Artificial Intelligence > Infinity Paradox

Infinity Paradox

By LEONARD MCGAVIN
Published: May 20, 2008

Infinity is an abstract concept of unboundedness which can perhaps be paralleled by the realm of our own Universe. In mathematics, infinity can take different forms depending on the situation in which it is used. It generally marks a limit of a value but it is not a limit in itself. It is paradoxical concept that allows the paradox - of there being no such thing as a final number - to not get in the way of some great concepts.

When looking at numbers, infinity pokes its head up quite often. There is positive infinity, negative infinity, infinity between 0 and 1 in a fractional sense - e.g. 0.0001..., and there is also infinity in a constant such Pi. Division by zero also seems to mess with the infinite. For the sake of sanity, the paradox that is infinity is abstracted out which allows mathematics get down to the nitty-gritty.

Can there be so many infinities as mentioned above or are they all belonging to same infinite but just viewed differently? The latter perhaps seems more likely. While not trying to jump to a conclusion too quickly, perhaps the infinite we are trying to model is that of the infinite we face in our environment - the Universe itself.

If Pi is a constant and an infinity at the same time, is not the value "1" much the same as Pi - e.g. 1.00000...?

It could be said that infinity still belongs to each value in mathematics but does it make any sense to focus on this fact or realise it in anyway? Perhaps not in most fields, but artificial intelligence might just be concerned with such issues.

Comments

1. bay on May 29, 2008

I think they are just many people's perceptions of the same concept. Of course it can be viewed and explained in so many different ways, like feelings and emotions, it is intangible and unending. How one person views happy is very different from how another person does, but in the end we are all trying to describe the same thing.

2. joshcombes on January 15, 2009

Hi Luke,

Here are some comments to add to the cauldron that is flickspin. Sorry they are not so coherent or eloquent .... time, for me, is a very finite resource :) Let me make a disclaimer first: I am not a mathematician, although I have studied some mathematics.

The number of digits that Pi has is equal to the number of natural numbers (counting numbers). In mathematics this notion is captured in what is called a cardinal number. The cardinality of the natural numbers is usually denoted as Aleph_0 (subscript naught). The unit interval (the numbers between 0 and 1) has a cardinality c = 2^Aleph_0 (two to the power of Aleph_0). Clearly these two infinities are different at the least in magnitude.

I am not sure if it makes sense to talk about a positive infinity being different to a negative infinity. The reason why is i can just relabel the axes in the opposite way. For example, look at an object that is on your left, then turn your body 180 degrees. What was on your left will now be on your right. It seems that positive and negative seem only to denote direction.

Pi is a constant. What does that mean? It means that it is more convenient to define a symbol that represents the number and use that symbol in our calculations. The term constant does not have any special connotations. I could equally define a constant, for example the greek letter alpha, to be equal to the number 1. There is no advantage to this. Constants like ‘e’ and ‘Pi’ are convenient definitions because they not be represented as a fraction.

Im not sure exactly what you are getting at with the ‘1.00000’ example but I will take a stab. Usually when scientists write ‘1.00 Volts’ the digits refer to the number of significant figures a quantity has been measured to. This reflects the confidence they have that their number is correct. If you are talking purely mathematically then 1 = 1.0=1.00=1.000 and so on. The adding of additional zeros does not change the number. If they are place holders for a digit other than zero then the number is not one. Just for further thought it can be shown that 0.9999...( point nine repeater) is equal to one. There is a wiki article on this :)

Ok bye for now.

Any Comments?


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