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| | ::::The book I'm reading (Charles Pinter's A Book of Abstract Algebra) seems to dodge the question - it has a section on inductive proofs, but it just starts with using integers and never mentions infinity. But the digits are countable, and go on for infinity, so they should be countably infinite - you should be able to match each digit to a digit in the countably infinite set of natural numbers, so that makes sense. | | ::::The book I'm reading (Charles Pinter's A Book of Abstract Algebra) seems to dodge the question - it has a section on inductive proofs, but it just starts with using integers and never mentions infinity. But the digits are countable, and go on for infinity, so they should be countably infinite - you should be able to match each digit to a digit in the countably infinite set of natural numbers, so that makes sense. |
| − | ::::I was trying to avoid the term "subset" as by axiom, a [[set]] can't contain itself. But I wasn't thinking of this as a set anyway, as there are no duplicates in a set, so the set of all digits in pi would just be the finite set <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0\}</math> after removing duplicates.--[[User:AlChesapeake|AlChesapeake]] 20:22, September 4, 2021 (EDT) | + | ::::I was trying to avoid the term "subset" as by axiom, a [[set]] can't contain itself. But I wasn't thinking of this as a set anyway, as there are no duplicates in a set, so the set of all digits in pi would just be the finite <--set <math>\{1, 2, 3, 4, 5, 6, 7, 8, 9, 0\}</math>--#> after removing duplicates.--[[User:AlChesapeake|AlChesapeake]] 20:22, September 4, 2021 (EDT) |
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| | The sentence in the article has been changed to | | The sentence in the article has been changed to |
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| | ::Your remark about a set of ordered numbers is not clear: did you mean natural numbers, real numbers, digits? And how do you insert pi into that set? | | ::Your remark about a set of ordered numbers is not clear: did you mean natural numbers, real numbers, digits? And how do you insert pi into that set? |
| | [[User:Blavek|Blavek]] ([[User talk:Blavek|talk]]) 22:07, September 4, 2021 (EDT) | | [[User:Blavek|Blavek]] ([[User talk:Blavek|talk]]) 22:07, September 4, 2021 (EDT) |
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| | + | :::I think you're making some implicit assumptions about [[infinity]] that restrain its power and scope. Infinity can be an infinite sequence of subseries which are themselves of infinite length. My suggestion about sets (which I meant to refer to digits) illustrates that. |
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| | + | :::By view pi as merely a sequence of digits that stretches on and on does not really capture all of pi's [[infinity]]. Infinity is not merely without end. It is also without enumeration. If pi does not repeat itself until after an infinite series of digits, then there is no problem with it becoming a pattern that would render it a rational number, which we know it is not. |
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| | + | :::Thanks for your comments about this, however, which is helping me clarify my thinking about it!--[[User:Aschlafly|Andy Schlafly]] ([[User talk:Aschlafly|talk]]) 22:50, September 4, 2021 (EDT) |
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| | ===Linking to this section from the article=== | | ===Linking to this section from the article=== |
| | Andy, do you really think that it is a good idea to link to the above section '''on a talk-page''' from the main article on [[pi|π]]? And please, read the discussion! (ups, that was the second sentence of this paragraph - you probably already started skipping...) --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 05:23, August 3, 2021 (EDT) | | Andy, do you really think that it is a good idea to link to the above section '''on a talk-page''' from the main article on [[pi|π]]? And please, read the discussion! (ups, that was the second sentence of this paragraph - you probably already started skipping...) --[[User:AugustO|AugustO]] ([[User talk:AugustO|talk]]) 05:23, August 3, 2021 (EDT) |