# RE: The seven step-Mathematical preliminaries

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> Date: Thu, 4 Jun 2009 15:23:04 +0200
> From: tor...@dsv.su.se
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Quentin Anciaux skrev:
>> If you are ultrafinitist then by definition the set N does not
>> exist... (nor any infinite set countably or not).
>>
>
> All sets are finite.  It it (logically) impossible to construct an
> infinite set.
What do you mean by "construct"? Do we have to actually write out or otherwise
physically embody every element? Why can't we think of a particular "set" as
just a type of rule that, given any possible element, tells you whether or not
that element is a member or not? In this case there's no reason the rule
couldn't be such that there are an infinite number of possible inputs that the
rule would identify as valid members.
>
> You can construct the set N of all natural numbers.  But that set must
> be finite.  What the set N contains depends on how you have defined
> "natural number".

How do *you* define "natural number", if not according to the usual recursive
rule that 1 is a natural number and that if N is a natural number, N+1 is also
a natural number? Hopefully you agree that there can be no finite upper limit
on possible inputs you could give this rule that the rule would identify as
valid natural numbers? I think your claim would be that simply describing the
rule is not a valid way of "constructing" the set of natural numbers. If so,
why *isn't* it valid? *You* may prefer to adopt the rule that we should only be
allowed to call something a "set" if we can actually write out every member,
but do you have any argument as to why it's "invalid" for the rest of us to
define sets simply as general rules that decide whether a given input is a
member or not? This seems more like an aesthetic preference on your part rather
than something you have a compelling philosophical argument for (or at least if
you have such an argument you haven't provided it).
Jesse
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