Quentin Anciaux skrev: > 2009/6/13 Torgny Tholerus <tor...@dsv.su.se>: > >> What do you think about the following deduction? Is it legal or illegal? >> ------------------- >> Define the set A of all sets as: >> >> For all x holds that x belongs to A if and only if x is a set. >> >> This is an general rule saying that for some particular symbol-string x >> you can always tell if x belongs to A or not. Most humans who think >> about mathematics can understand this rule-based definition. This rule >> holds for all and every object, without exceptions. >> >> So this rule also holds for A itself. We can always substitute A for >> x. Then we will get: >> >> A belongs to A if and only if A is a set. >> >> And we know that A is a set. So from this we can deduce: >> >> A beongs to A. >> ------------------- >> Quentin, what do you think? Is this deduction legal or illegal? >> > > It depends if you allow a set to be part of itselft or not. > > If you accept, that a set can be part of itself, it makes your > deduction legal regarding the rules.
OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? ------------------- Define the set B of all sets that do not belong to itself as: For all x holds that x belongs to B if and only if x does not belong to x. This is an general rule saying that for some particular symbol-string x you can always tell if x belongs to B or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for B itself. We can always substitute B for x. Then we will get: B belongs to B if and only if B does not belong to B. ------------------- Quentin, what do you think? Is this deduction legal or illegal? -- Torgny Tholerus --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
