Jesse Mazer skrev: > > > Date: Fri, 12 Jun 2009 18:40:14 +0200 > > From: tor...@dsv.su.se > > To: email@example.com > > Subject: Re: The seven step-Mathematical preliminaries > > > > It is, as I said above, for me and all other humans to understand what > > you are talking about. It is also for to be able to decide what > > deductions or conclusions or proofs that are legal or illegal. > > Well, most humans who think about mathematics can understand > rule-based definitions like "0 is a whole number, and N is a whole > number if it's equal to some other whole number plus one"--you seem to > be the lone exception. > > As for being "able to decide what deductions or conclusions or proofs > that are legal or illegal", how exactly would writing out all the > members of the "universe" solve that? For example, I actually write > out all the numbers from 0 to 1,038,712 and say that they are members > of the "universe" I want to talk about. But if I write out some axioms > used to prove various propositions about these numbers, they are still > going to be in the form of general *rules* with abstract variables > like x and y (where these variables stand for arbitrary numbers in the > set), no? Or do you also insist that instead of writing axioms and > making deductions, we also spell out in advance every proposition that > shall be deemed true? In that case there is no room at all for > mathematicians to make "deductions" or write "proofs", all of math > would just consist of looking at the pre-established list of true > propositions and checking if the proposition in question is on there.
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? -- 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 firstname.lastname@example.org 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 -~----------~----~----~----~------~----~------~--~---