To Infinity and Beyond - Bonus Content

I recently posted on the subject of Infinity.  Let's have some fun with what we learned there.  And, by "there", I mean:  http://sigma5.blogspot.com/2020/01/to-infinity-and-beyond.html.  Everybody has heard of the following conundrum:

What happens when an irresistible force meets an immovable object?

What do we mean by "irresistible" and "immovable"?  The obvious answer is "infinitely irresistible" and "infinitely unmovable".  Now that we are experts on infinity, let's see if we can shed some light here.

Mental pictures are helpful.  So let's picture our irresistible force as the Mongol Hordes and our immovable object as the Great Wall of China.  Neither, of course, is actually of infinite extent.  But we can take our mental picture of a large group of Mongol Hordes milling around and mentally replace it with a Horde with an infinite number of members.  Similarly, the Great Wall of China is long, very long.  But it isn't infinitely long.  But in our imagination we can extend our picture of it so that it is infinitely long.

Okay.  That's progress.  So what do we mean by "meets".  Well, in our mental picture we can now see this as "meets in combat".  Our infinitely large Mongol Horde attacks our infinitely long Great Wall of China.  That works, so we have made more progress.

So what happens when they meet, as in "fight it out".  Who knows?  But let's now ask the question in the context of our discussion of infinity.  How about this?  When an element of the Mongol Horde attacks an element of the Great Wall of China those two specific elements mutually annihilate each other.  In this context we can now define "winning".  Elements of each group meet and annihilate each other.  If one side has some remaining elements after this process is complete then that side wins.

And, more specifically, we can characterize combat as a process of bringing the two sets of combatants into "one to one correspondence".  If all the elements of one set can be brought into "one to one correspondence" with elements of the other set and if, after we are done, there are elements in the other set that are not matched, for which there is no "one to one correspondence", then that set is larger and that side wins.

Imagine our Mongol Hordes lined up in front of our Great Wall of China.  Say each Horde member occupies a file one yard wide.  And, to keep things fair, we divide our Great Wall of China into segments that are one yard wide too.  So in a particular one yard wide file we look to see if their is a Horde member and a Wall segment.

If both are present it's Horde to Wall and both are annihilated.  If there is a Horde member present but no Wall segment then the Horde member wins.  If there is a Wall segment present with no Horde member in front of it then the Wall wins.  If this was somehow real then the rest of the Horde could pour through any breach in the Wall.  But we aren't going to allow that.  Everyone has to stay in their assigned file.

And we assume that the Horde spreads itself out so as to cover as many Wall segments as possible.  If we run out of Horde members before we run out of Wall segments then that means there are more Wall segments and the Wall wins.  Similarly, if there are more than enough Horde members to cover every segment of the Wall that means there are more Horde members and the Horde wins.

We now have a complete mental picture of what's going on.  So what does go on?  The process of placing Horde members in front of Wall segments is just putting elements of the Horde set into "one to one correspondence" with elements of the Wall set.  So what happens depends on the rule we use to create our "one to one correspondence".

We went into this in some detail in our "Infinity" post.  It is easier to deal with Aleph Naught sets so let's do that.  It also seems right.  The number of members of our imaginary Mongol Horde is a natural number.  We start counting, one two, three, . . .  With a real Horde we would eventually get to the last Horde member.  Thus the highest "counting number" we would reach would be some finite, specific, number.  But we extended out Horde to infinity so we would end up with Aleph Naught Horde members.

In a similar manner, we could measure the length of the Great Wall in yards.  With the real Wall we would eventually reach the end.  So we would stop at a large, but finite and specific, counting number.  But, again, we extended the length of our imaginary Great Wall to infinity.  So we would decide that our great Wall was Aleph Naught yards long.

We could now cut to the chase.  But that's no fun.  So let's imagine that each side has a commanding General.  And that General's job is to come up with a "one to one correspondence" rule.  (We already know that trying to do it by hand, assigning a specific Horde member to a specific Wall segment, is impossible.  So, we'll have to use the "rule" method.)

In any case, let's say the Horde General says "I am going to count my Horde using Integers but the length of the Wall is obviously a natural number.  Then I will match each positive integer on my side with a natural number on their side.  So when we are done I will only have used up the positive integers.  That leaves zero and all the negative integers on my side unmatched.  So I win".

Sounds like a plan, right?  But if the Wall General is smart enough he can overcome this strategy.  He can win even if he lets the Horde General have Integers and he keeps Natural numbers.  What?  Well, he could propose the following rule:  Match the natural number "2" with the integer zero.  Then match the natural number "4" with the integer "+1".  Now match the natural number "6" with the integer "-1".  Keep going.

Using this method we can create a "one to one correspondence" rule that matches only the even numbers in the set of natural numbers with all of the numbers in the "integers" set.  After we have done this each and every integer will be matched up with an even natural number.  But all the odd natural numbers are unmatched and, thus, left over.  So the Wall General wins.

Our Generals would spend forever arguing about which "one to one correspondence" rule to apply.  And that's why mathematicians decided that the cardinality of two sets was identical if even one "one to one correspondence" rule could be found that matched all elements of one set with all elements in the other set.

So the answer to our conundrum is that both sides are evenly matched so both sides would be totally annihilated.  Unless, of course, one General could show that the cardinality of his set was Aleph One while the cardinality of the other General's set was only Aleph Naught.

See, wasn't that fun?  And here you thought that the mathematics of infinity had no real world applications.

The Iowa Primary

Some TV talking heads were droning away the other day when a particular exemplar made reference to "The Iowa Primary", not once but twice.  Taking heads are employed and given air time because of their expertise, right?  Except there is no such thing as an Iowa Primary.  What Iowa does is an entirely different type of political cat.  It's a Caucus.  I like to do posts on subjects where I can clarify and illuminate.  This boneheaded bungle alerted me to just such an opportunity.

And I'm not going to dive into the issues or spend time explaining what I like or dislike about any of the candidates.  I am going to stick with the mechanics.  What exactly is going on?  Why does this and the New Hampshire Primary even exist?  What are we to make of them?

The Iowa Caucus is the bastard stepchild of the New Hampshire Primary.  So what's the story behind that event?  It has been going on for a long time.  And for a long time I am convinced that the only reason it got any coverage at all was so reporters could write off their vacations.

New Hampshire is not far from New York City, still the headquarters of the US press.  North of NYC is Vermont, home of various ski resorts that New Yorkers like to frequent in the winter.  Next to Vermont is New Hampshire.  I am convinced that writers would head to Vermont for some skiing.

While they were there they would make a very quick detour over to New Hampshire.  They would later whip together a "New Hampshire Primary" story which they would subsequently file after they returned to work.  This ploy allowed them to charge all, or at least a big chunk of, their ski vacation off as a "business" expense.

So, for a long time the New Hampshire Primary survived by being kept alive by the cynical desire of multiple NYC based reporters to be able to write off their ski holidays as a "legitimate business expense".  Then 1968 came along.  Lynden Johnson seemed poised to easily cruise to re-election.  Then something happened that the press decided was newsworthy.

As expected, he won the New Hampshire Primary.  So that in itself was not especially newsworthy.  But it was a solid win, not the landslide that everybody expected.  A nobody named Eugene McCarthy finished a strong second.  This marked the beginning of the end.  Eventually, Johnson decided to not even run for re-election and that's how Nixon became President.

After that, the New Hampshire Primary became, and has remained to this day, a big fucking deal, at least in the eyes of the press.  And, seeing how well it had worked for New Hampshire, in about 1972 some political types in Iowa asked themselves "is there any way we can horn in on this?"  And thus the Iowa Caucus was born.

New Hampshire had deliberately positioned itself to be the first step along the road to the White House.  That was an integral part of the scheme, a scheme whose real goal was to raise the visibility of an otherwise insignificant state.  Iowa decided to try to jump in front of New Hampshire.

If they could pull it off it would definitely be good for Iowa and, by implication, bad for New Hampshire.  To cut a long story short, after years of squabbling, the two states eventually worked out a deal.  Iowa would be the first Caucus and New Hampshire would be the first Primary.  And both of them would work together to thwart plans by any other state to horn in on the whole "First" thing.

And, from a historical perspective, Iowa turned out to be a much condensed version of New Hampshire.  No one paid much attention in '72.  But in '76 an unknown peanut farmer from Georgia won the Iowa Caucus.  That peanut farmer went on to become President Jimmy Carter.  And with that, Iowa also became a big fucking deal.

But, as I said, a Caucus is a different kind of cat from a Primary.  A Primary is easy to understand.  It's just an election.  Someone wins.  Someone loses.  End of story.  Caucuses are much more complicated.  More accurately, what we are actually talking about something called a "Precinct Caucus".

Most states are broken up into precincts.  (The rest have something that amounts to the same thing.)  Everybody in a specific precinct votes in the same place.  And, on Caucus night, voters in Iowa go to their individual precincts and "caucus".  Iowa has 1678 precincts so they have 1678 caucuses.  Each precinct has about 370 Democratic voters in it.  (There are 1678 separate Republican Caucuses happening at the same time but, like everybody else, I am going to focus on the Democrats.)

That means that there will be 1678 separate events in Iowa on Caucus night.  And to participate you have to show up in person.  And you have to make it to wherever your assigned precinct is meeting.  There is no such thing as an absentee ballot in a Caucus.  If you don't show up at the right place at the right time you don't get a say.

So what happens at a caucus?  Well, given that it's a political event, a lot of hot air is expended.  But for our purposes what happens is that at some point in the evening attendees are segregated into groups, one group for each candidate that has at least one supporter present in that particular precinct.

Or voters can join the "uncommitted" group.  In a caucus you can support "none of the above" or, more accurately, "a candidate to be decided upon later".  It is also important to know that this is all done in public.  There is no secret ballot.  You literally have to stand for your candidate, which might be "uncommitted".  But you have to publicly pick a side.

Then the 15% rule kicks in.  If your group does not represent 15% or more of the attendees at your particular precinct caucus then your group must dissolve and its members have to join one of the "15% or more" groups.  Or, of course, you can just go home, leaving behind your ability to influence the process.  So when that's done, we are all set, right?  Not by a long shot.

What each caucus actually does is select delegates to the "district" convention.  Each delegate is committed to vote for whichever candidate (or, in the case of the uncommitted contingent, a candidate to be named later) they stood up for at the caucus.  The Precinct Caucuses are just the first step.  They are followed by "District Caucuses" and finally "State Conventions".

The process at the district caucus is pretty much the same except that even more hot air is expended.  But the delegates that actually show up again group by candidate.  The 15% rule is then applied.  The result, after a whole lot of bickering and the expenditure of vast amounts of hot air, is the election of a slate of delegates to go to the state convention.

The composition of the delegation mirrors the relative strength of support among district caucus attendees.  There is typically some falloff in attendance between the precinct caucuses and the district caucuses.  There is a whole "alternate" business.  But in practice it doesn't work.  So some delegates from some precincts don't show up.  And if you don't attend you can't vote.

This is followed by the State Convention.  Delegates selected at the district level tend to be more dedicated so there is usually very little falloff between the district caucuses and the state convention.  There, the same "group by candidate" business again happens.  The 15% rule is again applied.  Finally, delegates to the National Convention are selected.

A Presidential candidate is actually selected by the delegates attending the National convention.  There are rules governing how all the preceding steps operate.  But their entire job is to eventually feed delegates to the National Convention.  Depending on the year, the Iowa delegation will or will not closely track the numbers from the precinct caucuses.

If all this sounds exhausting, that's because it is.  Until relatively recently this process was designed to find and promote committed political types who would provide the blood, sweat, and endless amounts of time necessary to run a political party.  And, as such, it worked pretty well.  But then this whole "picking a Presidential Candidate" business got grafted on top of it.  This is supposed to be a better process than the old "smoke filled room" method previously employed.

So, at this point (focusing on Iowa for the moment) we are getting all kinds of polling and "informed speculation" about what will happen on Caucus night in Iowa.  All this can be indicative but that's it.  Iowa may come out the way the smart money thinks it will.  Or it may not.

Beyond that, the press will assume (because that's what they do every single time) that whatever numbers emerge from Caucus night in Iowa will at least tell the complete tale of which candidates will get how many delegates "from the Great State of Iowa" at this year's Democratic National Convention..

In reality, we won't know what the correct numbers are until the State Convention happens many months from now.  The better observes will monitor the district conventions to see what happens there because changes may creep in.  The process is pretty predictable from there on so extrapolation from that point is justified.

But let me circle back to Caucus night.  As I said, there are about 370 registered Democrats in each precinct in Iowa.  Should we expect 370 people to show up?  Of course, not.  How about, say 180, about half?  (That's close to the percentage of registered voters that voted in 2016.)  Probably not.  The actual answer is "nobody knows".

Washington State has, for the most part, been a Caucus state.  That's why I know so much about the process.  In off years attendance at the Precinct Caucuses I have attended has been in the single digits.  In hot years it has run as high as 40-60.  So 40-60 is a good guess for Iowa for this time around.  But it might go higher.  People are extremely revved up.

And, for context, if 20 people attend then 15% is 3 people.  Also, for context, it is impossible for more than 6 candidates to simultaneously hit the 15% threshold.  There are 12 Democratic candidates still in the race, at least according to the New York Times, so that means that half of them are going to get aced out.  And if there are 3 undecideds then that's another candidate who won't get anything.  And if a candidate gets 30% that means another candidate gets the boot.  So in most precincts three or perhaps four candidates will make the cut.

But a candidate can make up for a loss in one precinct if the same candidate does very well in another precinct.  Candidates tend to do relatively better in some places and worse in others.  So there should be some averaging out.  But the 15% rule and other rules are there to funnel most support toward a few candidates, maybe even only one.

By now it should be obvious that in Iowa it's all about the ground game.  A supporter who stays home might as well not be a supporter at all, as far as the results of the Caucus are concerned.  A campaign expecting to do well in Iowa must be good at getting their supporters to attend a caucus.  And absolute numbers don't matter.  A thinly attended caucus in one precinct sends the same number of delegates as a heavily attended caucus in another precinct.  And all that matters is delegates.

Carter put a lot of effort into Iowa while other candidates didn't.  He was able to identify people who would support him and who would stick with him all the way through the process.  He made sure they showed up at caucuses.  Other candidates didn't invest as heavily in their Iowa "ground game" and they ended up losing out.  But everybody has long since figured this out.

This "strong ground game in Iowa" strategy works best in years when the energy level is low.  If other candidates are coasting along then a lesser known candidate can sneak in and run away with Iowa even if he doesn't actually have that much support in the state.  If the supporters of other candidates stay home then it takes only a small group of dedicated followers to make a big splash.

But, given how much coverage is already focused on Iowa this year, all the candidates know that having a strong ground game in Iowa is more critical than ever.  A whole lot of second and third tier candidates are hoping that their Iowa ground game will catapult them up in the rankings.  If that happens we will know within twenty-four hours of the end of Caucus night.

On the other hand, if you are a top tier candidate and you don't do well in Iowa, expect to see a lot of "candidate in trouble" ink immediately spilled.  The good news for these candidates is that Iowa is not the whole story.  If you "bounce back" in New Hampshire then a poor showing in Iowa will soon be forgotten.  If you are a top tier candidate and you do poorly in both Iowa sand New Hampshire, then you are toast.  Or so the conventional wisdom has it.

Both Iowa and New Hampshire are atypical states.  They are small and rural and white.  They have hung on to their prominence because of their track records.  Historically, if you do bad in both state events, you are toast.  Given that they are so atypical, lots of people have advocated for diminishing their power.  There is a simple way to do this.  Tarnish their track record.  And this is the year that might happen.

Biden has been "the one to beat" for more than a year now.  He could do fairly poorly in both Iowa and New Hampshire (not a prediction, just a thought experiment).  He is currently polling extremely well in South Carolina, the fourth of the "big four" Presidential contests that come before Super Tuesday.  (I'll get to what Super Tuesday is in a moment.)  Besides the two states I have already discussed, Iowa and New Hampshire, the third "big four" state is Nevada.  If he actually does well in South Carolina and then follows that up with good success on Super Tuesday then he could end up with the nomination.

An even more interesting possibility is Bloomberg.  He has made no secret of his strategy.  He is, in effect, skipping all of the "big four" early contests entirely.  Not surprisingly, the "smart money" expects him to do badly in all of them.  Instead, he is betting heavily on doing well on Super Tuesday.

Super Tuesday happens exactly a month after the Iowa Caucus.  The Iowa Caucus is on February third.  Super Tuesday is on March third.  On Super Tuesday twelve states hold primaries.  Included in this list are Texas, California, and several other large states.  So it differs from the "big four" events in pretty much every way possible.  They are "one state at a time" contests.  Super Tuesday is a "many states at a time" event.  They are small, homogeneous states.  (At least Iowa and New Hampshire are.)   Super Tuesday includes large states and states with diverse populations.

Combined, the number of delegates at stake on Super Tuesday is enormous.  Iowa and New Hampshire, in particular, will each send small delegations of delegates to the Democratic National Convention.  They have such an outsized impact because of the press coverage they get and not because of either's actual direct effect on the outcome.  (BTW, the Washington State Primary is on March 10, a week after Super Tuesday.)

If Bloomberg cleans up on Super Tuesday he will be in good shape to snare the nomination.  He has been spending heavily for several weeks already.  And he has enough money to spend everywhere, not just in the "big four" early states.  If he then goes on to win the nomination (or if Biden follows the path I have outlined above) then he will have shown that there is a path to the nomination that does NOT go through Iowa or New Hampshire.  And that will substantially diminish their clout.

Finally, there is a little noted impact of caucuses.  Besides being responsible for each state's role in selecting Presidential Candidates, this whole Precinct - District - State business is also responsible for putting into place all of the official that run the local, regional, and state party apparatus.

In 1988 Pat Robertson took advantage of this in Washington State.  On the Republican side it was a low energy year so few people showed up on Caucus night.  That allowed his small but committed group of supporters to take control of all of the Republican party apparatus in this state..

They sent a slate of delegates consisting almost entirely of Robertson supporters to the Republican National Convention that year.  But they also put their people in place to run the Republican party in this state.  Their control lasted for many years.

Pat Robertson was a major figure in the religious right.  His bid to snag the Republican Presidential nomination went nowhere.  But the Republican party in this state fielded candidates that were closely associated with the religious right long after Robertson and his people faded from the national scene.

They were able to do very well in various regions in the state but pretty much struck out in state-wide elections.  Our last Republican Governor was John Spellman.  He left office in 1985.  It's not that the Democrats keep putting strong candidates up.  It's that, due to the control exerted by the religious right, for a long time Republicans put up very weak ones.

Their lock on the levers of power within the state Republican party has finally weakened.  But then the state has drifted decidedly blue in the past few years too.  This has allowed Democrats to continue to have great success in statewide races.  But if Republicans put up strong candidates, who knows what will happen?

Anyhow, you should now have a deeper understanding of what's what with the Iowa Caucus.  Is that going to affect the outcome?  Nope.  But it should better equip you to deal with the BS that is and will continue to be spewed by people who are supposed to know what they are talking about.  That's all I can hope for.

To Infinity and Beyond

This post is about mathematics but it is aimed more at the "I'm bad at math" crowd than it is at those having substantial expertise in the subject.  That doesn't mean the mathematically inclined won't enjoy the post.  They should.  At least, that is my hope.  On to business.

All of us are exposed to Arithmetic from an early age.  That is, the basic business of addition, subtraction, multiplication, and division.  Most of us got some exposure to Algebra somewhere along the line in school.  For many it represented a fork in the road.  My reaction to Algebra was "cool".  That put me irrevocably on the path toward nerdiness.  Others found it a slog, or worse.  They became part of the "I'm bad at math" crowd.

Beyond Algebra is Trigonometry.  And beyond that is Calculus.  The big four:  Arithmetic. Algebra, Trigonometry, and Calculus, provide a "degree of difficulty scale ranging from Arithmetic (easy) to Calculus (hard) that we are all familiar with.  But there are other kinds of math.

I wasn't exposed to set theory until I got to High School.  But it has now migrated down to elementary school, or so I'm told.  Geometry seems non-mathematical but generally gets lumped in with math.  Logic is also some kind of weird step-child of math.  So these oddballs, set theory, geometry, and logic, are kinds of mathematics that many people have at least some exposure to but which they also view as something other than "pure" math.

It turns out that once you get past the basic four there are lots of these oddball branches and offshoots of mathematics beyond the three I listed above.  And these other oddballs and offshoots are generally lumped together as "higher mathematics".  The general perception is that all of them are really hard.  After all, they are called "higher" mathematics because, with few exceptions, no one studies them until after they have mastered Calculus and Calculus is really hard.

And, for many kinds of higher mathematics, this is a totally accurate characterization.  Matrix Mechanics makes Calculus look like a walk in the park.  Yet it is the mathematical foundation of everything Particle Physicists do.  And that's why 99.99% of us are not Particle Physicists.  But there are also parts of higher mathematics that are not all that complicated.  They are just ideas that regular people never get exposed to.  One of them is infinity.  And that's what I want to expose you to.

We all have some idea what infinity is.  It's the number beyond all numbers, it's so big.  Or, to put it into more concrete terms, you start counting.  You go one, two, three, and so on.  You keep going and going and going.  And at some distance beyond the last number you count up to is the last number, infinity.  It's bigger than any number you can actually count to.  That's people's idea of infinity.

People have an intuitive idea that "infinity" is a weird number.  But they tend to underestimate its weirdness.  For one thing, it's not a specific number.  Say it was a specific number.  Call it "I".  Now add one to "I".  This "I+1" number is also a specific number and it's bigger than infinity.  But infinity is the biggest number there is.  So that's impossible.  So forget about the idea of infinity as "a number like any other number".  It just isn't.  It's these sorts of problems with infinity that make it hard to wrap your arms around it.

But mathematicians wrestled with all these kinds of issues and found a way to work with infinity anyhow.  They resorted to set theory.  A "set" is just a group of objects.  One of the fundamental attributes of sets, the mathematical construct, is that you need to have a way to determine if a particular object, called an "element", is in a particular set or not.  There are several ways to do this but I am only going to very briefly cover the two most common.

First, you can just list all of the elements of your set.  You can say "my set consists of 'A', 'B', 'C', . . ., and 'Z'".  The second way is to provide a rule:  "My set consists of all of the letters in the alphabet".  For many small sets the "list all the elements" method works just fine.  But we are going to be dealing with large sets so we will use the "rule" method.  And the elements of the sets we are going to be talking about will consist solely of numbers.   So no "the set of all Presidents" or "the set of all cars I have owned".

Another attribute of a set is the number of elements in the set.  It may be zero.  The set may be empty.  That is perfectly legal.  But all of our sets will have elements in them, lots of elements.  In fact, they will have an infinite number of elements in them.  And the technical term for the number of elements in a set is its "cardinality".

And we can compare sets.  Two sets may be equal (each consists of exactly the same elements) or unequal (there is an element that is in one set but not the other).  We can also have subsets (all the elements in one set are found in the other set) or supersets (some of the elements in one set are not in the other set but all of the elements in the other set are in the one set).

There is lots more that could be said about sets but that's all we need.  Oh!  There is a bit more we need to know about cardinality.  If two sets are equal (contain exactly the same elements) then the cardinality of the two sets is identical.  But what about sets that don't contain exactly the same elements?

Let's say that we have two unequal sets but we want to know if the cardinality of both sets is identical.  If we can make up a rule that associates exactly one element of the first set with exactly one element of the second set then we can say that those elements have been put into a "one to one correspondence".

And let's say that we make up a rule that creates a "one to one correspondence" for every single element of the first set with an element in the second set.  And let's say the same rule can be used to do the reverse, to create a "one to one correspondence" for every element of the second set with an element of the first set.  And to be clear, there is only one "one to one correspondence" for each element of each set.  We don't ever use the same element twice.

Anyhow, if we succeed in doing this then the cardinality of the two sets will be identical.  And you are probably saying to yourself at this point "this all seems needlessly baroque".  But it is necessary to deal with the fact that infinity is not a single specific number.  With that out of the way, let's go to work.

The "counting" numbers:  one, two, three, etc., are called "natural" numbers by mathematicians.  So if we create a set containing all of the natural numbers its cardinality will be infinity.  That seems like a lot of work to get somewhere that seemed obvious from the get go, but trust me, it's all necessary.  Because now the fun starts.

The next step up in the hierarchy of "numbers" from the natural numbers is to "integers".  Take each of the natural numbers, and add "plus" to its name and add it to a new list.  Then add the new number "zero" to the front of the list.  Finally, take each natural number in turn again.  But this time add "negative" to it's name before adding it to the list.  So, add "negative one", "negative two", "negative three", etc.  And let the negative numbers increase as we move left and also put them on the front of the list.

Now, instead of having a list of "natural" numbers starting at "one" and marching off to the right as they get bigger and bigger, we now have a list with a central number, "zero", and two lists of numbers, one marching off to the left and the other marching off to the right.  It's just that the list of numbers marching to the right now have a "plus" added to each name and the list of numbers marching to the left now have a "minus" added to each name.  That all seems straight forward.  But now let me throw a monkey wrench into the works.

Let's make a second set consisting of all of the integers.  The question is:  Is the cardinality of both sets (natural numbers and integers) identical or not?  The obvious answer would be "no".  The natural numbers are a subset of the integers.  We can form a "one to one correspondence" between each of the numbers in the set of all natural numbers and an element in the set of all integers.  Just match "one" from the set of all natural numbers up with "plus one" in the set of all integers.

Proceed step by step with "two" and "plus two", "three" and "plus three", and so on.  (This would take an infinitely long amount of time.  But that doesn't matter.  Obviously the process works.  And that's all we need.)  We end up with everything lining up nicely with respect to the "natural numbers".  But at this point we also have "zero" and all the negative numbers left over in the "integers" set, and they are not matched up with anything.  So obviously the cardinality of the set of all integers is greater than the cardinality of the set of all natural numbers.

There is just this one tiny little problem.  Take "zero" from the "integers" set and match it up with "one" in the "natural numbers" set.  Take "plus one" in the "integers" set and match it up with "two" in the "natural numbers" set.  Now take "minus one" in the "integers" set and match it up with "three" in the "natural numbers" set.  Keep going matching "four" in "natural numbers" to "plus two" in "integers", "five" in "natural numbers" to "minus two" in "integers", and so on.

It turns out that we can create a "one to one correspondence" between the "integers" set and the "natural numbers" set.  So what should we do?  If we do the "one to one correspondence" one way we decide that the cardinality of one set is greater than the cardinality of the other set.  If we do it another way we decide the cardinalities are identical.  And, if we are clever, we can even come up with a rule for creating a "one to one correspondence" that makes it look like cardinality of the "natural numbers" set is greater than the cardinality of the "integers".

What's really going on all goes back to the fact that infinity is not a single specific number.  And what mathematicians decided to do about that was to decree that as long as at least one rule existed that put every element of one set into exactly one "one to one correspondence" with every element of the other set and that as long as the "one to one correspondence" went both ways then the cardinality of the two sets was identical.  To say that this is not intuitive is a vast understatement.  But there it is.  But wait.  There's more.

"To infinity and beyond" was the signature line of Buzz Lightyear, a character in the movie "Toy Story".  Anyone who has seen the movie (and everyone should) knows that Buzz was not the sharpest tool in the box.  Still, Buzz brings up an interesting question.  Is it possible for anything to be "beyond" infinity?  And, spoiler alert, the answer turns out to be "yes".  What's beyond infinity turns out to be infinity.  But this second kind of infinity is distinct from the first kind.

Mathematicians call this first kind of infinity, the one based on integers and natural numbers, "Aleph Naught".  "Aleph" is the first letter of the Hebrew alphabet.  "Naught" is just British for "Zero".  So, Aleph Naught can be roughly translated as "A Zero".  And you will also sometimes see "Aleph Null" or "Aleph Zero" used.  They are just variants for the same thing and "Aleph Naught" is the form most commonly used by mathematicians, so it's the one I am going to use.

And mathematicians asked themselves:  "is there a bigger infinity than Aleph Naught infinity?"  Their early tries at finding one were a failure.  Like us, they started with natural numbers and moved on to integers.  Their next step was to try "rational" numbers.

Any number that can be represented using the combination of an integer and a fraction consisting of a natural number in the numerator and a natural number in the denominator is a rational number.  Minus twenty-seven and five thirty-sevenths is an example.  So we get all the integers plus all the numbers between two adjacent integers that can be created by making use of an additional fraction.

And there would seem to be a whole lot more rational numbers than there are integers.  There are literally an infinite number of rational numbers between each and every pair of adjacent integers.  And remember, there are an infinite number of integers.  So it would seem that the cardinality of the set of all rational numbers would be infinity multiplied by infinity.  So the cardinality of the "set of all rational numbers" surely must be greater than the cardinality of the "set of all integers".

Alas, it was not to be.  I am going to skip over the details but some clever mathematician found a way to create a rule that put the "rational numbers" set and the "integers" set into a "one to one correspondence".  So, nope.  The cardinality of the rational numbers is Aleph Naught.  But wait.  It gets worse.

What we have been talking about so far can be thought of as the "number line".  All of the rational numbers represent dots on a single straight line.  How about if we go multi-dimensional?  How about a "number plane", or a "number space", or even a "number hyperspace" with many, perhaps an infinite number of, dimensions?  Still, no joy.  It turns out that there is a rule that will put all those points in all those dimensions into a "one to one correspondence" with our original set of integers.  And that means the cardinality of the whole mess is still Aleph Naught.

So we're screwed, right?  There is only Aleph Naught.  Wrong!  We just haven't gotten creative enough yet.  Beyond the rational numbers are the "real" numbers.  It turns out that if you take a continuous line and plot all the points on it that can be described by rational numbers (integer plus a fraction consisting of a natural number in the numerator and a natural number in the denominator) not all the points on the line are marked.

The rest of the points on the line represent the location of various "irrational" numbers.  (The "real" numbers are what you get when you combine all the rational numbers with all of the irrational numbers.)  And there are lots of irrational numbers.  The most well known irrational number is pi.

And what we generally associate with pi tells the tale.  Pi is equal to 3.14159 and on and on forever.  In fact, it doesn't matter how many digits we list, the value will still not be exactly correct.  The exact value of pi, expressed as a decimal number, is a number with an infinite number of decimal places.

But when it comes to rational numbers there is a trick.  With this trick we can represent any rational numbers with complete accuracy.  And to do this we only need to list a finite number of digits.  We do this by breaking the decimal representation of a rational number into two parts.  Each of these parts contains a finite number of digits so the whole contains a finite number of digits.

To represent a rational number with complete accuracy in the usual way requires an infinite number of digits.  But the decimal representation of a rational number contains a quirk that is always present. With an irrational number like pi the digits never repeat no matter how far you go.  With a rational number the digits start repeating if only you wait long enough as you work your way along the number.  And once a particular number has gotten into the repeating part that's all it does forever after.

We can use this quirk of rational numbers to our advantage.  The process is pretty obvious.  We break the number down into a non-repeating part on the front and a repeating part on the back.  We list the non-repeating part as is. It will usually contain an integer sub-part to represent the "whole" part of the number and the part of fractional part that does not repeat.  (Either or both of these sub-parts may require listing no digits.)

This non-repeating part is supplemented by a repeating part.  The repeating part consists of a finite number of digits that are repeated an infinite number of times.  Mathematicians indicate this by drawing a bar across the top of the first set of repeating digits.  Then they just drop the rest of the number.  Note that we are getting rid of an infinite number of digits while retaining complete accuracy.  Nice trick, isn't it?

So, in example above (minus 27 and five thirty-seventh's) we would have "-27.".  That's the non-repeating part.  This would be followed by "135" with a bar over it (or so my calculator tells me).  I don't know how to get the blog formatting software to put the bar over the three digits so you'll just have to use your imagination.

As indicated above, it turns out that any rational number can always be rendered exactly in decimal form by splitting it into the non-repeat and repeat parts.  (BTW, sometimes the repeating part is just "0" repeated an infinite number of times.)  I'm not going to go into the process necessary to turn a rational number into a decimal number in this "non-repeating part plus and infinite number of repeating part" form.  Nor am I going into how to do the process in reverse.  Just trust me that it can be done.

Now we are in a position to note that an irrational number is one that has no part that repeats indefinitely when rendered as a decimal number.  Given two specific rational numbers rendered in decimal form it is easy to see how to construct lots of irrational numbers that lie between the two.  (Hint:  The rational numbers have repeat parts.)

Annoyingly, it is also easy to construct as many rational numbers as we want to, such that they lie between any two given irrational numbers.  You just note the early parts that coincide.  Then, starting with the decimal place where they diverge, you just toss in repeating sections that result in numbers that lies between the two.  This is another example of where this whole "not a single specific number" business makes life complicated.

Anyhow, a mathematician named Gregor Cantor came up with a way to crack the case by making use of the fact that real numbers can be represented accurately as decimal numbers if you are willing to include an infinite number of decimal places.  In 1874 he came up with a proof that a number bigger than Aleph Naught existed.

The easiest way to explain what he did is now referred to as the "diagonal argument".  Write down a list of all real numbers in decimal format.  Don't forget to include all of the infinite number of digits necessary to represent them with complete accuracy.  It doesn't matter what order the list is in.  It is just important that the list includes ever single real number.  (This is impossible to do in real life but this is what is called a "thought experiment".)

Now consider the first number in the list.  Change the first digit in the number to anything other than the correct digit and save it as the first digit of a new number we are constructing.  Move on to the second number.  This time change the second digit to anything other than the correct digit.  Again save this digit as the second digit of the new number we are constructing.

Move on through the list.  For each number we change the next digit along to be anything other than the correct digit and save it in the correct place in the new number we are constructing.  Again, this is impossible to do in the real world but the process is easy to understand.

When we are done (we never will be but, again, thought experiment) we will have constructed a number that does not exist anywhere in our original list.  (Remember, we constructed the new number so that it differed from every single number in the list of real numbers by at least one digit.)

But this is impossible.  Our original list was carefully constructed so that it included each and every real number.  This contradiction means that the cardinality of the set of all real numbers is greater than the cardinality of the set of all integers.  So there is an Aleph One and it is bigger than Aleph Naught.

So there are at least two infinite numbers.  There is Aleph Naught and Aleph One.  And it turns out that there is a way to construct Aleph Two and so on.  (I'm not going to go into the process because, frankly, I don't understand it.)

So there you have it.  There is a "beyond" beyond infinity.  "Infinity" presumably refers to Aleph Naught and there is an Aleph One that is "beyond" Aleph Naught.  You have now mastered some way cool mathematics that is so far beyond Calculus that you could say "it's infinitely far beyond Calculus".  So go forth and win bar bets and wow your friends and family at parties.

Pretty cool, hunh?