Re: Implementable Set Theory and Consistency of ZFC

On 22 Okt., 09:58, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
hagman wrote:
Of course your theory is "at least as correct" as ZFC in the sense
that if ZFC is consistant then so is your theory.

My theory _is_ consistent, unless you believe that the theory of natural
numbers is not consistent (because my sets are equivalent to naturals).


I believe that the theory of natural numbers is consistant.
One keystone of this my belief is the Axiom of Infinity in ZFC.

This does not imply that your theory is applicable as a base
for all subjects of mathematics and the "correct" theory to use
there.

I've never clained that. I've claimed exactly the _opposite_. Quotes:

a different strategy has been adopted in this article. Our purpose is
to obtain a set theory which is just a theory of sets, that is: _not_
suitable per se as a Foundation of Mathematics.

OK, I forgot that disclaimer.
But still your theory is only a theory of some sets so to speak
and if you are content with that small universe, then be happy with
it.
In my elementary scohool says, we learned some toy set theory
with urelements of two sizes, three shapes and four colours
(and no sets of sets).
What we learned was just a theory of sets of such pieces, that is:
_not_ suitable per se as a Fondation of Mathematics.


Each of our implementable (that is: hereditarily finite) sets uniquely
corresponds with a natural number (and {} = 0). And each natural number
uniquely corresponds with an implementable set. Thus our theory of sets
does NOT contain more sets than it contains numbers. Therefore it's NOT
suggested that an implementable set theory can serve as a foundation
for the whole of mathematics. It's rather suggested that sets are based
upon natural numbers. And that a good old statement still holds: 'Die
ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk'
(Leopold Kronecker - 1886). It's the WHOLE NUMBERS that are foundational
for Mathematics, not sets.

As an side note / exercise, construct a model of the following axioms:

[ -> ToDoList. Thanks ]

Han de Bruijn


.

Relevant Pages

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