29 Jun 10
What is infinity to you? Also, your reasoning behind it.
To me, infinity is not a single number, but it is fact a range of numbers, starting immediately after the largest number the human brain can comprehend and never ending. Thus, you cannot add, subtract, multiply, divide, mod, et cetera infinity. It is simply used for classification, ex. what is the size of the universe, it is infinity, implying that we cannot comprehend it's value.
29 Jun 10
Originally posted by ChaosPhoenix7I am sure that in some circumstances you can use the word 'infinity' to mean 'an incomprehensibly large number'.
What is infinity to you? Also, your reasoning behind it.
To me, infinity is not a single number, but it is fact a range of numbers, starting immediately after the largest number the human brain can comprehend and never ending. Thus, you cannot add, subtract, multiply, divide, mod, et cetera infinity. It is simply used for classification, ex. what is the size of the universe, it is infinity, implying that we cannot comprehend it's value.
But in mathematics that is not the case.
In mathematics, infinity is not a number at all. It is a symbolic representation of a concept.
I must also point out that there are different infinities. There are for example countable infinities and uncountable infinities.
Of course Wikipedia does a far better job than I can:
http://en.wikipedia.org/wiki/Infinity
It would be mathematically incorrect to say that the size of the universe is infinite - unless it truly is infinite ie boundless.
It is also not true that we cannot comprehend the size of the universe (if it is finite). We may have trouble visualizing the number of metres involved, but we would have no trouble with defining a new unit equal to the diameter of the universe (DU). Thus we might find the distance between one super cluster of galaxies and another was 0.001DU
So we would still be able to add, subtract, multiply and divide 1DU, so it does not have the properties you claim for infinity.
29 Jun 10
Originally posted by adam warlockI also realize that a bounded set can be infinite eg the set of Reals [0,1]
Mathematically speaking boundless and infinite aren't equivalent terms. The surface of a sphere is boundless and finite for instance.
I realize that they are not equivalent, but they are strongly related. You will see on the Wikipedia page for infinity that 'boundless' is used quite often.
But I think that in the context that I used it, it was correct, though I guess misinterpretation is still possible.
29 Jun 10
Originally posted by twhiteheadAll of that is very true, and I haven't seen the wikipedia article so I can't comment if they use the term boundless in a rigorous way or not, but the example you gave plus the example I gave show that even though they are related one has to be careful when using them in a precise mathematical sense.
I also realize that a bounded set can be infinite eg the set of Reals [0,1]
I realize that they are not equivalent, but they are strongly related. You will see on the Wikipedia page for infinity that 'boundless' is used quite often.
But I think that in the context that I used it, it was correct, though I guess misinterpretation is still possible.
One other thing I like in topology is that a set can be open and closed at the same time.
29 Jun 10
Originally posted by PalynkaBoundless in the case of a sphere I think is like this: Suppose you are on the equator of said sphere, like the Earth. It has 360 degrees as defined by geometry and trig.
What do you mean boundless? It's a bounded space, to my knowledge.
So you can start traveling on the equator and go through every one of those degrees but you can also keep going for an infinite # of trips around the equator thus making possible a trip of infinite duration, the finite nature of the Earth notwithstanding.
29 Jun 10
1 edit
Originally posted by sonhouseRight, but that's not a formal definition of boundedness.
Boundless in the case of a sphere I think is like this: Suppose you are on the equator of said sphere, like the Earth. It has 360 degrees as defined by geometry and trig.
So you can start traveling on the equator and go through every one of those degrees but you can also keep going for an infinite # of trips around the equator thus making possible a trip of infinite duration, the finite nature of the Earth notwithstanding.
29 Jun 10
3 edits
Originally posted by adam warlockThere's a bit of confusion here. You can have bounded sets without boundaries (like spheres) and unbounded sets with at least one boundary (like [0,\infty] ).
The surface of a sphere has no bounds (i.e. no limits): http://www.bartleby.com/173/31.html
What he means there is without boundaries.
29 Jun 10
1 edit
Originally posted by twhiteheadTopologically speaking the surface of a sphere is closed and not open. So in what sense do you say that that it is continuous and open?
I am still not convinced that a sphere is boundless. Surely, whatever co-ordinate system you use, there will be an upper bound and lower bound on each co-ordinate?
I would agree that it is continuous and open, but that is not the same as boundless.
This is not a matter of you being convinced. It's a matter of precise mathematical definitions.
29 Jun 10
Originally posted by PalynkaI don't think I understood you. The surface of a sphere does have a boundary. And a sphere does have a boundary too. I'm using regular topological definitions.
There's a bit of confusion here. You can have bounded sets without boundaries (like spheres) and unbounded sets with at least one boundary (like [0,\infty] ).
What he means there is without boundaries.
Is this a case of the same words being used in different contexts type of confusion? 😕