I find it interesting that to this day, not all equations involving infinity can be defined. Is this because the answers have yet to be discovered? Or, is there simply no solution? Wikipedia states that the following equations involving infinity are undefined:
+/- infinity multiplied by 0 = undefined
infinity multiplied by negative infinity = undefined
+/- infinity dividided by +/- infinity = undefined
1 to the power of +/- infinity = undefined
Friday, November 24, 2006
Sunday, November 19, 2006
Cardinalities
Infinity does not exist as a single number. Rather there are an infinite number of sets that represent infinite ranges of numeric values. Previously, we defined infinity as
1.The size of something infinite.Using the word in the context of sets is sloppy, since different infinite sets aren't necessarily the same size or cardinality as each other.
This definition presents the point that infinity does not refer to one size that can be measured. Rather infinity can have varying sizes. The range from -20 to infinity is a different size than the range of 10 to infinity. However, both are infinite in size. This phenomena can be understood through the concept of cardinality. Cardinality is defined as:
The number of elements in a set. If two setshave the same number of elements (i.e. there is a bijectionbetween them) then they have the same cardinality. Acardinality is thus an isomorphism class in the category of sets.
Cardinality helps to compare and quantify different sets of infinite numbers. This field was pioneered by the great German mathemetician, Georg Cantor. To learn more about Georg Cantor, go to http://en.wikipedia.org/wiki/Georg_Cantor.
Wikipedia (http://en.wikipedia.org/wiki/Infinity#Cardinality_of_the_continuum) talks about Cantor's contribution of cardinalities as a means of explaining infinity as follows:
A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its " proper" parts; this notion of infinity is called Dedekind infinite.
References:
infinity. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/infinity
cardinality. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/search?q=cardinality
1.
This definition presents the point that infinity does not refer to one size that can be measured. Rather infinity can have varying sizes. The range from -20 to infinity is a different size than the range of 10 to infinity. However, both are infinite in size. This phenomena can be understood through the concept of cardinality. Cardinality is defined as:
Cardinality helps to compare and quantify different sets of infinite numbers. This field was pioneered by the great German mathemetician, Georg Cantor. To learn more about Georg Cantor, go to http://en.wikipedia.org/wiki/Georg_Cantor.
Wikipedia (http://en.wikipedia.org/wiki/Infinity#Cardinality_of_the_continuum) talks about Cantor's contribution of cardinalities as a means of explaining infinity as follows:
A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its " proper" parts; this notion of infinity is called Dedekind infinite.
References:
infinity. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/browse/infinity
cardinality. (n.d.). The Free On-line Dictionary of Computing. Retrieved November 19, 2006, from Dictionary.com website: http://dictionary.reference.com/search?q=cardinality
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