[FOM] Dual of Dedekind-infiniteness.

Arnon Avron aa at tau.ac.il
Fri Oct 12 07:26:26 EDT 2007

On Wed, Oct 10, 2007 at 05:20:03PM +0200, Alex Blum wrote:
> I'm assuming it is the latter since you think that the notion of a 
> finite set  is more basic than the notion of a natural number. I wonder 
> if the latter is true for if one were to posit that a given set is 
> finite and yet is unfamiliar with the concept of a finite number of 
> objects, we might think that he failed to grasp what finite means. 

Not at all. By a "finite set" we originally and intuitively mean
"a set that can be constructed by repeating the operations
of joining one element in a time finitely many times". Here 
"finitely many" is a primitive notion,
not really reducible to anything else, and its understanding
is not connected to any precise quantative properties. The
concept of a finite cardinal number (which I guess is what you
mean here by a "standard integer") is far more complicated, and
frequently even students of math in the university do not
fully understand it.

> converse however is clearly untrue. 

To me the converse of what you say is clearly true.

> Following through with the 
> assumption that there is  a core concept, would it not be true that one 
> who asks if the number of stars is infinite or finite is asking in other 
> words if the number of stars is either infinite or is a(standard) integer?

Not at all. Your reformulation is more complex than the
original one. It makes sense only for someone
who knows that it is possible to associate with any finite set A
a unique integer #A so that #A=#B iff A is equipolent with B.
This is anything but trivial, even though most people take
it for granted on the basis of their experience with small
sets and because the language they speak implicitly
assume this (and "natural language"
are implicitely based on many assumptions. Some of this assumptions
are not even true). 

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