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.