infinity in numbers

[MAss]

also, that an exponential curve runs along the axis of time, whereas the simple conceptual number 0.999... exists independently of time...

so we can only ever see the curve completed, by considering the numeric example..

[Symptom777]

The idea is to show that by considering it in different ways we come up with different viewpoints; however in this case the tortoise is bluffing!

[Symptom777]

There are at least two infinities: countable and uncountable.
Countable sets can be mapped 1-1 with integers (which we can count), so are countably infinite. Uncountable sets can't be mapped 1-1 with integers, so are uncountably infinite.

[Bobbo]

If MAss wants to deal (mathematically) with "real" objects in the "real" world isn't integer (quantum?) math required to achieve exact results? I was taught that the calculus (his examples are at the threshold) yeilds approximation.

The way I understand it:
Fractions of things don't exist. If one cuts an apple into three equal parts, he is left with three objects. Each may have one third of the mass, volume, etc of the whole, but each is still a complete object.

If one "thing" is partitioned equally, the whole exists only as two or more parts, each part being one separate "thing". Ditto until further division is not possible. If one attempts to accumulate like elements into a set, and adds elements such that each new addition is smaller than the previous, the effort will be impossible unless all of the elements are composites of the smallest. Any "thing" thus achieved and recognized as a "whole" will be a composite of a finite number of elements.

It order to obtain a true graphical representation of the relationship or behavior of "real" objects, the scales must be in integer units and the plot must be dots not connected (discrete objects, discontinuous functions). One (1) will be the attainable lower limit and some finite number (the total of the elements to be considered or the duration of the exercise) will be the upper limit. Note that zero and the upper limit plus one are out of bounds and are not attainable.

MAss, if you're just playing with equations, fractions and number systems, I hope you have fun. I have. I would caution against believing that numbers and equations exist. Pretty to look at, but you can't hold one in your hand.

One "trick" I like is to calculate 1/7^2. It falls out as SUM (2 / 100)^x where x = 1 to whatever. I am amused that one (1) divided by "lucky 7" squared looks like some sort of degenerating geometric progression. I'm almost certainI used to know the high class name for this construct but I've been away from it for too long.

[Symptom777]

Originally Posted by Bobbo

One "trick" I like is to calculate 1/7^2. It falls out as SUM (2 / 100)^x where x = 1 to whatever. I am amused that one (1) divided by "lucky 7" squared looks like some sort of degenerating geometric progression. I'm almost certainI used to know the high class name for this construct but I've been away from it for too long.

That's cool - how come my maths' teacher never taught me that? Too bust doing topology by taking off his waistcoat without removing his jacket, I guess.