Re: Which axiom prohibits this kind of construction?

On Oct 14, 12:58 pm, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
MoeBlee <jazzm...@xxxxxxxxxxx> writes:
On Oct 14, 9:40 am, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
MoeBlee <jazzm...@xxxxxxxxxxx> writes:

How do we get that? I know there are various fixed point theorems for
certain kinds of ordered sets, but I'm not versed enough to know about
one that pertains to an operation symbol across all sets.

In this case, you can prove it by a simple construction.  Define

S_0 = {0}

S_{i+1} = F(S_i)

If you're using an ordinay definition by recursion theorem to define a
function S on w, then you need to have already proven the existence of
some superset of the range of the function CLASS F restricted to w.
(Or maybe there's another way to do it that I'm not recalling?)

To be perfectly honest, I'm not sure what point you're trying to make
here.  I assume this has something to do with the fact that I'm
defining a least fixed point for an operator, and not a function in
Set.

Since I just complained about another poster wasting my time on a dead
end, I don't wish to waste your time on my question if it is indeed
fruitless. My concern about your S might be easily answered by using
some a form of recursion other than the most ordinary recursion on w.
So, if this part of the exchange is not profitable for you, I don't
blame you for not following up.

That said, if I understand, you're defining a function S on w by
recursion.

Here's a typical formulation of a justifying theorem for such a
definition:

If f is a function from X into X and c in X, then there exists a
unique function S on w and into X such that
S(0)=c and
for all i in w, we have S(i+)=f(S(i)).

But your f (F in your case) is a function CLASS and is not merely from
some X into X. That does suggest transfinite recursion, I think. I'm
just saying that I'm rusty as to the details that would permit such a
transfinite recursion in the particular way you've formulated it (i.e.
only on w, and not on ordinals in general), which suggests that S (as
a set) is the restriction of some other function class defined by
transfinite recursion.

Now, as I mentioned I might be overlooking some other recursion
theorem (recursion on a PARTICULAR ordinal courtesy of a more general
transfinite recursion?) that allows not already having X at our
disposal. I am a bit rusty on that matter, so if you think it is a
tedious matter, but it is indeed routine, then I wouldn't blame you
for not following up.

But if you wish to follow up, to cut the matter short, it would
suffice merely to state the specific definition by recursion theorem
that justifies S.

But I don't see the issue.  Seems to me that the details of taking U
S_i can be settled with tedious reference to Replacement with some
appropriate recursion,

I'm not objecting to taking the union of the range of S once we do
have such a function S as you describe it. I'm just wondering as to
the admittedly tedious particulars that justify asserting the
existence of S as you describe it.

MoeBlee
.

Relevant Pages

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