A big name-drop during the middle of the Russell episode was the sad story of Georg Cantor and his insanity-inducing continuum hypothesis. Anyone unaware of Cantor and his contributions might want to look at this clip from the Dangerous Knowledge BBC documentary. I thought it provided a good visual explanation of higher levels of infinity. But perhaps they horribly oversimplified it for the sake of television -- mathematicians, share your thoughts!

If you like the clip, you can find the whole episode out on the wild web.

-Daniel Horne

Prior_Analytics says

I wont claim to be a mathematician, at work, I call myself an arithmetician, which is all I think anyone needs to start to understand the infinity.

On that point, let me take the liberty of tweaking the topic, as I think in the godatheist debate, the ‘concept of infinity’ is one that very few, if any, atheist ever quote properly.

note: I don’t know Georg Cantor, nor his mathematics, but I can gather enough of how he reasoned from the short clip.

A few observations.

1) Yes, infinity exists, take all numbers evenly devisable by 1024. This is an infinite set of numbers.

2) Yes, a hierarchy of infinities exist. You can accelerate to infinity faster by using all numbers ‘equally divisible by 512’, or by 128, or by 32, or by 1, or by .5, or by 0.015625.

3) You can also take any two numbers, divide them an infinite number of times. Between any two numbers within this set, you can start the process all over and still find another set of infinite numbers.

4) You can combine any two infinite sets of number, and create a 3rd set which also has an infinite set unique of the starting sets. For example, take all the rational numbers between 11 and 12. Then all of rational numbers between 21 and 22. Now multiply all of those in the first set with all of those in the second set and you will get an infinitely larger set of numbers that was not in either of the first two sets.

5) What is true for rational numbers is also true for irrational numbers, and all possible combinations of rational and irrational numbers. Again, take any two, divide the number between then an infinite number of times, and, you can repeat the process with any of the numbers in any of these sets.

6) I could go on for an infinity, but I think you get the point.

So where do atheists get the ‘concept of infinity’ wrong? In every debate that I have ever seen that debates the ‘infinity of god’ the assumption is made that infinity has no bounds, no limits, and is all encompassing.

But, this is a false concept of infinity. All rational numbers between 11 and 12 is an infinite set of numbers, but has very clear limits, very clear bounds and is anything but all encompassing.

Most atheists, use the term infinity, in a way that suggests that if the set of all rational numbers between 11 and 12 does not include the number 6, then it is not an infinite set of numbers. A conclusion, that could not be more false.

In fact, just to press the concept, it is possible to create an infinite number of unique and non-overlapping sets of rational numbers, using nothing more than the rational numbers between 11 and 12, and none of any of these infinite sets of infinite numbers, would include the number 6.

Likewise, on the topic of knowledge, there is an infinite number of unique sets of infinite knowledge available to anyone willing to do the work, by studying nothing more than all of the infinite numbers of any one of the infinite number of sets of both rational and/or irrational numbers between 11 and 12. And the fact that someone does not know the number 6, is insufficient to prove that their knowledge is not of an infinite set of facts.

And, infinite ability? The mere ability to know any one of the infinite sets of numbers between 11 and 12, would require an infinite capability.

It could not be done with every last computer in the World, it could not be done if you used every atom in the universe to create the largest database conceivable.

Take every last atom in the universe, build the largest most efficient database conceivable, and it could not hold even one complete set of an infinite set of numbers. Dispite the fact that an infinite number of these infinite sets exists between the numbers 11 and 12, and every other 2 numbers between negative infinity and positive infinity, counted in the smallest of conceivable increments.

And, even if you could build a database that could hold just one set with an infinite number of members, the fact that when you ran a database query for the number 6, and the results came back null, this would be insufficient proof that the set ‘did not have an infinite number of members’ within the set of all numbers within that database.

To disprove that god had an infinite number of powers, or to disprove a lack of infinite knowledge, one would need to show more than the fact that god didn’t know the number 6 nor could calculate 2×3=6.

In both the case of god, and the case of an infinite set of numbers, the exclusion of a particular item from the set, is insufficient to disprove it’s attribute of infinity…..

The only way to prove that something is not infinite is to prove that it is finite, and this can not be done by showing that something has even the smallest of limits or bounds.

-p_a

Michael M. Morbey says

Cantor in context – Quotes by Georg Cantor

http://www.braungardt.com/Mathematica/quotes_by_georg_cantor.htm

Tom McDonald says

Wow. They watch stuff like this on TV in the UK? Thems smartlike people.

Tom McDonald says

But with Cantor’s maths we can’t count all the beans!

Gary Geck says

Don’t forget my site garygeck.com ‘s series on Gerog Cantor http://garygeck.com/?page_id=44

Daniel Horne says

Too cool, Gary, thanks!

J. says

The reasonable person objects to Cantor’s neo-collections of infinities. Why? He just does. At least he asks… shouldn’t mathematics do things? Integrals work–the functions apply to real objects, to problems in the world, not a platonic realm. They help us build bridges, or rockets or CPUs. Who cares about the

mere abstractions (granted Russell was a bit guilty in that regard, with his love of pure mathematical logic, without regard to applications)

Steven Collins says

It’s not as though Cantor was just off having a wank one day and decided “Hey, I should study crazy infinities!” He discovered the transfinite numbers as a result of attempting to refine a theorem about the convergence of trigonometric series, a subject that certainly has some real-world value.

These ideas came up naturally in the course of attempting to describe various sets of real numbers (and this specific venture, descriptive set theory, is still very active, *and* applicable to other branches of math, today). The higher infinities are there, in the modern conception the real line; that conception may be wrong or misguided, but it’s the one that all the mathematicians and scientists and engineers use today.

Daniel Horne says

Very good point, Steven, I was not aware of that!

J. says

neo-platonic collections, that is. (anyway see Quine. not that he’s the final word, philosophically speaking–but did not accept the pure realism of Cantor)

Jorge Videla says

I’m an actuary with an UG degree in maths. “Higher infinities” is an extremely stupid idea. It has absolutely no use. Look up finitism.

Gary Geck says

J, I’ll address Quine in a future video in my series.