Re: Implementable Set Theory and Consistency of ZFC

On 15 Oct, 15:18, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
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?

It says that a model of ZF-infinity is not necessarily a model of ZF.

.

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: infinity
    ... > Yes, for your proposed mapping between N and P, ... the largest element would not map to the ... so our bijection never runs into any such endpoint and ... what is the last element in the mapping of naturals to evens? ...
    (sci.math)
  • Re: infinity
    ... In the latter case, there is a bijection, ... >> regarding the evens, and ask how many bits each set's elements has, the evens ... like the bijection between the naturals and the ... > infinite sets, there is no way to use a "running out of" argument, ...
    (sci.math)
  • Re: Galileos Paradox
    ... set theory. ... There are sets that are both infinite ... but without a bijection between them. ... the mapping: ...
    (sci.math)
  • Re: Uncountable sets in CZF?
    ... >bijection between N and R and thus P? ... numbers in a model V of ZFC, then there is a generic extension V ... first-order logic (or maybe additional axioms to the first-order logic, ... >naturals and the reals. ...
    (sci.math)
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