Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
General Math
Are there an infinite number of infinities?
Reply to thread
Message
[QUOTE="TeethWhitener, post: 6830097, member: 511972"] Generally, when people want to discuss sizes of sets, they use bijections. Basically, if you can think of a function from one set to another such that each input has one output and each output corresponds to one input (i.e., the function has an inverse), then the sets are the same “size.” This is called the cardinality of the set. So for instance: ##A=\{1,2,3\}## and ##B=\{4,5,6\}## have the same cardinality because there is a function ##f:A\mapsto B## that maps each input of ##A## to one, and only one, output of ##B##, namely ##f(a)=a+3##. For infinite sets, the same rules apply. Because we have a function from the set of all integers to the set of tenths of integers (namely, ##f(a)=\frac{a}{10}##), which has an inverse from the set of tenths of integers to integers (namely ##f^{-1}(b)=10b##), we say that the two sets have the same cardinality. In fact, it should be clear that this argument works for any finite decimal expression. They therefore have the same cardinality as the set of integers. A more complicated argument can show a bijection between the integers and all rational numbers. However, the set of real numbers has a strictly larger cardinality than the set of integers, which was a cornerstone result in modern mathematics first discovered by Georg Cantor. [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
General Math
Are there an infinite number of infinities?
Back
Top