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Topic: How simple Z is?
Replies: 2   Last Post: Apr 24, 2013 8:46 PM

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Graham Cooper

Posts: 4,417
Registered: 5/20/10
Re: How simple Z is?
Posted: Apr 24, 2013 8:46 PM
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On Apr 25, 5:54 am, Zuhair <zaljo...@gmail.com> wrote:
> On Apr 24, 8:58 pm, Zuhair <zaljo...@gmail.com> wrote:

> > On Apr 24, 9:50 am, Rupert <rupertmccal...@yahoo.com> wrote:
> > > On Tuesday, April 23, 2013 9:23:58 PM UTC+2, Zuhair wrote:
> > > > Pre-Z is the closure of all of what is provable from the logical
> > > > axioms of first order logic and the following axioms by the rules of
> > > > inference of first order logic (Hilbert style).
> > > > Comprehension: if phi is a formula then a set {x C A| phi} exists.
> > > > Infinity: Exist N. 0 in N & for all m. m in N -> {m} in N.
> > > > /
> > > > where C is the known "subset" relation.
> > > > It is nice to know that Pre-Z interpets Z. And that MOST of
> > > > mathematics can be formalized within it (through its interpretation of
> > > > Z of course).
> > > > Zuhair
> > > You use the symbol {m} as though it's guaranteed to have a unique referent, but you're not assuming extensionality. When you say {m} in N, do you mean (Ex)(x in N and (Ay)y in x <-> y = m) ?
> > Yes of course, I also use the symbol {x C A|phi} exists without any
> > guarantee for uniqueness (that's why I wrote a set and not the set in
> > comprehension), this can be used and it is understood to mean (Exist
> > s. for all x. x in s iff x C A & phi(x)) where s is not free in phi.
> > It is just a matter of abbreviated expression.

> > Zuhair
> The interesting thing is that almost all of mathematics can be seen to
> logically follow from such trivial non logical axioms, the first
> scheme that says that for any collection there is a collection of all
> parts of that collection, which is just a trivial constructive theme
> and it is intuitively appealing, however the other one that asserts
> the existence of some infinite set, albeit not that intuitive yet it
> is a very acceptable ideal, and actually it lies at the meta-level of
> logic itself, and it can be seen to be as a strong ideal that is
> necessary for easy mathematical reasoning actually, otherwise we'll
> need to replace it with weaker very complex statements that are not
> beautiful at all and not that motivating.

No its simplicity comes at a great price.

Comprehension: if phi is a formula then a set {x C A| phi} exists.

Z is a SUBSET theory, not a set theory.

That is why you need to add PAIRING and all the other AXIOMS

just to actually specify *some* sets A first!

because you cannot even define any set via a formula alone.


You simply don't have the fundamental mechanics (yet)

to handle a full set theory.

E(S) A(X) XeS <-> p(X)
E(S) A(X) XeS <-> p(X)


Instead you are STUCK with syntactic line by line predicate calculus

as your foundation of all formula.

Predicate Calculus SEEMS to work as a building block

but it only constructs F OR ~F

which is inherently useless since Formula must be F XOR ~F
to form a logic.

The *DREAM* of building up the Mathematical Universe V
from Predicate OR NOT(Predicate) OR BOTH
as foundation, with SET C SET C SET... on top is just
vague anatomical notes of the Skeleton.

Z does nothing on it's own, it's a syntax not a theory.


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