What does infinity mean in mathematics?

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Eva Abramson · Accepted Answer

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Different mathematical concepts of infinity can be found in different contexts and with different applications. One of the basic axioms of standard set theory is that there is an infinite set. In some formulations of set theory you would use the set of natural numbers #NN# as the standard example of an infinite set. Discussing "the set of Natural numbers" is essentially discussing "completed infinities"; if you begin to explore that concept, you might discover more than you anticipated. An infinite sequence of elements from a set #A# is a mapping from #NN->A#. For example, we could define a mapping that for each Natural number #n# gives us a Real number #a_n#. We would then have a sequence of Real numbers: #a_1, a_2, a_3,...# As an illustration, the formula: #a_n = 3 - 1/n# defines a mapping from #NN# to #RR# taking the values: #2, 5/2, 8/3, 11/4, 14/5, 17/6,...# The terms of this monotonically increasing sequence are all numbers between #2# and #3#. The limit of the sequence is #3# which is also known as the least upper bound of the sequence. Is it possible to enumerate all of the numbers between (say) #0# and #1# in one sequence? "No" is a pretty clear response. Georg Cantor, a mathematician, demonstrated that you could always create a number that wasn't on the list if you had a sequence of numbers like that. This suggests that there exist infinitesimals surpassing the quantity of natural numbers. This is a far more expansive topic than you may imagine.

Eva Abramson · Answer

undefined In mathematics, infinity represents a concept of being limitless or without bound. It is used to describe values or quantities that have no end or are unbounded.