[FOM] The Natural Language Thesis and Formalization

[Arnon Avron]

On Wed, Jan 23, 2008 at 12:35:45PM -0500, Steven Gubkin wrote:
>  3= {{{0}}}= {{{0},{0}}}= {{{0},{0,0}}}={<0,0>} 
> which is the function whose domain and codomain are both {0}. Is it 
> reasonable to conclude that there are natural numbers which are also 
> functions? 
> Clearly this "result" is an artifact of the way we 
> translate concepts into (and out of) ZFC. 

Is this result less reasonable than the result that given two real 
numbers, one of them is a subset of the other (as follows from the
definition of the reals as Deckind cuts)?

  As long as we do not know (or decide) what are the natural numbers
or what is a function, then we cannot answer whether there is a
natural number which is a function. Such a question makes sense
only in a framework in which concrete definitions are defined

 Besides, why go to this complicated example? It suffices to
take the example of the number 0. Assuming that it is identified 
with the empty set (something I believe to be quite reasonable
and intuitive). With this identification already 0 is both a 
natural number and a function. 

  Now I know from my expeience as a teacher that it is difficult
for students to understand that the empty set is a function
from itself to any other set. Usually they are convinced that
this is not an artifact of the way we define the notion
of a function (as a set of ordered pairs which satisfy the 
functionality condition) when it is pointed out to them that
otherwise the equation a^0=1 would not be true
(where exponentiation of cardinalities is defined
as is usual in basic set theory). 

  Note that the proposition "there are sets of natural
numbers which are also functions" seem no less counterintuitive
than "there are natural numbers which are also functions"
(and this proposition does not depend on the identification
of 0 with the empty set).

Arnon Avron