Re: abundance of irrationals

From: David Kastrup (dak_at_gnu.org)
Date: 01/27/05 

Date: Thu, 27 Jan 2005 12:51:24 +0100

mueckenh@rz.fh-augsburg.de writes:

> r.e.s. wrote:
>
>> You say that the number represented by Floor(pi*10^10^100)
>> does not exist, arguing that "it will never be available"
>> and that one can't "calculate with it".
>
> You see, there are some numbers, which may possibly be raised into
> existence, like floor(pi*10^10^50), which however do not yet exist for
> us humans (and I am sure nor for anybody else).

"Not yet exist" does not make sense in mathematics. The existence is
a consequence of axioms, and those do not come into being slowly.

> And there are some numbers which never will come into being because
> all of their information cannot be contained simultaneously in the
> universe.

The universe is not involved in axioms. Anyway, of course, _all_ of
the information of floor(pi*10^10^100) is easily contained in the
universe: I can write it down with about 20 characters. 3.14 is a
recipe for computation, and so is floor(pi*10^10^100) (because pi can
again be split into a recipe).

So you are just talking nonsense. If you want to get into some more
serious philosophical domain, all you have to notice that _every_
number for which you can write a symbolic definition down is a
computable number. And there are only countably many of those.

But the set of irrational numbers is much larger: the majority of its
members are not computable: still nobody can specify a _single_
noncomputable member of it.

Now _that_ is a philosophical problem: talking about properties of
numbers that one can't even specify singly in any manner: you only
encounter them as unnamed entities in proofs. And one works with the
properties of the continuum consisting of them.

And this _is_ an interesting question for mathematical philosophy:
should one create and work with models that give one entities one
cannot ever specify? floor(pi*10^10^100) _is_ a specification. 

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum

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