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April 20, 2011
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What really makes any statement (such as 2 + 2 = 4) true?
Is it possible for a truth to exist that absolutely cannot be known?
If so, what makes it true?
If nothing existed, would the truth still exist?
Would it still be true that nothing existed?
"I can conceive of nothing in religion, science, or philosophy, that is anything more than the proper thing to wear, for a while." ~ Charles Fort
September 19, 2009
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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. 
April 9, 2009
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Numbers are sets, at least usually...So saying it has a numerical structure implies sets are tacitly involved.
"it is easy to grow crazy"
September 19, 2009
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"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:
April 9, 2009
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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.
"it is easy to grow crazy"
September 19, 2009
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"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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April 9, 2009
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Well, I believe all numbers, hypernatural to surreal, all have equal ontological primacy, and they have equal ontological primacy as any set, meaning that numbers "exist" if and only if sets "exist".
The definitions, axioms, and rules of inference are the cruxes of the matter. What is usually assumed is that once you have definitions and once you select axioms to assume (for the sake of an argument) and once you decide what rules of inference are allowed (such as modus ponens), the aggregate of all theorems "generated" by the axioms/definitions/rules of inference is set in stone, so to speak.
If someone defines oaweincaeonvadje, lists axioms that oaweincaeonvadjes satsify (which is a description about how oaweincaeonvadjes behave), and accepts a collection of allowed rules of inference, then the aggregate of all theorems (i.e., "true statements") about oaweincaeonvadjes follows. If there is no definition of oaweincaeonvadjes and/or no axioms describing how oaweincaeonvadjes behave, then literally every statement about oaweincaeonvadjes is entailed, including the ones I believe and the ones you believe. That's just a general principle called vacuous truth.
"it is easy to grow crazy"
September 19, 2009
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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:
April 9, 2009
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Godel's theorems don't worry me as much as that the word set is undefined.
"it is easy to grow crazy"
September 19, 2009
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"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.
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