- #1
rmberwin
- 13
- 1
I came across the following argument that attempts to show that the notion of infinite decimal numbers is incoherent. Try adding these two numbers:05.4123482100439884...
16.3482518100560115...
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21.760600020?999999...By the Axiom of Choice, "There exist arbitrary infinite sequences of numbers", the two numbers being added are arbitrary, that is, not computable. In the sum, because of the series of 9's the 10th digit after the decimal is indeterminate. It could be either a 0 or a 1, depending on whether there is eventually a carry-over, which would 'cascade' to the left in the sum. The claim is that by using arithmetic (that is, an algorithm) the sum cannot be derived, and therefore infinite decimal numbers don't exist. I'm not a mathematician, but here's my thought. If by the AoC, "Arbitrary infinite sequences exist", then the sum, even if not calculable, still exists, because the AoC merely asserts the existence of certain infinite sets, not how to derive them. So the argument is faulty.
16.3482518100560115...
___________________
21.760600020?999999...By the Axiom of Choice, "There exist arbitrary infinite sequences of numbers", the two numbers being added are arbitrary, that is, not computable. In the sum, because of the series of 9's the 10th digit after the decimal is indeterminate. It could be either a 0 or a 1, depending on whether there is eventually a carry-over, which would 'cascade' to the left in the sum. The claim is that by using arithmetic (that is, an algorithm) the sum cannot be derived, and therefore infinite decimal numbers don't exist. I'm not a mathematician, but here's my thought. If by the AoC, "Arbitrary infinite sequences exist", then the sum, even if not calculable, still exists, because the AoC merely asserts the existence of certain infinite sets, not how to derive them. So the argument is faulty.