Exploring Infinity: Proportional Subsets in an Infinite Set

[Imparcticle]

By definition, an infinite set is a set whose subset is proportional to the set which contains it.
The sequence of numbers whether it be expressed as (n+1) or not, is infinite. Then (I am veturing into grounds I know little of here...) I am guessing it is safe to say that the numbers (n+1< or = to 10 <0) or 1, 10 and 1 through 10 are subsets of the the infinite sequence of numbers. If that is true, then by the definition I stated in bold, it is also true that the numbers (n+1< or = to 10 <0) are proportional to the infinite set which contains the aforementioned numbers.
Okay, I am 99.9% sure I'm wrong here mainly because I have no knowledge of set theory besides what it is, and the definition of "infinite" according to a NOVA special. In addition, I may want to highlight that I am merely a freshman in high school so please don't make your explanations too complex for a student of geometry such as myself. I was just curious about this so I'm asking.

thanx.

 

[matt grime]

That isn't *THE* definition of infinte. That is the definition of Dedekind infinite.

By " the sequence of numbers", what are you referring to? What sequence, what do sequences have to do with anything? Do you just mean the set of Natural numbers?

(n+1<=10<0) makes no sense. 10 is strictly less than 0. What do you mean?

What does proportional mean? In bijective correspondence? Then say so, I think you need to state things far more clearly.

A set is Dedekind infinite iff there exists a bijection from it to a *proper* subset. That is it can be put into 1-1 correspondence with a proper subset of itself.

N, the *set* of natural numbers is infinite with this definition since the map

n->n+1 is an bijection from N to N\{0}, ie the natural numbers and the natural numbers less zero.

 

[Imparcticle]

sorry about the late reply...

That isn't *THE* definition of infinte. That is the definition of Dedekind infinite.

what's Dedekind infinite?

By " the sequence of numbers", what are you referring to? What sequence, what do sequences have to do with anything? Do you just mean the set of Natural numbers?

all real numbers. (for example, 1,2,3,4...)

what's bijective correspondence? I mean proportional like the relationship shown here: 1=1. I'm not too familiar with set theory, so I am unaware of what degree of clarity I should describe something. Please just ask questions and I'll answer and thus learn about how clear I should be.


That's all I have time for right now. I will see if I can continue tomorrow.

 

[matt grime]

sorry about the late reply...

what's Dedekind infinite?

Erm, a set is dedekind infinite if there is an injection from it to a proper subset of itself, just like i said, and just like you said. Properly, a set is infinite if it is not finite. It's easy to show this is the same as Dedekind infinite provided you use a certain technical axiom that some people feel is best avoided. (the axiom of choice, for completeness). obviously a set possessing an injection to a proper subset of itself is infinite, but it's harder to show that an infinite set must have such a map.

all real numbers. (for example, 1,2,3,4...)

That isn't the set of real numbers, it's the set of natural numbers.

what's bijective correspondence? I mean proportional like the relationship shown here: 1=1. I'm not too familiar with set theory, so I am unaware of what degree of clarity I should describe something. Please just ask questions and I'll answer and thus learn about how clear I should be.

You need to learn the meanings of the following terms:

injection (aka one to one)
surjection (aka onto)
bijection (one to one and onto)

as well as the idea of cardinality for (infinite) sets.

I'd suggest google and wolfram were your best bet since these are well known terms and there's no need to go into another explanation of them here.

proportinal relationship has no commonly understood definition as far as i know in this area.