On Saturday, 8 October 2022 at 02:00:14 UTC+2, Mostowski Collapse wrote:
schrieb am Sonntag, 20. Juni 2021 um 16:19:11 UTC+2:
> > Let U be the universe of all numbers.
> Thats something for wonky man. He also makes a recurring
> error to start modelling the universe of discourse, as a
> set inside the universe of disourse.
What are you even talking about?! Indeed, he may have
"logical" problems, but the one who keeps throwing and
thriving on just word salad is you.
So, if I read/unpack it at face value:
Firstly, to the point you raise above, the "universe of
discourse" here is more than just sets, no doubt on that,
or we would not write "ALL(s)-such-that-s-is-a-set" and
would just write "ALL(x)". Rather, in that universe, there
must be things called sets, thanks the predicate Set(x).
That said, the axiom is stating that "for every set 's', there
exists an *object* 'a' (i.e. not necessarily a set, as stated!)
such that 'a' not in 's'".
Now, should sets here only contain other sets (as they
standardly do, and as I guess the intention here was),
via some other assumption/axiom not shown here, then
the axiom above is poorly written to begin with, as an 'a'
not constrained to be a set could not appear on the LHS
of that epsilon to begin with.
So, let's rather consider the two cases:
1) if 'a' is constrained to be a set, the axiom ends up
stating the indicated "there is no set of all sets"... or
does it?? In a well-founded theory of sets the axiom
is indeed and necessarily true since unique candidate
for such 'a' would be the set of all sets itself, but then
*it* is no element of any set not even itself, so no cigar;
with non-well founded sets I suppose it becomes
possible to have a set of all sets, and the above is a
"contingent" axiom of the kind "thou shall not have a
such thing" and the consequences come down
2) if 'a' instead is an arbitrary element from the domain
of discourse, the axiom is vacuously true if there actually
exist objects that are not sets in the universe; otherwise
(I have not given it much detailed thinking, better double-
check all of the above.)
> 2 Set(nat)
> 3 Set(nat) => EXIST(a):~a ε nat
> U Spec, 1
> 4 EXIST(a):~a ε nat
> Detach, 3, 2
> Oopsi, what is this thing outside of N?
> Whats is the classical way, for example in FOL, to not
> be subject of this paradox, the wonky man paradox?
To begin with, don't write gibberish. Indeed, what are you
even trying to say: that 'nat' is "a universe"?? Of course it
is *not*, i.e. at least as long as there exists in the universe
things, even sets!, that are not in fact natural numbers.