Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:

But whatever it means, since a model of ZFC is also a model of
ZFC - Infinity, you seem to be claiming that you can build an infinite
set using only the first four axioms.

But the reverse is not true: a model of (ZFC - Infinity) is not a model
of ZFC(Infinity included). So I'm building only (ZFC - Infinity), with
those first four axioms: extensionality, empty set, pairing, union. All
finite sets can be build with these 4 axioms (but there are infinitely
many of these finite sets, like with the naturals).

In other words, you are *not* speaking about every model of ZFC - Infinity, despite what you claimed.

And just as I suggested.

Have no idea what you are talking about, with your "every model".

Han de Bruijn

.

Relevant Pages

  • Re: ZFC in 4 Axioms.
    ... Ax.1) Extensionality: As in ZFC ... Ax.4) Infinity: As in ZFC ... theory, not axioms. ... E!vAy yev. ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... Han de Bruijn writes: ... four axioms are necessary for a constructive build of all sets. ... ZFC - Infinity and not all sets of ZFC. ...
    (sci.math)
  • Re: For All x
    ... How do we interpret ~EzAx xez? ... Would it mean that the set of all finite sets does not exist? ... ~EzAx xez is a theorem of ZFC with or without the axiom of infinity. ... It's even a theorem of certain theories weaker than ZFC without the ...
    (sci.logic)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... ZFC - Infinity, you seem to be claiming that you can build an infinite ... set using only the first four axioms. ... many of these finite sets, ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... But the reverse is not true: a model of (ZFC - Infinity) is not a model ... those first four axioms: ... many of these finite sets, ...
    (sci.math)