sci.logic 2007-12-08

Consider sampling a real number by flipping coins to get a sample between zero and one. Then, each coin flip is both: a refinement of all previous samples, and: the initial element of a new sample. Then, if a rational number is sampled, its repeating terminus is sampled infinitely many times, so the point is marked infinitely more darkly than if an irrational was sampled, where then infinitely many irrationals each get a much fainter mark.

What if an argument that the assumption that there is truth implies there is a predicate "true", and then in quantifying over objects, they all satisfy it so there is unrestricted comprehension. Then, in building sets, if there is some supertheory of ZFC containing ZFC's axioms as thus truisms, with the unrestricted comprehension over sets in ZFC, then there's a universal collection, a collection of all the elements of ZFC. Yet, then Cantor/Russell etcetera as paradoxes follow. Otherwise there's not a universal quantifier (with considerations of around three different kinds of "universal" quantifiers, indicating various accords or lack thereof with the transfer principle, for any / for each / for every / for all.)

Among reasons I think ZF is inconsistent, consider any theory that has as its elements of discourse those elements of ZF (casually referring to ZF as a collection of sets defined by the set-theoretical non-logical/proper axioms), and as well some other elements. Now, quantifying over those elements with "x is a set in ZF", then that collection is the Russell set, so the set containing "x: x is a set in ZF" is the Russell set so ZF contains an irregular set. Otherwise there's no universe (in a broad sense).

People seem quick to accept that the Russell set contains unspecified elements but few address the objects of Peano Arithmetic notionally having a similar concern. Consider Burali-Forti, that the order type of ordinals would be an ordinal so there is no collection of all ordinals in ZF, in terms of a difference between "for any", "for each", "for every", and for "all". For each ordinal, its order type is an ordinal, for all ordinals, their order type is not a set. That's basically in distinction of the transfer principle and making shorthand the notion of arbitrarily extended induction, particularly those structures that are only primitively distinguishable among themselves via induction, a memoryless two-step process.

It seems those natural objects bootstrap themselves (hoist by their own bootstraps) into a framework where they have a synthetic interface. That is to say, the natural integers form naturally, as a consequence there is infinity, and the universe, and only after where there is the complete framework of all objects is it possible to synthetically distinguish two integers. They do so from nothing.

ZF has no universe. In that sense ZF isn't, for example, a Cantorian set theory, where Cantor wanted both a universe and infinite powerset incongruence in his theory. Those two were found incompatible, thus the universe was discarded. There definitely, by definition, specifically, is a universe where the domain of discourse is no other thing. No theory exists in a vacuum.

Then, in consideration of which axioms of ZF might be false, I think the axiom of infinity is incorrectly stated, because a variety of fundamental theorems of a set-theretical infinity, among all possible set theories, would have that infinity is non-well-founded, and in large structures their grandness presupposes their identity. Then, that would lead to the notion that ZF's axiom of regularity is as well so not-necessarily-true: false.

There are no universal truths in a theory without the universe, and where there's a universe it's THE universe.

Ross