Re: An uncountable countable set

David Marcus wrote:
Ross A. Finlayson wrote:
David Marcus wrote:
Ross A. Finlayson wrote:
David Marcus wrote:
I thought you said there was a contradiction in ZF. In the context of
ZF, the Burali-Forti argument shows that there is no set of all
ordinals, but does not lead to a contradiction. So, do you still say
there is a contradiction in ZF? If so, what is it?

{For any x: x is a set} = emptyset <=/=> {For any x: x is a set} = U
(V, L)

That says, for any x, that's the empty set, and, for any x, that's the
universal set, it seems sufficient to show the universe non-empty.

There is no set of ordinals nor cardinals in ZF. Yet, because there's
the axiom of infinity, those infinite ordinals/cardinals as there ever
would be are claimed to exist, basically where they're all hereditarily
finite, those ordinals of the cumulative hierarchy.

Sorry, but I don't follow. Are you saying this is a contradiction within
ZF? By "within" I mean that ZF proves this contradiction.

You seem like you know what you're talking about, which is good.

Basically, yes, I say the existence of the (universal) quantifier,
where the word universal is in parentheses because there's the mutually
implicit existential quantifier, that the existence of the universal
quantifier in ZF lead to illustration of a contradiction derivable from
ZF.

OK. But the Burali-Forti argument does not (as far as I know) lead to a
contradictionn derivable from ZF. So, what is the contradiction that you
can derive from ZF?

I have some other arguments along those lines as well, of similar tack,
where I advocate axiom-free natural deduction, as a return of sorts to
a more "naive" set theory with post-Cantorian acknowledgement.

That is to say, there are thousands of pages more of my opinion about
these matters readily available.

--
David Marcus

Hi David,

I just mentioned Burali-Forti to see if you had heard of it. Basically
the consideration is that because of the mechanistic structure of the
ordinals, the set of ordinals, or counting numbers extended to the
hyperfinite, because of the mechanistic structure of the ordinals the
set of ordinals would also be an ordinal.

That's an example of where the transfer principle holds and what is
true for elements of the set is true for the set, here, in terms of set
membership, definition.

So, in ZF there is no set of ordinals. That's basically the very same
argument as that there is no complete infinity, in for example the
finite ordinals also known as the whole, counting, or natural numbers,
that the set of those can not exist, for, the successor in natural
generation of the mechanistic ordinal is yet another mechanistic
ordinal.

So, looking at the Burali-Forti "paradox" illustrates how unresolved
"paradoxes" earlier in development of definition, in this case about
infinity, reexhibit themselves under further analysis of the system.

I hope that was not misguiding, about inconsistency in ZF. I think the
existence of a universe is implicit with the existence of a quantifier.
ZF claims a quantifier, with "restricted comprehension", as has been
discussed here before. Yet, even the definition is couched in terms of
"unrestricted comprehension."

An axiomless theory still has logical axioms, it just doesn't have
proper, or illogical, axioms.

Ross

.

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