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?
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