Re: Inconsistency of the usual axioms of set theory
- From: lwalke3@xxxxxxxxx
- Date: Fri, 20 Mar 2009 21:50:05 -0700 (PDT)
On Mar 20, 5:12 am, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
lwal...@xxxxxxxxx writes:
Then I pointed out that even with Aatu's proof, someoneYou seem to be confused.
would eventually question whether the existence of any
infinite set implies the existence of omega. And sure
enough, this is exactly what Little has done.
Tommy insinuated that the *definition* of infinity (not the axiom, but
the definition) "leads" to N. Evidently, he meant that one can prove
N exists just by defining infinity. (If he meant something else, I
don't know what.)
Tim is asking something else entirely. He is asking whether a
particular axiom of infinity (different than the most usual axiom)
implies the existence of N.
I thought that Little asked his question about an alternate
axiom in direct response to tommy1729's post because
tommy1729 was talking about that same axiom.
You do understand that Dedekind-infinite is not the same term as
infinite, right?
Of course! Indeed, I address this in my earlier post below:
[W]e don't even need it to be a Dedekind infinite set as
Ullrich has given, but even under the weaker assumption
that a Tarski infinite set exists -- then one can prove
that omega exists.
Aatu's proof only requires that the set be Tarski infinite,
not necessarily Dedekind infinite. In ZF+~AC there exist
T-infinite sets that are D-finite, but in ZFC every
T-infinite set is D-infinite.
.
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- Re: Inconsistency of the usual axioms of set theory
- From: lwalke3
- Re: Inconsistency of the usual axioms of set theory
- From: Jesse F. Hughes
- Re: Inconsistency of the usual axioms of set theory
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