The history of infinitely large quantities runs like a dark thread through the work and writings of the great mathematicians of the past 2000 years. For Greek thinkers such as Aristotle, as well as for their Hellenistic successors like Eratosthenes, the idea of an infinitely large set was incoherent or “irrational.” While all of these writers often used the notion of an “infinite” (or, more precisely, an “indefinitely long”) process, they all rejected the idea that a completed set could contain a number of items that wasn’t finite.

FractalThis situation, with the idea of infinity acting as a kind of backdrop to mathematical thinking, persisted until the work of mathematician Georg Cantor (1845-1918). After Cantor’s work was based on a simple conception: Two sets are the same size if, and only if, their members can be placed into a one-to-one correspondence with each other. So the sets {Moe, Larry, Curly} and {Groucho, Chico, Harpo} are the same size because we can define a correspondence (or, in math terms, a 1-1 function) from one to the other:

In cases of finite sets, this is obvious — so obvious that we simply use “cardinal numbers” such as 3 to represent the size of all sets like the two shown here. The “cardinality” of a set is, then, just the number you reach if you count up all its members.

One set is smaller than (i.e., has a smaller cardinality than) another set if you cannot find a one-to-one correspondence from it to the second set that doesn’t leave out some of that second set’s members. For example, if we add Zeppo to set B, then A is smaller than B, because Zeppo would be “leftover” from any one-to-one correspondence that included the other three Marx brothers.

As soon as we start thinking about larger sets, though, things get interesting. For example, consider these two sets:

A = {0, 1, 2, 3, 4, 5, … }

B = {0, 10, 20, 30, 40, 50, … }

On the one hand, B is clearly a subset — and a proper subset — of A. That is, every member of B is also a member of A, but there are many members of A that are missing from B. So, in this sense, A is larger than B.

But consider the following function:

Clearly, if two sets’ being the same size means that you can “pair up” their items in a one-to-one correspondence, then A and B are actually the same size — which means they have the same cardinality. This result, that a set can be the same size as one of its parts, is not too surprising. After all, since A and B are both infinitely large, we are comfortable saying that they are the same size.

What Cantor did next, though, quite literally rocked the foundations of mathematics — especially because he did it in an area (sets) that had always been considered very elementary, even childishly simple.

There is a very simple set that can be formed from any given set, called its “power set,” which consists of all subsets of the original set.

For example, if A = {Moe, Larry, Curly}, then the power set of A is

P(A) = {ø, {Moe}, {Larry}, {Curly}, {Moe, Larry}, {Moe, Curly}, {Larry, Curly}, {Moe, Larry, Curly}}.

Notice that each set in P(A) is a subset of A and, more importantly, that P(A) is larger than A. A has a cardinality (or “size”) of 3, while P(A) has cardinality 8, which is 23. If you experiment with other sets, you’ll see that if a finite set has size n, the power set of that set will have size 2n.

Now, for an infinite set A, it is obvious that the power set, P(A), will be at least as large as A itself — for one thing, for any element x in A, there will be a corresponding set {x} which is a member of P(A). What we might expect, as with the case of A and B above, is that A and P(A) are the same size — after all, they’re both infinitely large, right?

Cantor, though proved this shocker: In fact, the cardinality of P(A) is larger than the cardinality of A! That is, he proved that any one-to-one mapping, starting from A and going to P(A) will “leave out” many, many, items of P(A) — as if we had left out infinitely many Zeppos in the case of A and B above! The conclusion is inescapable: The two sets, A and P(A), are both infinitely large, but P(A) is strictly larger than A. Cantor referred to the “sizes” of sets like P(A) as “transfinite” numbers, and his result means that there are many of these, and that they continue to grow — forever. Why? Well, once you have Cantor’s Theorem, which says that P(A) is always larger than A, then you can take the power set of the power set of A — i.e., P(P(A)) —  and get a larger set.

So we have a series of sets:  A is smaller than P(A), which is smaller than the P(P(A)), which is smaller than P(P(P(A))), and so on. If A itself is transfinite, then you have an unending sequence of transfinite sets, each one larger than the next.

The existence of this sequence was, as noted, a shock to the mathematical world — so much so that some great figures (Poincaré, for example) suggested that Cantor was out of his mind, or even a sinister character! For the past 100 years, however, Cantor’s work has blossomed into a vast and respected field within the boundaries of mathematical research.

Further reading:

Enderton, Herbert: Elements of Set Theory

Halmos, Paul: Naive Set Theory

Suppes, Patrick: Axiomatic Set Theory

larry-headshotAbout the Author
A former math teacher in Georgia, Larry Coty is now USATestprep’s Math Content Team Leader. He has two daughters and resides in Tucker, GA.

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Do schools in low–income areas perform badly on tests?  What about schools in areas where only a small percentage of adults have a college degree?  Or schools in communities where most residents rent rather than own?

These are questions we hoped to answer as we studied data from over 8,000 students in 42 randomly-selected schools. Below are details on our research.

Methodology and Sample Size

Test scores were collected from 8,031 middle school and high school students in 42 randomly selected schools from 5 states. These schools are in both major metropolitan areas and rural communities.  The average per capita income for the zip codes of the 42 schools is $25,102, or about 12% less than the national average. The test scores came from the subject areas of Reading, Science, ELA, and History.  Per capita income data came from  Data on owner-occupied housing came from  Data on college degrees came from the WashingtonPost.

After compiling and analyzing the test scores, we did not find what we expected – at the outset. While there are some exceptions, most data indicates that low income correlates with lower test scores.

Little or no research has been done on the correlation between schools in areas with a high percentage of rentals versus a high percentage of owner-occupied dwellings.  Also, we have been unable to find any previously published data on the correlation between the percent of college-educated adults in a community and that community’s test scores.

The test scores used in this article have been taken from assessments on USATestprep’s platform. Data on income, college education, and owner-occupied housing is for each school’s physical zip code.

Below are the schools’ test scores by income, percent of college-educated adults, and percent of owner-occupied housing.   

Test Scores by Per Capita Income

6th-12th Grade Test Scores by Income

Test Scores by Percent of College-Educated

Test Scores by Percent of College Educated Adults in Community

Test Scores by Owner-Occupied Housing

Test Scores by Percent of Owner-Occupied Housing in Community


Teacher pay, by state, is easy to find online.  However, some states have higher costs of living than others. For example, $50,000 in California spends much differently than it does in Tennessee.  With that in mind, which states have the best- and worst-compensated teachers, given cost of living and taxes?  With the help of data from the Bureau of Labor Statistics, Forbes, and other sources, we’ve compiled a list of the best and worst states for teachers – not by pay, but by “adjusted pay.” You can download the spreadsheet with salary, cost of living, and tax data here.  We’ve even adjusted the middle school teacher salary figures from each state to reflect the cost of living and taxes, so that you can compare “apples to apples”.

Teacher Pay and Standard of Living in the US5 Highest Paying Cities for K-12 TeachersElement3rv1Element4rv1ElementFooterE1-1ElementFooterE4ElementFooterE5ElementFooterE3ElementFooterE2

We posted this on our Facebook page but it was too good not to share on our blog, as well.

Radiolab offered up a look at the power of suggestion, “white lies”, and placebo on this show that originally aired in 2007. They examined not just the “Placebo Effect” but the use of hypnotism on patients and the power of white doctor’s coat. We thought some of this might be useful in a science or psychology class, but it is definitely enthralling stuff regardless of your area of expertise. This episode turned out to be one of those shows that you listen to even after you’ve reached your destination. Click this link to go to the episode’s home page.

I was doing a yahoo search on USATestprep and discovered this interesting question posted in YAHOO! answers. A student asked “How to cheat on usatestprep???”  Someone posted a fantastic response to the question. Here is a screen capture of the post.

grow.jpgWe have often gotten requests from teachers for specific questions to be added to the database. For this reason, we have recently developed the “Help Grow the Database” feature. This new feature allows teachers to add questions in an easy, step-by-step process. These questions are then internally reviewed and added to the online system for students to start using.

You will find this listed as the second item at the top right under “Other Items” after you log in as a teacher. We hope you enjoy this new feature, and we look forward to receiving your submissions.

We know that the AHSGE will soon switch to Biology only. In looking at the DOE AHSGE page, the same science standards are listed that have been there for years. If anyone knows what standards will be implemented or can provide some information, we can get started and make sure we are ready as soon as the state switches over. Contact us at if you have any information