2009/6/14 Torgny Tholerus <tor...@dsv.su.se>: > > Quentin Anciaux skrev: >> Well it is illegal regarding the rules meaning with these rules set B >> does not exist as defined. >> > > What is it that makes set A to exist, and set B not to exist? What is > the (important) differences between the definition of set A and the > definition of set B? In both cases you are defining a set by giving a > property that all members of the set must fulfill.

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Yes and one fulfil it according to the given rules the other not. I would add that your "excercise" is inconsistent from the start, whatever a set is, your argument is contradictory whatever the rules are. > Why is the deduction legal for set A, but illegal for set B? There is > the same type of deduction in both places, you are just making a > substitution for the all quantificator in both cases. That's all the point of puting rules and checking that something is correct or not according to it. 1+1=3 is false according to PA... that doesn't mean you couldn't find a rule or mapping that would render this statement true ***regarding the chosen rules***/ Regards, Quentin > -- > Torgny Tholerus > >> >> 2009/6/13 Torgny Tholerus <tor...@dsv.su.se>: >> >>> 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 >>> >>> >> >> >> >> > > > > > -- All those moments will be lost in time, like tears in rain. --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---