Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
Drexel University or The Math Forum.



Re: How simple Z is?
Posted:
Apr 24, 2013 8:46 PM


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: > > > > PreZ 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 PreZ 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 Aphi} 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 metalevel 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.
Herc  www.BLoCKPROLOG.com



