Re: Inconsistency of the usual axioms of set theory

lwalke3@xxxxxxxxx writes:

On Feb 22, 7:55 pm, Tim Little <t...@xxxxxxxxxxxxxxxxxx> wrote:
On 2009-02-22, amy666 <tommy1...@xxxxxxxxxxx> wrote:
davids definition is not wrong.  but it leads to N.
so im not "babbling".
Can you prove from ZF without the axiom of infinity, that "there
exists a Dedekind-infinite set" implies the existence of N?
Or are you just babbling?

OK, I was away from sci.math back in late February, thus I
haven't keep up with this thread.

In another thread, Aatu proved that in the theory
ZF-Infinity, one can prove that "there exists an infinite
set" implies "omega exists." MoeBlee confirmed the
correctness of the proof, which depends on the axioms of
Powerset and Replacement Schema.

Then I pointed out that even with Aatu's proof, someone
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.

You seem to be confused.

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.

Tim's question does not make Tommy's strange claim retroactively
respectable.

You do understand that Dedekind-infinite is not the same term as
infinite, right?

--
Jesse F. Hughes
"Basically there are two angry groups. I am a harsh force of
one. Against me is a society of mathematicians. So far it's been a
draw." -- JSH gives another display of keen insight.
.

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