On 06/04/2013 2:34 PM, fom wrote: > On 4/6/2013 11:48 AM, Nam Nguyen wrote: >> On 05/04/2013 12:20 AM, fom wrote: >>> On 4/5/2013 12:57 AM, Nam Nguyen wrote: >>>> On 04/04/2013 10:55 PM, fom wrote: >>>>> >>>>> Who knows what is and what is not -- even >>>>> in the simple realm of mathematics -- claims >>>>> a certain knowledge that is revealed rather >>>>> than discerned. >>>> >>>> So, since Godel, is the knowledge of the natural numbers >>>> a revealed or discerned one? >>>> >>>> Revealed by whom? Discerned from what? >>>> >>> >>> I thought you claimed to be a relativist. ??? >> >> I am, by at least the 3rd Principle "Principle of Symmetry (of >> Non-Logicality)" mentioned in: >> >> http://groups.google.com/group/sci.math/msg/20bb0e7c183ae502?hl=en >> > > What appears to be a problem with your principle is > that one does not know what is and what is not provable > to begin with.
You seem to misinterpret the principle, which is actually a logical one.
Assuming that the formula A is neither a tautology or contradiction, it's impossible to conclude A or to conclude ~A from (A \/ ~A). Therefore it's _relative_ to your choice to choose which of A, or ~A be your axiom. Ditto for the dichotomy (A \/ B): it's relative to which of A and B you'd choose. This is in the realm of syntactical proof via rules of inference.
In the realm of language structure construction, _some_ descriptions (i.e. the definition of structure instance) about infinity will have to be incompletely described, defined. And in such incomplete definition of the structure you can't not verify the truth value of A or of B, if you assume the dichotomy (A \/ B) is true to begin with. A relativity of _freely_ assuming either A or B be true is thus born.
If you don't want a chance for the relativity to exist, your should _NOT_ have an _axiom_ of the form of a dichotomy, even in disguised form such as a trichmotomy.
Fwiw, Shoenfield's axioms of PA has an equivalence of a trichmotomy axiom: (x<y \/ x=y \/ y<x). Thus, even from a textbook from an author, you could already see _some_ signpost of mathematical relativity.
> > To establish the "relativity" of which you speak, one > must demonstrate a model for a given statement and and > a model for that given statement's negation.
No. One simply just needs to prove the set of ordered-pairs _intended to be a structure_ is incompletely complying to the definition of language structure.
If a non-compliance exists, there exists a relativity. _It's only a matter of definition compliance_ and It's that simple.
>>> The history of mathematical logic is entwined >>> with a philosophical perspective that is collapsing >>> from its own analyses. Goedel argued for >>> idealism and platonism. That is not the logicism, >>> logical positivism, or predicativism that tried >>> to ground mathematics on the basis of realism >>> and otherwise characterized the era in which he >>> worked. >> >> So why do people react negatively with the _logical notion_ that the >> concept of the natural numbers Godel used is a relativistic concept? >> >> > > Logic is a complicated topic: > > http://plato.stanford.edu/entries/logic-ontology/#DifConLog > > http://plato.stanford.edu/entries/logical-consequence/#3 > > And, there are many more wonderful entries at that > site.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.