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| | For example, the existence of infinity implies the existence of its two opposites, [[zero]] and [[negative infinity]]. Zero is the amount that is infinitely small. The existence of an infinite [[God]] implies the necessary existence of the opposite: the void described in [[Genesis]]. Negative infinity, as implied by the existence of positive infinity, is the numeric equivalent of the [[Hell]] mentioned frequently in the [[New Testament]]. | | For example, the existence of infinity implies the existence of its two opposites, [[zero]] and [[negative infinity]]. Zero is the amount that is infinitely small. The existence of an infinite [[God]] implies the necessary existence of the opposite: the void described in [[Genesis]]. Negative infinity, as implied by the existence of positive infinity, is the numeric equivalent of the [[Hell]] mentioned frequently in the [[New Testament]]. |
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| | == Teachings about infinity by Jesus == | | == Teachings about infinity by Jesus == |
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| | So infinity might arise in statements like these: | | So infinity might arise in statements like these: |
| − | *<math>\frac{1}{0} = \infty\ \ </math>NO! This isn't allowed! Infinity is not a number, and division by zero is illegal! | + | *<math>\frac{1}{0} = \infty\ \ </math> Mathematics does not allow division by zero, so this is technically not allowed unless does as a limit, as follows: |
| − | *<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math>Yes, sort of. One could say that "the limit is infinite", since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity", one would have to say that this is the limit from the right. The limit from the left is "minus infinity", that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above. | + | *<math>\lim_{x\to 0}\frac{1}{x} = \infty\ \ </math> Yes, one could say that "the limit is infinite," since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity," one would have to say that this is the limit from the right. The limit from the left is "minus infinity," that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above. |
| − | *<math>\int_0^1\frac{1}{x} = \infty\ \ </math>Infinity has a special meaning for integrals. | + | *<math>\int_0^1\frac{1}{x} = \infty\ \ </math> Infinity has a special meaning for integrals. |
| − | *<math>\|\mathbb{Z}\| = \infty\ \ </math>The cardinality of the integers is infinite. | + | *<math>\|\mathbb{Z}\| = \infty\ \ </math> The cardinality of the integers is infinite. |
| − | *<math>\mu(\mathbb{R}) = \infty\ \ </math>The (Lebesgue) measure of the reals is infinite. | + | *<math>\mu(\mathbb{R}) = \infty\ \ </math> The (Lebesgue) measure of the reals is infinite. |
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| | In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general. | | In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general. |