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More Information on Why "Infinity" Does Not Exist in the Context of Any Number System

This page provides supplementary information to the page of explanations on the question "does infinity exist?".

No "infinity" concept exists in the context of any number system, if by number system one means a collection of concepts that have operations like addition and multiplication the way familiar numbers do, operations which obey the usual properties of arithmetic.

One way to see this is to think, what would infinity minus 1 be? It couldn't be a finite number, since no finite number plus 1 equals infinity. So it must be infinite, and this would mean

infinity - 1 = infinity .
From this one can immediately see that the rules of arithmetic must be violated, since if they held one could subtract infinity from both sides to conclude that -1 = 0, which isn't true.

Therefore, there is no number system which possesses the usual rules of arithmetic and in which infinity exists. In other words,

Infinity does not exist, if by "exist" one means in the context of a number system.

A side note:

One should note that these arguments show there's no such thing as a number system with a single "infinity" concept. But there's nothing to stop us constructing a number system containing "infinitely large numbers" which are bigger than all the usual numbers; it's just that there would have to be many of them and no single one of them could be called "infinity".

A fairly straightforward example is the set of all polynomials. They're a number system (you can add, subtract, and multiply any two polynomials), and if you were to try to put an ordering on them (some system for saying which polynomials are larger than which), the most natural way to do so would be to consider the degree 1 polynomials to be "larger" than the degree 0 polynomials. So a polynomial like "x+2" would be considered larger than any constant (no matter how big that constant was). Therefore, in this number system, the polynomial "x+2" would be an "infinitely large number" because it's bigger than all the ordinary numbers that have no x in them. But it would be quite unjustified to call it "infinity" because it's just one of many degree 1 polynomials, and there's no reason to single it out with a special name rather than, say, x+1 or x+3 or 2x-17.

This page last updated: September 1, 1997
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