
Re: Tim Chow in Forcing for dummies
Posted:
Nov 11, 2017 4:27 AM


Am Samstag, 11. November 2017 02:13:23 UTC+1 schrieb FredJeffries:
> > > He is saying that every model of ZF contains the actual, true, straight from the holy volcano omega. > > > > Even given that, I chose to engage him on the facts that the reals, that is the powerset of the naturals, can differ from model to model. > > Even more than that, the empty set can differ from model to model.
No. According to ZF, there is one and only one empty set because a set is defined by its elements. See also Wikipedia: "the empty set is the unique set having no elements". Further see Zermelo (1908), p. 263: Es gibt nur eine einzige Nullmenge.
It is irrelevant whether you name it 0 or { } or $ or empty set. These names are independent of any model.
> Because the epsilon relation (is an element of) differs from model to model.
That is irrelevant to the form of the sets that are defined precisely. Inside and outside the question whether a = b is uniquely defined because it is always uniquely defined. > > FROM OUTSIDE, the empty set in a particular model can look like it contains things:
In a model of ZF, in particular of Axioms IV and VII the empty set is defined by 0 or { }. > > Let M be out model of set theory. Let $ be something that does nor exist in M. > > Now, let's create a new model from M, call it M'. M' take all of the sets of M and all the epsilon relations. Now, just sneakily insert $ into each set, but do not change the relations between the sets. (No, $ is NOT an urelement). > > So in M', the empty set will be {$}.
That is not a model Axiom VII, because there the form of the empty set is uniquely specified to be 0.
> But the model M' "doesn't see" that $.
Perhaps the author of this nonsense cannot see that this is contradicted by Axiom VII? Of course you can denote the empty set by $, but that must not change from inside or outside, because otherwise the model is not a model.
> It's not what the "sets" in the models actually ARE. It's only how they are related to each other via epsilon that is important.
That is wrong. At least for a model of Axiom VII which uniquely specifies what the sets actually are 0, { }, {{ }}, ...
> Just like it doesn't matter what the actual elements of a group are, it's only the multiplication table that counts.
Wrong again. The group axioms do not specify the elements. Axiom VII however uniquely specifies a minimum set of elements 0 and with a also {a}.
> And the relations in M' are exactly the same as in M. The content has changes the the forms are identical (isomorphism).
Axiom VII does not only specify relations! It specifies sets. Otherwise no infinity would be reached. > > {$} in M' has each and every property of the empty set that can be proved from the axioms of set theory.)
No. According to Axiom VII of set theory the empty set is defined as different from any set containing something and is unique. > > But The Professor cannot see this simple fact because, to him, there is one and only one empty set and it is {} and doesn't contain anything.
To me the Axiom is the only arbiter.
> And THAT empty set MUST be in every model and generate omega in every model and therefore omega must be the same in every model (THAT doesn't follow either, but that's for another time).
That follows from Axiom VII.
> Because THAT is his ridiculous, outofcontext exegesis of the holy writing of Zermelo.
There is no exegesis required. There are the words, clearly visible. See p. 266f of Zermelo (1908). > > > Of course, your knowledge of set theory and logic far excedes mine. > > I don't know diddly about set theory.
But you should nevertheless be able to recognize that the set 0 is uniquely specified in Axiom VII, that is the difference to group axioms, usually concealed by "logicians". > > > However, it doesn?t take much to challenge the Great Professor. > > The Professor isn't interested in challenges.
Challenges and simple lies are different things.
Regards, WM

