Re: Inconsistency of the usual axioms of set theory

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.

Therefore tommy1729 was not babbling. Aatu has proved in
ZF-Infinity that if an infinite set exists -- and we
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.

I know that the OP, Hosseiny, was attempting to prove
that ZFC is inconsistent. Since Hosseiny's attempted
proof centered around the Axiom of Infinity, it might be
accurate to call him a finitist. And many finitists are
often considered to be "cranks," especially if they try
to prove that ZFC is inconsistent.

Recently I've been part of a heated debate with Little
about what exactly makes a "crank," what makes a
"standard analyst" (or set theorist), and so on. So I
can't be sure whether Little would consider either the
finitist Hosseiny or the mereologist tommy1729 to be
so-called "cranks" at all.
.

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