Re: Implementable Set Theory and Consistency of ZFC

On Oct 10, 4:00 pm, David R Tribble <da...@xxxxxxxxxxx> wrote:
4. [p.15]
Adding the "Killer Axiom" (which I've only seem it called this
by you) to ZFC creates a system inconsistent with the
Axiom of Foundation. How does this show that ZFC is
inconsistent?

I think I see what HdB is trying to accomplish here. At this
point, he is not yet attempting to show the inconsistency of
ZFC, but rather:

Z-Infinity+~Infinity |- Foundation

(where |- denotes "turnstile"), or more to the point, that
his "Killer Axiom" is _equivalent_ (in Z-Infinity+~Infinity) to
~Foundation somehow. Obviously (abbreviating as KA)
we have KA -> ~Foundation. Indeed, if KA (i.e., that
forall a(a = {a}), we have that every set is a singleton
containing itself as an element. Such a set is known as a
Quine atom, and is mentioned in other sci.math threads
dealing with non-wellfounded set theory. But note that
Extensionality alone is powerless to prove that a Quine
atom must be unique. It is possible for there to be two,
three, or even infinitely many distinct Quine atoms.

But unfortunately for HdB, it is not the case that
~KA -> Foundation at all. For example, there may exist
two distinct sets a, b such that a = {b} and b = {a}. Now
~KA would still hold, since neither of these sets contains
itself as an element, but clearly a = {b} and b = {a} would
contradict Foundation.

We may even have an infinite descending chain:

x_n = {x_(n+1)}

once again, implying both ~KA (since none of these sets
contains itself as an element) and ~Foundation. Notice
that ~Infinity doesn't imply that infinite descending chains
don't exist. After all, ~Infinity doesn't prevent there from
being infinite _ascending_ chains -- the chain of von
Neumann natural numbers is infinite ascending. (Of course
the transitive closure of x_0 would be infinite, but no one
ever said that the transitive closure of a set must be a set --
indeed the proof of the existence of transitive closures
requires, believe it or not, the Axiom of Infinity!)

And so we show that since Foundation is not provable in
Z-Infinity+~Infinity+KA, it's not provable in Z-Infinity+~Infinity
either, and so HdB is mistaken here.

.

Relevant Pages

  • Re: infinity
    ... > were inconsistent, then it is only because ZF itself is inconsistent. ... There are minimal extensions to ZFC which do deal ... >>> No, ZF is inconsistent, and infinite sets are equivalent. ... > We can perform this sum only thanks to Countable Additivity. ...
    (sci.math)
  • Re: Why Regularity?
    ... "Naive comprehension is the principle that, ... this set theory is not heading at a Russell set as ZFC does. ... I think if I add this axiom, then I will currenty that is a set ... The axiom is inconsistent all by itself. ...
    (sci.math)
  • Re: Why Regularity?
    ... "Naive comprehension is the principle that, ... this set theory is not heading at a Russell set as ZFC does. ... I think if I add this axiom, then I will currenty that is a set ... The axiom is inconsistent all by itself. ...
    (sci.math)
  • Re: Small Set Theory,Updated.
    ... Ax2) Comprehension: Ex x is P_defined ... Ax3) Infinity:As in ZFC ... is NOT a theorem in this theory, unless the theory is inconsistent. ... On the contrary, this Axiom is inconsistent, with the rest of the ...
    (sci.math)
  • Re: Why Regularity?
    ... "Naive comprehension is the principle that, ... this set theory is not heading at a Russell set as ZFC does. ... I think if I add this axiom, then I will currenty that is a set ... The axiom is inconsistent all by itself. ...
    (sci.math)