Re: Contradicrtion-free mathemattics (The new nonstandard analysis

>The three axioms are all listed in my first post of Jan. 5. At any rate, here are the
>axioms of the new real number system R*, +, x.

>1) R* contains the basic integers 0, 1, . . ., 9.
>2) The addition table
>3) The multiplication table.

Those aren't axioms in the sense that anyone else uses the word. As
far as anyone can tell by reading statement 1, R* is exactly the set
{0,1,2,3,4,5,6,7,8,9}.

I want a formal axiomatization in the sense of mathematical logic. I
want the signature of your theory and the axioms written as formal
sentences in predicate calculus using that signature. Only then can
you claim that you have actually stated the axioms.

You ignored my question about what you consider a proof to be. What
inference rules do you allow? How can you draw a conclusion that isn't
an axiom already?
In ordinary arithmetic, addition is associative and so (d* + x) - x =
d* + (x-x )= d* + 0 = d*, so (d* + x) - x = d* = 0 if d* + x = x.
That's because the real numbers are a group under addition...
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My construction is built on full critique of foundations and the real number system and is fully explained in my posts. But let me highlight some important points. 

I require that every mathematical space and its concepts (symbols, terms, etc) are well-defined by a consistent set of axioms. I reject undefined, ill-defined and ambiguous concepts because they are sources of contradictions. External rules of inference are not well-defined by the axioms. The rules of inference must be specific to the mathematical space and well-defined by its axioms. Proofs rest entirely on the axioms.

Since every mathematical space is well-defined only by its axioms distinct mathematical spaces are independent and any proposition involving concepts from both is ill-defined, nonsense. In particular, any proposition including undefined or ill-defined concepts is ill-defined, ambiguous, nonsense. They are the the undecidable propositions in the sense of Godel.

Regarding d* it is a new real number, not a real number, and has special properties much like that of an infinitesimal: epsilon + x = x.

E. E. Escultura
.

Relevant Pages

  • Re: Contradicrtion-free mathemattics (The new nonstandard analysis
    ... The three axioms are listed in my first post of this thread. ... They well-define the mathematical space called the new real number system and they well-defined the rules of inference as well. ...
    (sci.math)
  • Re: Contradicrtion-free mathemattics (The new nonstandard analysis
    ... You haven't given nearly enough detail for anyone to find a contradiction.* Please: ... Start with a list of the undefined notions, then give a formal statement of each of the axioms. ... The three axioms are all listed in my first post of Jan. 5. ...
    (sci.math)
  • Re: Contradicrtion-free mathemattics (The new nonstandard analysis
    ... >The three axioms are all listed in my first post of Jan. 5. ... I want a formal axiomatization in the sense of mathematical logic. ... sentences in predicate calculus using that signature. ...
    (sci.math)
  • Re: Torkel Franzen on truth
    ... And they communicate ... if all I give you is the signature. ... The axioms are what matter, and, in fact, even THEY could be phrased ... which are the only OTHER way to clarify what the predicate MEANS. ...
    (sci.logic)
  • Re: Continuum hypothesis
    ... set of axioms, T_0. ... entails absence of any inputs to the first-order-language construction ... IS NO P IN THE SIGNATURE, how can you get P into the language? ...
    (sci.logic)
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