On Jan 1, 6:55 am, Zuhair <zaljo...@gmail.com> wrote: > Add a new primitive to the language of ZFC, > this primitive is the binary relation "exemplifies" denoted by > the infix dyadic symbol ~
Zuhair, you are trying to define one nebulous term (mathematics) with another nebulous term ("form"). In fact, "form" is so nebulous that it doesn't even have a definition itself, so now you have to define what IT is. That's not how definitions work.
You define something using familiar, well understood, accepted terms and concepts. Then we immediately gain the wealth of knowledge of firsthand experience using these terms and concepts. And personal experience can even give meaning to something inherently unfathomable. What does it mean to think? That is fairly nebulous. But since we all do it most of the time, we can accept references to "thinking".
That is why I keep pointing out that you have not established the essential properties of your system - that it defines exactly math, no more, no less: there is no intuitive meaning to your terms because you are coining them as you go along. You even use odd syntax where $ appears in an expression next to other punctuation characters.
Both Hilbert and Einstein tell us any theory can be explained in simple terms if we understand it enough. That is because understanding means to equate something with our past experiences which we know so well and are so comfortable with. Not introducing another undefined term. Undefined term Math = = Undefined term Form = = ???
> Define: > > R is a form defining relation iff > R is an equivalence relation & > For all x. ~x=0 -> For all s. Exist y. y R x & s in TC(y) > > TC(y) refers to the 'transitive closure of y" > defined in the usual manner as the minimal > transitive superset of y. > > To the axioms of ZFC add the following axiom scheme: > > Forms: if R is a binary relation symbol, then > > [R is a form defining relation -> > For all x. Exist! $: For all y (y ~ $ <-> y R x)] > > is an axiom. > > Each $ here is said to be a form defined after R and x. > / > > Now the idea is that For any theory T if T can be > interpreted in ZFC+forms in a manner such that > each object in T is interpreted as some form in > ZFC+forms and all primitive relations of T are > interpreted by defined relations in ZFC+forms; > then T is a mathematical Theory. > > Example: Peano Arithmetic "PA" is interpretable in > in ZFC+forms by an interpretation where all objects > of PA are interpreted as forms each defined after > equivalence relation bijection and some set in > the following manner: > > Define (natural number): # is a natural number iff > (Exist x. x is finite & for all y. y ~ # <-> y bijective to x) > > It is just straightforwards to interpret all primitive predicates of > PA by 'defined' relations in ZFC+forms. > > So PA is a mathematical Theory. > > This is what I paraphrase as "PA supplying a consistent > discourse about form". What I mean by "consistent discourse" > is being interpretable in a consistent extension of logic, > here ZFC+forms is the consistent extension of logic and > since PA is interpretable in it then it provides a consistent > discourse of its objects and since its objects are interpretable > as forms in ZFC+forms then PA is said to provide a consistent > discourse about form. Any theory that is so interpretable > in any consistent theory that can define forms is accordingly > said to be a theory that provides a consistent discourse > about form and thus it is "MATHEMATICAL". > > So roughly speaking this account views mathematics as being > about "Logic of Form". > > Para-consistent (inconsistency tolerant) discourse might be > interesting if proves to be indispensable by the use > of consistent discourse, and thus theories supplying > para-consistent discourse about form would be > designated also as MATHEMATICAL. Actually I tend to think > that any discourse about form other than the trivial discourse > of proving everything if proves indispensable by use of other kinds > of discourse, then it would be mathematical > > So Mathematics can be characterized as: > > Non trivial discourse about form. > > The difference between mathematics and science is that the later is > about establishing the TURE discourse about the objects it negotiates, > while mathematics is about supplying any NON TRIVIAL discourse about > the objects it negotiates (i.e. forms) whether that discourse is what > occurs in the real world (i.e. True) or whether it doesn't (fantasic). > Mathematics supplies a non trivial discourse of form, so it supplies a > language about form, and as said above more appropriately put as > supplying: Logic of form, whether that logic is consistency based or > paraconsistent. The truth of that, i.e. reality matching of the > discourse about those forms is something else, this really belongs to > a kind of physics rather than to mathematics. The job of mathematics > is to supply the necessary language that enables us to speak about > forms, not to validate its truth. > > Anyhow that was my own personal opinion about what constitutes > mathematics. > > Zuhair