> Date: Sat, 13 Jun 2009 11:05:22 +0200
> From: tor...@dsv.su.se
> To: everything-list@googlegroups.com
> Subject: Re: The seven step-Mathematical preliminaries
> 
> 
> Jesse Mazer skrev:
> >
> > > Date: Fri, 12 Jun 2009 18:40:14 +0200
> > > From: tor...@dsv.su.se
> > > To: everything-list@googlegroups.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.

It's well known that if you allow sets to contain themselves, and allow 
arbitrary rules for what a given set can contain, then you can get 
contradictions like Russell's paradox (the set of all sets which do not contain 
themselves). But what relevance does this have to arithmetic? Are you afraid 
the basic Peano axioms might lead to two propositions which can be derived in 
finite time from the axioms but which are mutually contradictory? If so I don't 
see how allowing only a finite collection of numbers actually helps--like I 
said in an earlier post, the number of propositions that can be proved about a 
finite set of numbers can still be infinite. I suppose it might be possible to 
make it finite by disallowing propositions which are created merely by 
connecting other propositions with the AND or OR logical operators, but it's 
still the case that if your largest whole number BIGGEST is supposed to be at 
least as large as some numbers humans have already conceived--say, as large as 
10^100--then there is no way we could actually write out all possible 
propositions about these numbers that follow from some Peano-like axiom system 
to check manually that no two propositions contradicted each other (do you want 
to try to calculate 10^100 + A and A + 10^100 for every possible value of A 
smaller than 10^100 to verify explicitly that they are equal in every case?) 
So, it seems that unless you want to make your universe of numbers *very* 
small, you have to rely on some sort of mental model of arithmetic to be 
confident that you won't get contradictions from the axioms you start from, 
just like how people are confident in the non-contradictoriness of the Peano 
axioms based on their mental model of counting discrete objects like marbles 
(see my comments in the last paragraph of the post at 
http://www.mail-archive.com/everything-list@googlegroups.com/msg16564.html ).

Jesse
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