Re: Implementable Set Theory and Consistency of ZFC

On 31 Okt., 15:59, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Jesse F. Hughes wrote:
Here's what I think you *can* say: Statements (5)-(8) are true in your
model and your model is canonical in a sense (I think it's minimal).
It is not the case that (5)-(8) are therefore *entailed* by (1)-(4) in
any sense of the word that I understand.

It seems that I can learn something from you here. Please explain to me
the words "canonical" and "minimal" and what they mean in this context.

Han de Bruijn

I think I have shown in another part of this thread:

The HdB model is described by a (meta) set N and
a relation e (defined by some bit operations).
If we have any other model consisting of a set N' and
a relation e' such that (1)-(4) are valid then
there exists a uniquely determind fucntion f:N->N'
such that for all x in N and all y' in N'
y' e' f(x) if and only if there exists an y in N
such that y'=f(y) and y e x.

Define f recursively:
Let x be any set in N (essentially x is a natural number)
and assume we have defined f(y) for all y<x and that
x has n elements.
All these elements are numbers <x, hence f is defined for them.
i) n=0: Then x is the empty set.
By assumption, N' has an empty set 0'.
Let f(x) = 0'.
(This uses EMPTYSET)
ii) n=1 or n=2: Then x is the pairing of two (possibly
identical) sets y1,y2.
f(y1) and f(y2) are already defined.
In N' there is a set {f(y1),f(y2)}.
Let f(x) = {f(y1),f(y2)}.
(This uses PAIRING).
iii)n>2: Let y be one of the elements and z=x\{y} (which
exists in N).
Then f(y) and f(z) are already defined.
Let f(x) = U{{f(y),f(y)},f(z)}
(This uses PAIRING and UNION).

Uniqeness is clear in all three cases.
It is interesting to note that EXTENSION of (N',e')
was not really used, but without it (meta) choice
seems necessary.

hagman

.

Relevant Pages

  • Re: A probably really silly and kooky idea about set theory.
    ... (By Empty Set and Complements, ... finite von Neumann naturals. ... exists by Empty Set, Pairing, Union.) ... Powerset -- much less that there exists a bijection ...
    (sci.math)
  • Re: A simple understanding of sets.
    ... But pairing and extensionality are kind of different from each other. ... Pairing is a consequence of Replacement if you are in Z *F* C. ... and the powerset of the powerset of the empty set has 2 elements. ...
    (sci.logic)
  • Re: Axiom of union
    ... So from where do You get the existence of elements? ... From empty set and pairing. ... math with this set theory as a fundament. ...
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  • Re: Implementable Set Theory and Consistency of ZFC
    ... I have shown that, once accepted the truth of Empty set, Extensionality, ... Pairing and Union, it follows that Substitution, Specification, Choice, ...
    (sci.math)
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