After setting the foundation on set theory by working on number sets and proving that the set R of real numbers is larger than the set N of natural numbers, Georg Cantor started working on a puzzling issue that was later called the Continuum Hypothesis. This deals with the following question “Is there a set of cardinality larger that set N and smaller that set R (the continuum)?”
Cantor believed that the answer to the question is negative but he did not managed to prove it and no one else did. So 150 years later this belief is formed as a hypothesis and it is still open, for most people.
Perhaps the most complete article on the hypothesis may be found here, it explains in details the whole issue its implications and the progress so far.
Researchers mainly use logic and axiomatic set theory on their attempts to prove the hypothesis. The research efforts focus on applications of the Zermelo-Fraenkel axioms with the axiom of choice; known as ZFC axiomatic system. The problem on this approach is that the hypothesis has been proven by Gödel and Cohen to be independent of ZFC. Since then, the scientific community has been divided in two parts. Scientists like Cohen believed that independence imply undecidability and the hypothesis has no answer. Others, like Gödel, believed that the ZFC system is not appropriate of resolving the hypothesis and some new system of axioms should be adopted.
This post is about some remarks that I have about the hypothesis and the ZFC application on it. The belief that ZFC independence results undecidability clearly implies that the ZFC system is universal on the sense that the properties of every possible set may be effectively described by the system. As ZFC is an axiomatic system and some of its axioms rely on intuition its universality should not be taken for granted.
For instance, the well-ordering theorem that is equivalent to the axiom of choice states that in every non-empty set there is one least element based on some property of the set. If we consider a set of uncomputable numbers how can we claim that any element of the set is the least element? It would be fruitful to investigate new axiomatic systems or try to provide stronger evidence about the validity of the currently used axioms.
Now let us suppose that the hypothesis is false and there is set H the cardinality of which lays between N and R. Set H is uncountable and its elements are uncomputable.
Set H is uncountable as it is larger than N. Considering the Turing machine model as a universal model of computation and as the set of all Turing machines is countable there are more elements in H than Turing machines so there are elements of H that do not correspond to a Turing machine and as a result they are uncomputable. There are uncomputable numbers in R as well and these considerations imply that a system or a theory that could be used to prove or disprove the existence of H should be able to handle uncomputable numbers and their properties.
Cantor believed that the answer to the question is negative but he did not managed to prove it and no one else did. So 150 years later this belief is formed as a hypothesis and it is still open, for most people.
Perhaps the most complete article on the hypothesis may be found here, it explains in details the whole issue its implications and the progress so far.
Researchers mainly use logic and axiomatic set theory on their attempts to prove the hypothesis. The research efforts focus on applications of the Zermelo-Fraenkel axioms with the axiom of choice; known as ZFC axiomatic system. The problem on this approach is that the hypothesis has been proven by Gödel and Cohen to be independent of ZFC. Since then, the scientific community has been divided in two parts. Scientists like Cohen believed that independence imply undecidability and the hypothesis has no answer. Others, like Gödel, believed that the ZFC system is not appropriate of resolving the hypothesis and some new system of axioms should be adopted.
This post is about some remarks that I have about the hypothesis and the ZFC application on it. The belief that ZFC independence results undecidability clearly implies that the ZFC system is universal on the sense that the properties of every possible set may be effectively described by the system. As ZFC is an axiomatic system and some of its axioms rely on intuition its universality should not be taken for granted.
For instance, the well-ordering theorem that is equivalent to the axiom of choice states that in every non-empty set there is one least element based on some property of the set. If we consider a set of uncomputable numbers how can we claim that any element of the set is the least element? It would be fruitful to investigate new axiomatic systems or try to provide stronger evidence about the validity of the currently used axioms.
Now let us suppose that the hypothesis is false and there is set H the cardinality of which lays between N and R. Set H is uncountable and its elements are uncomputable.
Set H is uncountable as it is larger than N. Considering the Turing machine model as a universal model of computation and as the set of all Turing machines is countable there are more elements in H than Turing machines so there are elements of H that do not correspond to a Turing machine and as a result they are uncomputable. There are uncomputable numbers in R as well and these considerations imply that a system or a theory that could be used to prove or disprove the existence of H should be able to handle uncomputable numbers and their properties.
