Recent content by Adam Mclean

Apparently in his book 'A decision method for elementary algebra and geometry', 1948, Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable. This seems at first glimpse in contradiction to Godel's Incompleteness Theorem, but of...

There is very clear article by David Pierce on Model theory at http://www.math.metu.edu.tr/~dpierce/philosophy/Kant/model-theory-gen.pdf

I wonder if I have perhaps been seeing the problem of the Incompleteness of formal mathematical systems the wrong way round. It seems that Godel Incompleteness can only be expressed and defined in axiomatic systems built on first-order logic. For axiomatic systems built on second-order...

Fraenkel and Bar-Hillel use this term within their description of the Church-Rosser undecidability Theorem which says Elementary arithmetic is essentially undecidable. They expand on this "Rosser was able to prove... that every consistent extension of first order arithmetic is...

Surely one can produce many models of the integers within the reals. The Real numbers 1.000, 2.000, 3.000,... are only one of many possible models of the integers within the reals. 0.1, 0.2, 0.3,... would be another (though one needs to define successor and division...

Regarding how one axiomatic theory can inherit undecidability from another, I have just read the following in Fraenkel and Bar-Hillel 'Foundations of Set Theory' : "The scope of undecidability proofs was greatly increased when it was proved ... that an essentially undecidable and finitely...

This was my question that opened this thread. As far as I understand the axiomatic system of the Real numbers are consistent. I am trying to understand how Godel's Incompleteness theorem does not work for the reals. I am accepting that the Reals have been proved consistent.

Could I please intrude further on your generosity by asking you to explain this a little further, or perhaps point to a book or web page that might help me understand this point.

First-order logic is complete - Gödel's completeness theorem Formal system of arithmetic is incomplete - Gödel's Incompleteness theorem. Arithmetic is undecidable - Church's Theorem or Thesis (not yet proven) Theory of real numbers is decidable - Tarski's theorem Have I got that right ?

You are right. By this counter-example, the defining of unique successor does not lead to incompleteness. Although not a sufficient condition it could still be a necessary condition for incompleteness. We can also see that Presburger arithmetic as it does not define multiplication has no...

Does not the decidability of a formal axiomatic system imply the completeness of the axioms ?

Can I ask how one can construct a model for the real numbers from the integers ? For real numbers there is no concept of the successor of a number, whereas this is the key concept for the integers. Indeed the idea of successor is one of the Peano axioms for the integers. It seems to me...

I think that the complex numbers, quaternions, octonions, and so on, are merely defined as extensions of the real numbers, with additional axioms which define the i, j, k and the way of doing additions and multiplication of complex numbers or quaternions. Thus they will not change the...