Hal Ruhl wrote:

A kernel of information is the that information constituting a particular potential to divide.

The All contains all such kernels.

The All is internally inconsistent because it contains for example a complete axiomatized arithmetic as well as an infinity of other such kernels of information.

So a set of all statements generated by an axiomatic system would qualify as a "kernel of information"? Even if you allow inconsistent axiomatic systems (as opposed to just consistent but incomplete ones), I still don't see why this makes the All inconsistent. After all, an axiomatic system is just a rule for generating strings of symbols which have no inherent meaning, such as "TBc3\". It is only when we make a mapping between the symbols and a *model* in our head (like 'in terms of my model of arithmetic, let T represent the number two, B represent addition, c represent the number three, 3 represent equality, and \ represent the number five') that we can judge whether any pair of symbol-strings is "inconsistent". Without such a mapping between symbols and models there can be no notion of "inconsistency", because two meaningless strings of symbols cannot possibly be inconsistent. And if we do assign symbol-strings a meaning in terms of a model, then if we find that two strings *are* inconsistent, that doesn't mean the symbols represent an inconsistent model, it just means that one of the statements must be *false* when applied to the model (for example, the symbol-string 7+1=9 is false when applied to our model of arithmetic). The model itself is always consistent. So unless you believe that inconsistent axiomatic systems represent true facts about inconsistent models, I don't think you can say the All must be inconsistent based on the fact that it contains rules which generate false statements about models as well as true ones.

So a set of all statements generated by an axiomatic system would qualify as a "kernel of information"? Even if you allow inconsistent axiomatic systems (as opposed to just consistent but incomplete ones), I still don't see why this makes the All inconsistent. After all, an axiomatic system is just a rule for generating strings of symbols which have no inherent meaning, such as "TBc3\". It is only when we make a mapping between the symbols and a *model* in our head (like 'in terms of my model of arithmetic, let T represent the number two, B represent addition, c represent the number three, 3 represent equality, and \ represent the number five') that we can judge whether any pair of symbol-strings is "inconsistent". Without such a mapping between symbols and models there can be no notion of "inconsistency", because two meaningless strings of symbols cannot possibly be inconsistent. And if we do assign symbol-strings a meaning in terms of a model, then if we find that two strings *are* inconsistent, that doesn't mean the symbols represent an inconsistent model, it just means that one of the statements must be *false* when applied to the model (for example, the symbol-string 7+1=9 is false when applied to our model of arithmetic). The model itself is always consistent. So unless you believe that inconsistent axiomatic systems represent true facts about inconsistent models, I don't think you can say the All must be inconsistent based on the fact that it contains rules which generate false statements about models as well as true ones.

Jesse