A. Wolf wrote:
>> I'm well aware of relativity. But I don't see how you can invoke it when
>> discussing all possible, i.e. non-contradictory, universes. Neither do I see
>> that list of states universes would be a teeny subset of all mathematically
>> consistent universes. On the contrary, it would be very large. It would
>> certainly be much larger than that teeny subset obeying general relativity or
>> Newtonian physics or the standard model of QFT in Minkowski spacetime.
> You said: "So universes that consisted just of lists of
> (state_i)(state_i+1)... would exist, where a state might or might not
> have an implicate time value."
> I was trying to express that the universe in which we reside isn't
> separable into a set of lists of states. It's more mathematically
> complex than that.
> Some mathematical models are self-contradictory, and some are not.
> This is true regardless as to how you formulate a foundation of
> mathematics, and it forms the basis for understanding and proving
> mathematical truths. I believe that a mathematical structure complex
> enough to capture the entire set of events that define a universe must
> be non-self-contradictory to be a truthful model for that universe.
> There are mathematical structures which are self-contradictory because
> they are predicated upon axioms which ultimately contradict
> themselves; these structures are not well-defined and cannot be a
> basis for existence. Such a basis would make existence itself
> ambiguous, because all things would have to exist and not-exist at the
> same time, and not in the quantum way--with no discernable structure
> or foundation at all.
> I'm not certain what you're trying to argue, but it seems like you
> think that anything you can imagine must have a well-founded
> mathematical basis...?
So long as it is not self-contradictory I can make it an axiom of a
basis. It may not be very interesting mathematics to postulate:
Axiom 1: There is a purple cow momentarily appearing to Anna and then vanishing.
but by the standard that everything not self-contradictory is mathematics it's
just as good as Peano's.
>You can imagine all you like, but it won't
> bring into being a universe where Godel's incompleteness theorems
> don't hold, for example. The fundamental things that we know about
> mathematics itself transcend any particular realization of the
I'm arguing that "all mathematically consistent structures" is itself an ill
defined concept. A mathematical structure consists of a set of axioms and
of inference. So I supported my point my giving an example in which the set of
axioms is an infinite set of propositions of the form "state i obtains at time
i" where "state i" can be any set of self-consistent declarative sentences
whatsoever. I leave the set of rules of inference empty - so there can be no
contradiction inferred between states. Then according to the theory that all
mathematically consistent structures are instantiated (everything exists) this
set exists and defines a "universe" just as well as general relativity or
quantum field theory (perhaps better since we can't be sure those theories are
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