Re: The set of All sets

zuhair wrote:
> MoeBlee wrote:
> > Herman Rubin wrote:
> > > In article <1134419772.606126.252290@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > MoeBlee <jazzmobe@xxxxxxxxxxx> wrote:
> > > >zuhair wrote:
> > > >>> Now All ZFC axioms contains a term called "set"
> > >
> > > >No axioms of ZFC mention 'set'. One can define the predicate 'is a set'
> > > >or even omit having such a predicate. Nothing in plain ZFC depends on
> > > >having a predicate 'is a set'. One should not confuse informal and
> > > >intuitive mention of 'sets' with 'is a set' as a formal predicate of
> > > >the langugae. The only primitives of the language of ZFC are the
> > > >2-place predicates 'equal to' and 'is a member of'. (We can even define
> > > >'equal to' and have only 'is a member of' as the only primitive if we
> > > >like.) Anything else can be definied precisely and without circularity.
> > >
> > > >MoeBlee.
> > >
> > > In ZF, everything is a set. In ZFU, there are individuals
> > > also, so everything is either a set or an individual.
> > >
> > > In NBG, everything is a class, and a set is a class which
> > > is an element of a class. Similar remarks to the above
> > > apply to NBGU. The axiom of choice has no effect in either.
> >
> > I agree with all of that and nothing that I posted contradicts it. (I
> > say that only because I don't know whether you intended for your post
> > to be an addition to what I said or a correction of it.)
> >
> > Thanks,
> >
> > MoeBlee
>
> hmmm.............., let's see:
>
> axiom of extensionality
> states that:-Two "SETS" are the same if and only if they have the same
> elements.
>
> What is the meaning of SETS here.
>
> About predicate logic you are using , non of ZFC axioms mentions that
> explicitely.
>
> Zuhair

Hi Zuhair, "MoeBlee",

The predicate logic is basically beneath ZFC.

ZFC has what are called axioms, the axioms of ZFC. Those are also
called "non-logical" or "proper" axioms.

The "logical" "axioms" are combinations of true and false via union and
intersection and their results as true or false, i.e. the truth tables.

T and T => T
T and F => F
F and T => F
F and F => T

T or T => T
T or F => T
F or T => T
F or F => T

T nor T => F
T nor F => F
F nor T => F
F nor F => F

T xor T => F
T xor F => T
F xor T => T
F xor F => F

or

~ = and or nor xor

T ~ T T T F F
T ~ F F T F T
F ~ T F T F T
F ~ F T T F F

Actually those aren't the logical axioms per se, they're just not very
different from them. Also, they don't have the same definition as the
standard ones, where for example the column for "and" would be TFFF and
for "or" TTTF, and they generally have a different representation and
the above might just be seen as wrong. In electronic digital logic the
terms to consider are AND, OR, NAND, and NOR, and XOR and XNOR.
They're all a result, or derivative from, tautology, where tautology is
an equality or identity, and "not".

So then the "logical" axioms are basically not under discussion in most
considerations of the "non-logical" axioms. They're basically assumed
to hold because of the "law of the excluded middle", LEM or the LEM,
that something is either true, or false, and not both, and not neither.
Some statement of a subject x and predicate P or P(x) is either true,
or false, there is no middle ground, the middle is excluded. That's
the law of excluded middle, for if something were both true and false
it would confuse the truth tables.

Then, there are various considerations of indeterminate values that are
not necessarily true nor false, some call such a notion U for
Undeterminate.

In the West, truth tables are normally blamed on De Morgan, and
sometimes Boole or Venn. Those are important, key, and critical, and
used implicitly in all decision-making. Then the undeterminate values
are a fringe thing that started in some formality with Kleene and
Lucasiewicz.

http://en.wikipedia.org/wiki/Truth_table
http://www.google.com/search?q=%22logical+axiom%22

There's some good information in that, and almost undoubtedly some bad.

Ross

.

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