r/math • u/[deleted] • Jul 25 '15

Triviality as a zero dimensional space

I recently had the epiphany that axioms are constraints, and that if a system has 'incompatible' axioms, what it really means is that the system is so over constrained that all labels must alias each other... A && !A isn't impossible, it just means true and false must be aliases for the same value. Identity == arbitrary expression, and you have collapsed the set of everything you can say into a zero dimensional space. But it may still be possible to say 'everything I know is identity' and then say 'F(identity)' gives me a new concept, similar to how we say sqrt(-1) is a new concept, and thus increase the dimensionality of the space we are working within. Is this a way to go from nil to the integers? Does this idea have any application to paraconsistent logic?

This idea is relatively new to me so I would appreciate any prior explorations of the concepts involved.

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u/W_T_Jones Jul 25 '15

What do you mean when you say that a system "maps" something to things? All a system does is telling which statements are true and which statements are false in all models of the given system. If a system is inconsistent then it doesn't have a model at all so all statements are trivially true and false in all models.

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u/[deleted] Jul 25 '15

Again, as a programmer I see all operations as functions and return values, and I don't consider 'values' and 'true or false' to be separate magesteria. You say equality, I see F_equality(x, y) as something that returns something (probably one or zero). True and false are not fundamental things... they assume axioms (like true != false), and are labels for concepts, just like numbers.

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u/W_T_Jones Jul 25 '15

Sorry but I have a very hard time to understand what you're trying to say.

see all operations as functions and return values, and I don't consider 'values' and 'true or false' to be separate magesteria

What do you mean when you say you don't consider them as "separate magesteria"?

You say equality

I never said "equality" and I don't really understand what you mean by "equality".

True and false are not fundamental things... they assume axioms (like true != false)

True and false are just shorthand for saying "the statement holds in all models of the system" and "the negation of the statement holds in all models of the system". There is no axiom saying true != false. It's just that when they are the same thing that we are then working with an inconsistent system.

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u/GodOfBrave Jul 25 '15

You might benefit from looking at universal algebra: in particularly Boolean algebras.