Structure:

http://en.wikipedia.org/wiki/Structure

Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities.

The universe displays patterns that strongly suggest it has a numerical structure. That structure is not dependent on the notion of a "set" though.

"at1with0" wrote: Numbers are sets, at least usually...So saying it has a numerical structure implies sets are tacitly involved.

Sets are still in their infancy as far as being a foundation for numbers. Cantor introduced the set concept in the 19th century but numbers existed before then. :ugeek:

"at1with0" wrote: I see what you're saying but if numbers aren't based on sets then you're gonna end up with multiple primitive, undefined notions -- I would think that minimizing the number of undefined notions is a good thing.

I will keep that in mind. :thumbup:

Reality might display an informational bit structure that reduces to 1 or 0 as its most fundamental or atomic elements.

The Continuum and other esoteric number ideas like imaginary numbers, pi, and such, might be represented by the uncollapsed wave-function or a "1". The finite natural numbers, and rational numbers can be the collapsed wavefunction, a "0".

These are my primitive and exploratory axiom thoughts that may or may not lead the thirsty theoretical horse to the trough... :wall:

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Whatever axioms we choose, there are going to be problems...

http://en.wikipedia.org/wiki/G%C3%B6del ... s_theorems

If mathematics structure is analogous to a well tailored suit, then there will always be loose threads hanging out of it no matter what axioms are chosen... :wall:

"at1with0" wrote: Godel's theorems don't worry me as much as that the word set is undefined.

I am not sure if I 100% understand Christopher Langan's solution to the liar paradox :geek:

http://en.wikipedia.org/wiki/L.....ris_Langan

Chris Langan

Chris Langan in his work The Theory of Theories states:

... Consider the statement “this sentence is false.” It is easy to dress this statement up as a logical formula. Aside from being true or false, what else could such a formula say about itself? Could it pronounce itself, say, unprovable? Let’s try it: "This formula is unprovable". If the given formula is in fact unprovable, then it is true and therefore a theorem. Unfortunately, the axiomatic method cannot recognize it as such without a proof. On the other hand, suppose it is provable. Then it is self-apparently false (because its provability belies what it says of itself) and yet true (because provable without respect to content)! It seems that we still have the makings of a paradox…a statement that is "unprovably provable" and therefore absurd.

But what if we now introduce a distinction between levels of proof? For example, what if we define a metalanguage as a language used to talk about, analyze or prove things regarding statements in a lower-level object language, and call the base level of Gödel’s formula the "object" level and the higher (proof) level the "metalanguage" level? Now we have one of two things: a statement that can be metalinguistically proven to be linguistically unprovable, and thus recognized as a theorem conveying valuable information about the limitations of the object language, or a statement that cannot be metalinguistically proven to be linguistically unprovable, which, though uninformative, is at least no paradox. Voilà: self-reference without paradox! It turns out that "this formula is unprovable" can be translated into a generic example of an undecidable mathematical truth.