijection. This means that each number from one set can be matched up with a number in another set.

Conclusion

In summary, when adding infinities together, you get another infinity. However, this new infinity will have the same cardinality as the previous one. This is because it will be possible to make a bijection with the original infinities.

On the other hand, there are infinities which are so large, that their cardinality (known as 𝖈) is still considered to be a hypothesis, and anything between the smallest cardinality and the largest one is still a mystery to be proved.

This shows how hard it is for our finite minds to imagine a concept as large as infinity.

Bibliography

Alfeld, Peter. (1996). Why are there infinitely many prime numbers?. Available: https://www.math.utah.edu/~pa/math/q2.html .

Crowston, Robert. (2011). Hilbert’s Hotel. Available: https://nrich.maths.org/5788 .

Jaksich, John. (2013). Infinity is Weird. Available: https://skullsinthestars.com/2013/11/14/infinity-is-weird-how-big-is-infinity/ .

Marianne. (2013). Maths in a Minute: Countable Infinities. Available: https://plus.maths.org/content/maths-minute-countable-infinities .

Mcgregor, Peter. (2008). A glimpse of Cantor’s Paradise. Available: https://plus.maths.org/content/glimpse-cantors-paradise .

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Summary of the Rugby World Cup 2019

Last September commenced the start of the Rugby World Cup 2019 in Tokyo, Japan, where the top twenty countries battled it out for a claim of the title. The tournament is rugby’s centrepiece event, becoming the big stage that rugby players from all over the world dream of reaching.

This was the first year where the tournament would be hosted in an Asian country and the Japanese succeeded in proving their place amongst rugby’s top nations. Dominating over top performing countries, such as Ireland and Scotland, the Japanese team rightfully earnt their place in the quarter-finals, which was seen as a significant achievement for the host nation. A lot of admiration can be taken from the fighting spirit of Japan, from entering the tournament as historically not the strongest team but fuelled from the encouragement of its home crowd spirit and pride for its nation- Japan preserved through some challenging matches in the group stage.

Despite not making the semi-finals, the Japanese performance should be seen as the silver lining of the tournament. Immense amount of pride and honour should be taken away by the Japanese players, as their performance has inspired a nation to take a love to the sport and hopefully promise a successful new generation of boys and girls for the future of rugby for Japan.

written by Lucy – hockey rep

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Selected for Lions Hockey

Huge congratulations to Amelie and Lauren for being selected for the UK Lions hockey! Lions hockey is where talented young boys and girls from England, Ireland, Scotland and Wales come together to play competitive hockey both home and abroad. Over the years this has run many of the players have been selected to represent countries at junior and full international level, therefore being an awesome opportunity for the girls. We are all hugely proud of them!

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GDST Hockey Final Training Day selections

Next Friday, our sports captain – Emily, is going to Northampton High to attend the final selection day for the GDST Hockey team with the hope of reaching the final squad which will be touring Loughborough in February. We are proud of her making it already through the first selection and wish her luck for the final selection!

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Is empathy the most important thing we can teach our students?

Mr Daniel Addis, Head of Academic Scholarship and Teacher of Classics at WHS, looks at the purpose of education, and asks whether empathy could be the key skill students should develop in an academic environment.
Whenever one considers what education is for, there are several arguments that immediately sprout up. There is a ‘Scholar Academic’ (SA)[1] perspective that suggests there is a key set of knowledge that students need to know in order to be upstanding members of society. The ‘Social Efficiency’ (SE) model argues that it is skills that are imperative to learning in order to prepare students for life in the workplace, whereas the ‘Learner Centred’ (LC) model suggests that content is immaterial; students should have the opportunity to study whatever they desire to benefit themselves. Finally the ‘Social Reconstruction’ (SR) model suggests that education’s main imperative is to facilitate the creation of a more just society, based on the balance between different groups, whether that is racial, class-based, or other forms of segregation.

The true answer presumably lies in a combination of these different models, but I would argue that empathy is the link upon which all of them rely. Empathy is the key knowledge, the important skill, the centre for the learner, and the methodology through which we can create a more-just society.

Nussbaum, in her excellent work Not for profit: Why democracy needs the humanities[2], discusses how empathy requires three things.

  1. A child who knows how to do things for themselves
  2. Recognition that total control is neither possible nor good, that the world is a place in which we all have weaknesses and we need to find ways to support one another
  3. An awareness of solidarity and the idea that we are not alone

Each of the 4 models of curriculum I mentioned have part of these three aspects in them. In the SA it is a solidarity gained by the shared experience of learning the same key material along with understanding of the past that demonstrates the lack of total control in the world; in the SE it is developing children’s ability to do things for themselves; in the LC it is also the development of the students’ ability to do things, along with supporting them individually; and in the SR it is the concept of solidarity amongst peers, and support of others. The fact that part of if not all of these key facets of developing empathy are in each curriculum model demonstrates how important it is to students’ education.

Though all the models have different aims, aspirations, history, ideology, and conceptual understanding, Empathy runs through them all. This is eminently understandable. In E.D Hirsch’s book Cultural Literacy: what every American needs to know[3], he lists facts, figures and great works of literature that he considers a key part of any western education. The list has a strong historical bias, requiring students to learn the history of culture and society. This develops a cultural empathy, understanding of where we have come from, as well as helping us understand and relate to other cultures our students might come across in future.

Photo by Tatiana Vavrikova from Pexels

According to Nussbaum “seeing how another group of intelligent human beings has cut up the world differently… gives a young person an essential lesson in cultural humility.”[4]  Coming across something different which requires greater study and further analysis helps students to understand their experiences as it is comparatively more different to their own. The fact the traditions and models are more unfamiliar allows students to develop analytical tools they can use in other spheres. Their analytical ability is honed further as it is used by the student dissecting more peculiar practices. When considering the greater intellectual difference, the similarities become more poignant, and the nature of combined human experience can imbue students with an awareness of solidarity between peoples, something required for empathy.[5]

Image by Tumisu from Pixabay

From a pastoral perspective, empathy is something that is an obvious focus to develop in young people, but a blending of the academic and pastoral is important in this setting. It is only by intellectually engaging with alternative information and different perspectives through academic learning that empathy is fully developed. It is enhanced through questioning, analytical rigour and searching for deeper meaning rather than, as can be the case in PSHE, something that is assumed. We all consider ourselves to be empathetic but by questioning information from other sources comes a stronger empathy, not a facile, ethereal thing, which can lead to more substantial change.

If we look more closely at some of the material covered in class, we can question the inherent benefit. What is the purpose of learning about life in Chaucer’s England? Would it make much difference in someone’s life to know about life in a Roman household? Perhaps the facts themselves are not important. But by comparing their own experience with others, students can gain a more concrete understanding of the beneficial aspects of their own life. This, in turn, can help them understand other cultures around the world, other people, new information which will prove a vital skill for their later life. With the rate the world is changing, being able to intellectually adapt and understand the needs of others is one of the core skills our students need to possess.

Whilst I do believe that some knowledge is inherently beneficial (I would hardly be a Classicist if I didn’t!), it is important to remember the overall purpose of what we do at school. By putting empathy at the front and centre of the learning experience, we not only develop analytical ability, but we also develop better people who can utilise a different perspective, challenge assumptions and develop their understanding of others. In this way, we give them the tools to change the world, building on our shared past, in order to develop our best future.

References

[1] The four terms I use for curriculum ideologies are found in Schiro, M. S. (2013) Curriculum Theory: Conflicting Visions and Enduring Concerns 2.e.; Sage Publishers. p. 4. There are other terms used by other authors but these four are the clearest.

[2] Nussbaum (2010) Not for profit: Why democracy needs the humanities. Princeton, NJ: Princeton University Press. p. 97

[3] Hirsch, E.D. (1988) Cultural Literacy: what every American needs to know; Incidentally this was the key text upon which the 2015 UK governmental education policy was based.

[4] Nussbaum (2010), 90

[5] Ibid, 97

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Did the Great Depression influence the response to the 2008 Financial Crisis?

Lauren, Year 13, discusses whether the Great Depression influenced the response to the 2008 Financial Crisis.

During the Great Depression, wages were cut for workers which led to a reduction in demand. This stemmed in the bankruptcy of thousands, as the stock market went into free fall after the Wall Street Crash.  Between 1929 and 1932 more than 100,000 businesses went bankrupt, and around 11,000 banks stopped trading. When these banks shut down, savers lost all of their money so they could no longer buy consumer goods. This reduction in demand resulted in the redundancy of many workers, ultimately creating a further decline in the level of aggregate demand. Thus, the economy entered a downward spiral.

President Hoover interpreted the Depression as hypothetical notion, a normal business turndown, rather than a solidified and evidenced occurrence.  Consequently, when an attempt to take action was made it was a little too late. Following the Great Depression, regulations were altered, and economic policies restructured all across the world. The economic system was redesigned to avoid a repeat of this disaster and the levels of government spending were increased.

After the Great Depression, it was often assumed that there would not be another economic downturn of such major proportions as it was believed that the lessons that had been learned then could be applied to any future crisis, protecting the future from such economic turmoil. However, perhaps the lessons learned were not enough to ultimately fend off the Financial Crisis of 2008 in which there had been a rise of non-bank institutions which were not regulated to the same extent as commercial banks, concerning loans.

Several measures were put into place in order to alleviate the effects of the crisis. In the USA, loans from the Federal Reserve were enforced, and the US even tried a Keynesian fiscal stimulus in early 2008 to ‘jump-start’ the economy, but this wasn’t successful enough because the stimulus was too small, at only about 1% of GDP.

A matter of interest to many economists is how the crisis was dealt with in the UK under the Labour government, because they continued to spend significantly in the immediate aftermath of the Financial Crisis. This helped to ease the initial impact because it reduced the economic downturn, but the rate at which the national debt was shooting up was dangerous.


The coalition government slashed public spending after 2010, damaging public services and holding back economic recovery after the crisis. Although it would have been wrong to ignore the huge government deficit inherited from Labour, it could be suggested that Osborne should not have cut spending on infrastructure and capital to such a degree, allowing the UK to invest to boost productivity.

In conclusion, had the enormous intervention by governments not happened, the impact of the Financial Crisis would have been significantly greater. This means that it can be argued that the Great Depression did, to some extent, influence the response to the 2008 Financial Crisis as it persuaded governments to intervene quickly and at great expense in order to avoid a repeat of events in the 1930s. However, it can be argued that these policies were not as successful as envisaged due to the complexity of new financial instruments.

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Sahar Khodayari, the ‘Blue Girl’

The meaning and significance of sport can be lost in our manic day-to-day lives. At school, we are fortunate to be offered a plethora of opportunities to participate in a large variety of sports. Wherever we turn, there are many role models to whom we can look up to; Olympians, sport stars, even some of our friends. Sometimes, even the daily news reports may seem to be saturated with sporting success. This makes it easy to forget that today many women are still actively prevented from participating in, or spectating, certain sporting events.

After the Islamic revolution, political and legislative changes were introduced in Iran. For example, since 1981, women have been prohibited to attend men’s football matches. Since then, many women have protested, as well as tried to enter football stadiums dressed as men, which generally resulted in their arrests and subsequent prison sentences.

Sahar Khodayari was a young Iranian football fan, a graduate with degrees in English and Computer Science, who paid a very high price for her love of sport. Nicknamed the ‘Blue Girl’, blue being the colour of the football team she supported, on the 2nd September 2019, set herself alight, in protest against the current rules of Iran. She died a week later in hospital.

Sahar Khodayari was arrested six months earlier in March, while attempting to enter a Football Stadium to watch a football match of the Asian Champions League. She was dressed as a man, but got caught by a security guard and arrested for “violating the prohibition”.  For six months she awaited her trial, but on the day of 2nd September, the judge was not available to appear at court, therefore no verdict was passed. However, Sahar heard from other sources that she should expect to receive a six to twelve months prison sentence. She then took the decision to self-immolate. After leaving the court house, she poured petrol on herself and set herself on fire.   

These events have sparked a series of debates and pressure has been placed on Iran to lift the controversial ban. Many Iranians took part in online campaigns, urging influential sporting organisations to ban Iran from participating in international competitions. Eventually, on the 10th of October 2019, Iranian women were permitted to attend the World Cup qualifier match against Cambodia. The match took place at the Azadi Stadium in Tehran with a turnout of 3,500 women.

Ostensibly, this seems like a success; the media reported the event with headlines such as ‘Women attend first match in decades’ and ‘Thousands of Iranian women watch football match for first time’. However, women who attended the match were still segregated from men within the stadium, sitting in a separate area on allocated seats. Neverthless, it is also worth mentioning, that, it had only been just over a year ago, that in June 2018 similar headlines appeared stating ‘World Cup 2018: Women finally allowed in Iranian football stadium’. On that occasion, women were permitted to be present in the Azadi Stadium while the football match, physically taking place in Russia, was shown live on big screens in the stadium. However, the privilege of being able to spectate the sport ceased as soon as the World Cup came to an end. When FIFA (International Federation of Association Football) applies pressure on Iran, they appear to comply and permit some women to watch a high profile football match live in a stadium. These images circulate on the internet, then journalists report on the momentous changes and what these mean for the women of Iran. Unfortunately, however, the rules revert back as soon as media attention drifts away and focuses on other news elsewhere in the world.

Although this doesn’t detract from the success of 3,500 women being allowed to watch the World Cup qualifying game on the 10th October, it is important to ensure that Sahar Khodayari is not forgotten, because her fight has not been won yet.

Meanwhile, instead of viewing sport as an inconvenience, trying to avoid participation or flicking over sport news and sports channels, we should perhaps pause and remember those who lost their lives due to their love of sport.  We don’t all have to love it, but we should all appreciate that we have the choice and we are encouraged to embrace it. Knowing that women are still punished for and prevented from participating in sport, makes me feel sad and humbled.

written by Izabella – swimming rep

Sources:

https://www.bbc.co.uk/news/world-middle-east-49646879

https://www.bbc.co.uk/news/world-middle-east-50002658

https://www.theguardian.com/world/2019/oct/10/thousands-of-iranian-women-watch-football-match-for-first-time

https://time.com/5683512/fifa-women-iran-football/

https://www.bbc.co.uk/news/blogs-trending-44561909