Re: Principles of Induction in non well-founded set theories?

On Oct 4, 2:58 am, Adam Burley
<ajbur...@xxxxxxxxxxxxxx> wrote:
On Oct 3, 1:27 pm, hagman <goo...@xxxxxxxxxxxxx>
wrote:

(1) For any two members of X, these two
members
are comparable by the membership relation, or
equal.
(2) Any subset of X has a "least element"
(i.e.
an element Y such that for any member Z of X, we
have
Z = Y or Z is not a member of Y).

nonempty subset!

(3) Any element of X is a subset of X.

Since you leave out Foundation, there might in
principle exist a set X
such
that {X} = X.
Such an X is an ordinal.

No it's not. From his definition of 'ordinal' and
without using
regularity, we prove that no ordinal is a member
of
itself.

I am afraid that our friend "hagman" is right. The
set he constructs, which is called the Quine Atom by
the way, is an ordinal by my definition. However this
is easily fixable - remove the words "Z = Y or" from
point (2).

Right, my mistake.

But it seems to me there is another problem with your
formulation.

Now you have: "Any subset of X has a "least element"
(i.e. an element
Y such that for any member Z of X, we have Z is not a
member of Y)."

But I think that should be "for any member Z of the
nonempty subset
[...]".

Again, a rather stupid mistake on my part. You are right, of course.
.

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