Brian Tenneson skrev:
> This is a denial of the axiom of infinity. I think a foundational set
> theorist might agree that it is impossible to -construct- an infinite
> set from scratch which is why they use the axiom of infinity.
> People are free to deny axioms, of course, though the result will not
> be like ZFC set theory. The denial of axiom of foundation is one I've
> come across; I've never met anyone who denies the axiom of infinity.
>
> For me it is strange that the following statement is false: every
> natural number has a natural number successor. To me it seems quite
> arbitrary for the ultrafinitist's statement: every natural number has
> a natural number successor UNTIL we reach some natural number which
> does not have a natural number successor. I'm left wondering what the
> largest ultrafinist's number is.
It is impossible to lock a box, and quickly throw the key inside the box
before you lock it.
It is impossible to create a set and put the set itself inside the set,
i.e. no set can contain itself.
It is impossible to create a set where the successor of every element is
inside the set, there must always be an element where the successor of
that element is outside the set.
What the largest number is depends on how you define "natural number".
One possible definition is that N contains all explicit numbers
expressed by a human being, or will be expressed by a human being in the
future. Amongst all those explicit numbers there will be one that is
the largest. But this "largest number" is not an explicit number.
--
Torgny Tholerus
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Received on Thu Jun 04 2009 - 16:05:00 PDT