Re: Implementable Set Theory and Consistency of ZFC

Brian Chandler wrote:

Indeed. Let's prove that Han's latest Toy Set Theory is inconsistent.
It must include at least some finite sets, and algebra based on it
will include at least some finite fields. So construct a bijection
between the set {0, 1, 2, ..., n-1 } for n some conveniently small
number, and the elements of the finite field of order n. Also consider
the obvious bijection (within Toy Set Theory) between the number n and
{0, 1, 2, ... n-1}. Then there is a bijection from the number 6 to the
finite field of order 6, which does not exist (by a fairly elementary
proof I won't bore you with). Collapse of Toy Set Theory.

This counter argument relies on a precise definition of "bijection". OK?

While it is true that there is a bijection between the number n and the
finite field of order n, the situation with hereditarily finite sets and
natural numbers is not quite like this. In fact, sets in _implementable_
set theory are _identical_ with natural numbers (: N_0). Can we likewise
say now that finite ordinals are _identical_ with naturals, in classical
mathematics? I think so, because they are "defined" like that. In short:

Hereditarily finite sets = naturals : implementable set theory
_ Mainstream mathematics : naturals = finite ordinals

So the naturals are a common factor in two theories. And they join the
finite ordinals (that is: axiom of Infinity) with the "set of all sets"
in implementable set theory. The latter does not exist, though.

Doesn't that say something? Isn't there an analogous pattern, somewhere
in common model theory?

Han de Bruijn

.

Relevant Pages

  • Re: Implementable Set Theory and Consistency of ZFC
    ... So construct a bijection ... and the elements of the finite field of order n. ... Collapse of Toy Set Theory. ... say now that finite ordinals are _identical_ with naturals, ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... You STILL seem to be confused between how people describe set theory ... and both can be bijected with the naturals. ... Using set theory doesn't require calling cardinality "size". ... same as admitting a bijection. ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... >>What I am saying is that the first order ZF set theory axioms don't ... >>prove the existence of uncountably many subsets of the naturals. ... In set theory, to say that a set S is uncountable is to ... ZF proves that there does not exist a bijection between P ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... >>What I am saying is that the first order ZF set theory axioms don't ... >>prove the existence of uncountably many subsets of the naturals. ... In set theory, to say that a set S is uncountable is to ... ZF proves that there does not exist a bijection between P ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... It must include at least some finite sets, and algebra based on it ... So construct a bijection ... and the elements of the finite field of order n. ... Collapse of Toy Set Theory. ...
    (sci.math)