[FOM] Question about Set Theory as a formal basis for mathematics

[Aatu Koskensilta]

On Feb 26, 2006, at 6:03 PM, Andrea Proli wrote:
> I did not understand what do you mean by "the concepts necessary to
> understand its description, are not in defined by ZF(C) but must be
> understood the same way one understands what the natural numbers are". 
> You
> mean that natural numbers are not formally defined based on set 
> theory, in
> Number Theory for example (I don't know, I'm guessing)?

Sure they are, but the only foundational significance of these 
definitions is that they show that natural numbers can be coded as 
objects of this or that system not already including them as 
primitives. But it is of absolutely no help in understanding what the 
natural numbers are to learn that they are the finite von Neumann 
ordinals (which they aren't, really). What a natural number is, what 
finite means and all the baggage that comes with these concepts must be 
understood before we can even speak of any formal systems let alone 
models thereof.

Similarly, before one can understand such locutions as "this or that is 
a model of ZF(C)" and "there is a such and such model of ZF(C)" one 
must already understand what set theoretic talk means. And if one is to 
accept such assertions, one obviously needs to accept various set 
theoretic principles (which may or may not be provable in ZF(C) when 
formalized, if possible, in the first order language of set theory). 
But the inert bunch of first order axioms ZF(C) is does not provide 
either understanding or acceptance by itself. Rather, we accept ZF(C) 
because we understand the concepts and accept the principles, 
formulated using these concepts, formalized in ZF(C). The formal 
theory, a mathematical object, and its models, also mathematical 
objects, are important only when we study some of the properties of 
these principles and concepts, e.g. what can and what can't be 
expressed in terms of the concepts and what can and can't possibly be 
proven on basis of the principles. These mathematical objects - ZF(C) 
and its models - don't tell us what "set theory is about", and model 
theory - as done in the framework in set theory - is equally silent 
about that.

> What are the involved "concepts" you talk about?

"Set", "powerset", "iterate" and so forth.

> I would say that variables can also assume value {0, {0}}, for 
> example, which is not generated by only powerset  operator and empty 
> set.

But it is. {0,{0}} is in the powerset of {0,{0}} and appears at stage 3 
in the cumulative hierarchy

  V_0 = 0
  V_1 = P(V_0) = {0}
  V_2 = P(V_1) = {0,{0}}
  V_3 = P(V_3) = {0,{0,{0}},{{0}},{0}}
      .
      .
      .
  V_w = {x | x in V_n for some n} = union of V_i for i in w
      .
      .
      .

(where w = omega, the first infinte ordinal).

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus