Okay this is just a musing I'm having and it might be already talked about or proven or whatever.
If G is an infinite set and H is an infinite subset of G, can a bijection occur between G and H?
In this case G does not equal H.
For example the set R, of real numbers and Z the set of integers. Both sets are infinite, however R has a higher level of infinity than Z, obviously. So can a bijection exist between one such infinite set and its infinite subset?
Ah but for example:
0<x<1 where x is an element of R, is an infinite set. Then consider N, the subset of R of natural numbers. N has a cardinal number d, but the infinite set 0<x<1 has a cardinal number c that is greater than d as the set is nondenumerable. Therefore there couldn't be a bijection between these two sets.
I get what you are trying to say about X-A also being infinite like X, and an infinite set was actually defined once as a set that is equivalent to a proper subset of itself.
My point is simply that the argument doesn't always work. A scale of arithmetic infinites exists.
I understand up to here correctly.However.....
What do you mean the argument doesnt work? If there is a bijection between two sets, doesnt it mean that they have the identical cardinal numbers?I get what you are trying to say about X-A also being infinite like X, and an infinite set was actually defined once as a set that is equivalent to a proper subset of itself.
My point is simply that the argument doesn't always work. A scale of arithmetic infinites exists.
I dont understand what doesnt work... Can clearly spell it out for us?
Thanks
I never said there was a bijection between the two sets. I'm just having musings upon bijcetions between infinite sets.
Here's an example: The set of all algebraic numbers is denumerable. The set of complex numbers is not denumerable. Thus there must exist complex numbers that are not algebraic. These are transcendental numbers. One example is e.
I'm just voicing some inner thoughts on the crazy infinite set theory.