Is it really possible for there there be different sizes of infinity? Ricky Chubbs
IntroductionI decided to take advantage of the sample topicsand I found one that really caught my eye. Sizes ofinfinity? How can you compare two things that areinfinite? How would you measure infinity? Is itreally possible to have different sizes of infinity? ∞ - ∞ = 2? :S
What is infinite and finite?Infinity essentially means limitless, as oppose to it’sopposite anything finite, infinity is unbounded.Infinite can represent just about anything, but justto make things a little simpler we’ll just be treatinginfinity as a set of numbers. Infinity and well.....Not infinity :P
Georg CantorGeorg Cantor is one of the main contributors to the theory of having sizesof infinity. Instead of accepting the traditional theories concerninginfinity, in the 19th century he and some other mathematicians formulatedthe set theory. The set theory suggests that the cardinality of different setsof infinity may differ. George Cantor invented a collection of transfiniteordinal and cardinal numbers that would describe the order type and anumber set that doesn’t cover an absolute infinity.
CardinalityCardinality represent the number of elements in aset. For example:We have the set [1, 2, 3,]. What is the cardinality?The Cardinality Is 3! There are three numbers in theset, therefore, the cardinality is 3.But now how will this apply to infinite sets. More examples: If we have [2, 4, 6, 8,] Cardinality is equal to 4 If we have [3, 4, 5,] Cardinality is equal to 3
Cardinal and Ordinal numbersI mentioned Georg Cantors Cardinal and Ordinalnumbers earlier but I thought I improve of that justto help understanding.An ordinal number would be the order or way ofcounting the number within a set.A cardinal number is the number of elementswithin the set. Example using the positive set of integers (natural numbers) Ordinal number under the traditional order is: ω Cardinal number is: Aleph Null (unfortunately I couldn’t find the appropriate symbols)
Dedekind’s ApproachAnother very accomplished mathematician concernedwith the set theory is Richard Dedekind. Dedekind’sapproach takes place mainly by using one to onecorrespondence in order to formulate a size using thecomparison of the 2 sets. The way I like to think aboutit is to think that you’re partnering each number upwith itself then eliminating them. Which ever set hasthe most remaining cardinal numbers was the biggersize of infinity. Set 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 Set 2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23... Bigger
Still a little weird? Imagine having two enormous containers one holding apples and the other holding oranges. If you remove one apple and one orange at a time, at the end, we will either be left with apples, oranges, or nothing. Which ever crate contained more will have leftovers and it’s this set that will be greater in size. I know it’s a poor drawing :PNotice the same amount of apples and oranges are removed yet some oranges remain in thecrate. This obviously shows that there were more oranges than apples.
Countably infinite?The are many different sets of infinity, the smallestone being the infinite set of natural numbers;(1, 2, 3, 4). The infinite set of natural numbers are aset of ordinal numbers that can easily becounted, although cardinality may not be a specificnumber this set could be counted one after anotherwithout a problem. One the other hand we alsohave sets of infinity that are too extensive to count. Few examples: Infinite set of natural numbers is countable Infinite set of both positive and negative integers is countable Infinite set of all the real numbers is uncountable.
Reals vs. Naturals As you may know both real numbers and natural numbersare sets of infinity when unbounded. Cantor used theargument that there is a higher cardinality for an infinite setof all real numbers than there are is for an infinite set of allnatural numbers. Since the natural numbers miss both therational numbers and the negative integers, the infinite set ofreal numbers is obviously greater in size the natural numbers.Just for the sake of saying it, there is an infinite amount ofirrational numbers just between the numbers 1 and 0. And I think we just answered our question. So far the theory looks good, it makes sense and I personally agree, it could be possible to have different sizes of infinity.
ConclusionIn conclusion we now have a basic understanding ofhow there can possibly be more than 1 size ofinfinity. All this theory points out that there aredifferent ways to interpret infinity that will havedifferent values. I understand think I’ve covered thebasic concepts of the set theory of infinity, and I’vediscovered yes it is possible that there is more thanone size of infinity.
Main SourcesBrowser: Internet ExplorerSearch Engine: GoogleWikipedia: Infinity, finite, cardinality.http://www.math.grinnell.edu/~miletijo/museu m/infinite.htmlhttp://www.scientificamerican.com/article.cfm? id=strange-but-true-infinity-comes-in- different-sizes