Jesse Mazer skrev:
> > Date: Sat, 6 Jun 2009 16:48:21 +0200
> > From: tor...@dsv.su.se
> > To: email@example.com
> > Subject: Re: The seven step-Mathematical preliminaries
> > Jesse Mazer skrev:
> >> Here you're just contradicting yourself. If you say BIGGEST+1 "is then
> >> a natural number", that just proves that the set N was not in fact the
> >> set "of all natural numbers". The alternative would be to say
> >> BIGGEST+1 is *not* a natural number, but then you need to provide a
> >> definition of "natural number" that would explain why this is the case.
> > It depends upon how you define "natural number". If you define it by: n
> > is a natural number if and only if n belongs to N, the set of all
> > natural numbers, then of course BIGGEST+1 is *not* a natural number. In
> > that case you have to call BIGGEST+1 something else, maybe "unnatural
> > number".
> OK, but then you need to define what you mean by "N, the set of all
> natural numbers". Specifically you need to say what number is
> "BIGGEST". Is it arbitrary? Can I set BIGGEST = 3, for example? Or do
> you have some philosophical ideas related to what BIGGEST is, like the
> number of particles in the universe or the largest number any human
> can conceptualize?
It is rather the last, the largest number any human can conceptualize.
More natural numbers are not needed.
> Also, any comment on my point about there being an infinite number of
> possible propositions about even a finite set,
There is not an infinite number of possible proposition. You can only
create a finite number of proposition with finite length during your
lifetime. Just like the number of natural numbers are unlimited but
finite, so are the possible propositions unlimited but finte.
> or about my question about whether you have any philosophical/logical
> argument for saying all sets must be finite,
My philosophical argument is about the mening of the word "all". To be
able to use that word, you must associate it with a value set. Mostly
that set is "all objects in the universe", and if you stay inside the
universe, there is no problems. But as soon you go outside universe,
you must be carefull with what substitutions you do. If you have "all"
quantified with all object inside the universe, you can not substitute
it with an object outside the universe, because that object was not
included in the original statement.
> as opposed to it just being a sort of aesthetic preference on your
> part? Do you think there is anything illogical or incoherent about
> defining a set in terms of a rule that takes any input and decides
> whether it's a member of the set or not, such that there may be no
> upper limit on the number of possible inputs that the rule would
> define as being members? (such as would be the case for the rule 'n is
> a natural number if n=1 or if n is equal to some other natural number+1')
In the last sentence you have an implicite "all": The full sentence
would be: For all n in the universe hold that n is a natural number if
n=1 or if n is equal to some other natural number+1. And you may now be
able to understand, that if the number of objects in the universe is
finite, then this sentence will just define a finite set.
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