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Cantor: The Uncountable Continuum, part 1

Non-Quantitative ChangeLaRouchePAC

Cantor: The Uncountable Continuum, part 1

The problem of comparing different domains of possibility different infinites is a difficultone. We'll make use of a powerful tool developed by Georg Cantor, for the demonstration ofdifferent sizes (cardinalities) of infinites. This video covers the content of this section, but it alsogives away some answers. It's up to you whether you'd rather start with the video or the webpage.

To get into Cantor's concept, we'll start with some experiments you can participate in. To start, gothrough the following images in order. For each one, figure out whether there are more uppercaseor lowercase letters:

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Im sure you noticed that the examples got easier and easier, first through the introduction ofcolor, then by coupling, and then by arranging them neatly. In the last example, it is very easy tosee that there are more lowercase than uppercase letters, because the g is all by itself, without acompanion uppercase letter. Try looking through the examples again. In the first two examples,you probably counted up all the uppercase letters and lowercase letters separately, and comparedyour totals. In the third and fourth examples, since the letters are coupled, you just have to look forany solo letters, which makes the extra q easy to spot. And in the fifth case, there's no trouble atall: the lonely g stands out.

Now, were going to expand the experiment just a little bit, as we move from finite numbers of

objects (in this case, letters), to infinite numbers of objects, as we look at numbers themselves.

Counting infinites

Lets start with this set of numbers. As you can see, there isnt enough space to write them all, butthere are enough to give you the idea. Now, how many numbers are there in the idea of the seriesillustrated here?

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Thats right, an infinite number. Now, lets compare that to another unending group of numbers,the even numbers:

Again, the number of even numbers is... infinite! Would it be possible to ask whether its the sameinfinite? It seems reasonable to think that the number ofall numbers is larger, twice as big, as thenumber ofeven numbers. After all, there are all those odd numbers that arent found in the secondset! Heres where a difference between finites and infinites comes in the way we count thingscan change how many we have. What if we align the numbers like this:

Now, every whole number has its even number partner, just like the uppercase and lowercaseletters in the fifth letter example above. But unlike that example with letters, here there is nosolitary g, no extra whole number without a partner. That is, we can't point to any whole numberwithout its even double, or any even number without its half. Since all numbers are in such

partnerships, there cant be more whole numbers than even numbers! Pretty shocking, isnt it? Ofcourse, we are dealing with infinites, so perhaps the rules have to change.

Next:Cantor: The Uncountable Continuum, part 2

Sections

Welcome

IntroductionWhat is Quality?Cantor: The Uncountable Continuum, part 1

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Cantor: The Uncountable Continuum, part 2Bernhard RiemannComplex MagnitudesMultivalued Functions : IMultivalued Functions : IIDirichlet's PrincipleConnectivityEconomic ReflectionsFarewellBy the Author

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