Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

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

Not in my article:

http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf

Look, what you said was simply wrong. You said (piecing things together):

Only the first four axioms are necessary for a constructive build of
all sets of any "implementation" (model?) of ZFC - Infinity.

In the first case, it's not at all clear what this statement means.

It's quite clear! Once you take the effort (but .. wow, _now_ I'm asking
something) to _read_ and absorb what I've actually written.

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).

Obviously, you do not believe that, so I am sure you did not say what
you mean.

Han de Bruijn

.

Relevant Pages

  • 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: 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: infinitely many nns = infinite nns?
    ... First, you would have to STATE SOME AXIOMS, ... Here, "infinitely many" is aleph-0, which is "actual infinity." ... Similarly, the "fact" that all the numbers are "finite" means that all numbers have values LESS THAN aleph-0, that the natural numbers are limited to "potentially infinite" values. ... To be honest, it won't SURPRISE me one iota if you claim that there are some natural numbers with infinitely many digits, but that they, too, have only finite values. ...
    (sci.logic)
  • Re: infinitely many nns = infinite nns?
    ... You asked for my axioms, and I really want to use the standard ones, so the question is mostly pointless, but I thought I would give clearer definitions of how I am using actual and potential infinity. ... "Aleph-0" DOES NOT EXIST outside some LOGICAL context ... "Actual infinity" is a term from informal natural-language ...
    (sci.logic)