|
|
Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 11, 2013 1:53 PM
|
|
On Feb 10, 2:47 am, fom <fomJ...@nyms.net> wrote:
> On 2/9/2013 6:19 PM, Charlie-Boo wrote:
>
> > On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote:
>
> >> How do you see Logic and Set Theory as being the same?
>
> > Both are concerned with mappings to {true,false}. A propositional
> > calculus proposition is 0-place. A set is 1-place. A relation is any
> > number of places. (A relation is a set - of tuples.)
>
> > So you have the same rules of inference: Double Negative, DeMorgan
> > etc. apply to propositions and sets.
>
> > To prove incompleteness, Godel had to generalize wffs as expressing
> > propositions to expressing sets when the wff has a free variable.
>
> Hmm...
>
> This is naive set theory (which you have stated
> as being fine with your views).
>
> I view set theory as being about the existence
> of mathematical objects. Naive set theory failed,
Failed meaning? There is nothing wrong with naïve set theory.
A. A wff maps SETS to SETS. E.g. if P(x,y) is a set then (exists
M)P(M,x) is a set.
B. x ~e x is not a set.
C. x = y is a set.
D. For any set M, x e M is a set.
How much of ZF can you prove from this? LOTS!
There are 3 kinds of formal systems:
SIMPLE: Correct and well-designed e.g. Propositional Calculus,
Combinatory Logic and CBL.
COMPLEX: Correct but can be smaller (Occam's Razor is not being
satisfied) e.g. Peano Arithmetic is just any logic in which TRUE, ADD,
MUL and their complements are representable.
HAIRY: False because all the primitives used in the hairy part are not
axiomatized nor are their properties even known. A correct
formalization is to axiomatize each primitive and all the primitives
are in fact primitives - small e.g. ZF axioms are sometimes very hairy
and guilty of this. There are no axioms for functions and other
concepts contained with the harriest ZF axioms.
C-B
> in part, because of something in Aristotle--do not
> negate "substance". Do not get me wrong. I am
> not planning to run out and buy a number 2 while
> I pick up my next Turing machine....
>
> The problem, however, is that the connection of
> mathematics to any metaphysical truth (if such
> a statement can be sensible) requires that the
> objects represented in physics books (material
> objects) correspond with some sort of mathematical
> notion. So, while mathematics is abstract,
> there must be some sort of interpretation that
> accounts for its apparent ability to model
> real-world situations.
>
> Either physics is a collection of mathematical
> hallucinations or there is a better explanation
> of set theory.
|
|
