Re: Why? [was Re: Cantor`s powerset theorem is false?]

apoorv wrote:
I am afraid I am still not able to see how you reach ~w={w] without the
use of Separation

Remember 'w' is being used as a constant, not a variable, so 'w' (read
'omega') stands for the set of natural numbers. We do not claim to
prove, without the axiom of regularity, Ax~x={x}, as we've discussed
that that sentence is independent of ZF without the axiom of
regularity. But as to w in particular, we do prove ~w={w}. And just to
prove the existence of w, we use the axiom of separation. I didn't
claim that the axiom of separation is not needed here.

and how you can separate a subset out of an entity such as z={z} .

I don't know what specific problem you see. (1) A set has subsets
whether the set obeys regularity or not. For example, z above has at
least the empty set and itself as subsets, no matter what else it may
have as subsets. (2) In regards the proof that ~w={w}, there is no set
z such that z={z} comes into play in that proof.

(Even
Koskensilta seems to have a different perception than you on the
applicability of Separation to entities not a priori delared sets.)

I haven't read any post by Koskensilta that shows that he and I
disagree as to what the axiom schema of separation is or what rules of
inference for first order logic to apply. The axiom schema of
separation applies just as I stated it, which does not require any
detour into consideration of what are "a priori declared sets"
(whatever that might mean).

But
let me think this over and if I eventually agree with you (get
enlightenment ), I will happily revert back to this forum (and make
your week!).

But just thinking it over is not what is needed. What is needed is to
understand first order predicate logic and the axioms of set theory.

As to the motivation, I find it 'natural' to think that if the
relationship of 'belonging to' is iterated an infinite number of times,
than the resultant entity 'belongs to' itself ( 'an onion with an
infinite number of layers').

Then it's good that set theory does not embody your natural thinking,
since otherwise we'd have an inconsistent theory.

If A(k+1)={A(k)}, then either A(w)={A(w)}
or A(w) does not exist.

I don't know what A you have in mind such that A(k+1) = {A(k)} and what
relevence it has to the present discussion. If it's the successor
operation that you have in mind, then the formula is S(k+1) = k+1 u
{k+1} = S(k) u {S(k)}. But that is not a definition of the successor
operation but rather is a consequence of the definition as applied to
k, for all k such that k is a successor ordinal. w is not a successor
ordinal, but the successor operation applied to omega is w u {w}, just
as the successor operation applied to any x is given by the definition
S(x) = x u {x}.

Since w involves precicely such an infinite
iteration,

What is PRECISE is the mathematics. What is IMPRECISE is your
characterization of the mathematics as "involving infinite iteration".
We did not define w as depending on an "infinte interation". We defined
w as the least set that is successor inductive. The mathematics issues
from THAT definition, which is precise, not from imprecise notions as
to "infinite iterations".

I find it almost an irresistible proposition that w e w.

I don't. I find the notion to be utterly counter-intuitive. But the
mathematics does the resisting anyway. The axioms prove ~wew, and there
has been no demonstration yet seen that the axioms are inconsistent.

So,
I am looking at the logical formal constructs to understand why this
is not the case or why it could be the case.

Understanding the logic is understanding first order predicate logic
and the formulation of the axioms in a first order language.

MoeBlee

.

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