Talk:Cantor's diagonal argument/Arguments: Difference between revisions

Revision as of 14:37, 3 April 2025

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:# ''“Why mention the decimal interpretation of bit sequences? It does not help at all.”'' <br/> Because it's an encyclopedia, not a university lecture script, and its aim is to present facts and concepts to a widest audience. Whenever possible it's good to use a language and examples comprehensible for an 'ordinary reader', whose mathematical bacground may be as low as a mid-high school. A decimal notation is known to everyone, so this actually helps an 'ordinary reader' to grasp the idea of the injection we consider. If we talked about a binary interpretation, most of them would switch their thinking off, just because 'I have no idea what they're talking about'. Of course we may also include higher-leverlevel, more abstract and general examples – and this is in the box – but the basic description should be as simple as possible (but not simpler!).

:# ''“The assumption is that the infinite list of infinite bit streams is an enumeration of the reals in the unit interval. Which it is only in the base-2 interpretation!”'' <br/> No, and no. There is no such assumption, and that is not true for base-2 interpretation. The first ‘no’: we do not ''assume'' the list is an enumeration of anything (except the enumeration of binary sequences.) Instead, we ''prove'' (i.e. conclude at, not start from) it ''can be put into equivalence to'' an enumeration of numbers from some set. The second ‘no’: the equivalence is not direct or immediate even with binary representation of numbers, and that's because of tails of repeating digits: a repeating one in 0.101111... makes the representation equivalent to 0.110000... with repeating zero, hence the two ''different'' binary strings “101111...” and “110000...”, when simply transformed into binary representation of real numbers, become the same number. Hence the list of binary strings is ''not'' “an enumeration of the reals in the unit interval” by itself. We make a special trick with interleaving double representations, and only after that we get a desired bijection (“enumeration”).

: [[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 06:56, 21 May 2019 (UTC)