
Re: Uncountability of the Real Numbers Without Decimals
Posted:
Dec 7, 2013 2:43 PM


On Saturday, December 7, 2013 10:04:31 AM UTC8, fom wrote: > On 12/7/2013 9:57 AM, Albrecht wrote: > > > On Saturday, December 7, 2013 4:37:33 PM UTC+1, fom wrote: > > >> On 12/7/2013 4:22 AM, WM wrote: > > >> > > >>> Am Freitag, 6. Dezember 2013 23:18:03 UTC+1 schrieb Virgil: > > >> > > >>>> In article <f22fac9b9a7a41f699534705e02c1215@googlegroups.com>, > > >> > > >>>> > > >> > > >>>> WM <wolfgang.mueckenheim@hsaugsburg.de> wrote: > > >> > > >>> > > >> > > >>> > > >> > > >>>>>> Maybe you should look into it, and see what I did say actually means. > > >> > > >>>> > > >> > > >>>>> > > >> > > >>>> > > >> > > >>>>> I know it. It is nonsense. There is no model of ZFC. > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> > > >> > > >>>> That is just onemore shortcoming of WM's wild weird world of > > >> > > >>>> > > >> > > >>>> WMytheology. Outside of that queer place there are any number of models > > >> > > >>>> > > >> > > >>>> of ZFC. > > >> > > >>> > > >> > > >>> A model of a theory shows that the theory is free of contradictions. Is ZFC free of contradictions? > > >> > > >>> > > >> > > >> > > >> > > >> Yes. > > > > > > So you know more than anybody else in the world. > > > > > > > I know it is important that Boolean algebras are > > not the faithful syntactic models of the logical > > calculus: > > > > http://arxiv.org/pdf/quantph/9906101v3.pdf > > > > > > > > I know that it is important to have an understanding > > of how a consistent firstorder theory would > > express itself based on the notion of equivalence > > lattices rather than Boolean lattices: > > > > http://math.stackexchange.com/questions/526335/largecardinalsandpartitionlattices > > > > > > > > I know that a consistent logic is associated with > > open topologies and I know how Frege's advances to > > a compositional logic relate the syntax of such > > deductive systems to a topology that may interpret > > firstorder consistency with respect to open filters. > > > > The important notion is feeble compactness and it > > is a characteristic property of minimal Hausdorff > > topologies. With a feebly compact representation, > > a language may still represent inconsistent sets > > of formulas. > > > > https://groups.google.com/forum/#!original/sci.math/bpHMUSiYFBc/vu5F57GmTf4J > > > > > > > > I know that topology has fixedpoint theroems based > > upon continuity assumptions and that the logic with > > which I interpret the theory is a fixedpoint logic > > based on specialization preorders constructed from a > > strict, transitive order, > > > > http://en.wikipedia.org/wiki/Transitive_closure#In_logic_and_computational_complexity > > > > > > > > I know that topological semantics have been extended > > to firstorder languages and that necessary denotations > > correspond with continuous denotations, > > > > http://people.ucalgary.ca/~rzach/static/banff/awodey.pdf > > > > > > > > I know that the semantics of classical term logic is > > associated with specialization preorders and that the > > specialization preorders associated with sober spaces > > have the necessary fixedpoint characteristics, > > > > http://en.wikipedia.org/wiki/Specialization_%28pre%29order#Definition_and_motivation > > > > http://en.wikipedia.org/wiki/Sober_space > > > > http://en.wikipedia.org/wiki/Directed_complete_partial_order#Properties > > > > > > I know that the recursivelygenerated semantics of > > my primitive relation generate the needed pointed > > set relation for the pointed directed completed partial > > order discussed in the link above, > > > > > > AxAy( x irreflx y <> ( Az( y irreflx z > x irreflx z ) /\ Ez( x > > irreflx z /\ ~( y irreflx z ) ) ) ) > > > > accepted: { a < b, a < c } > > omitted: { b < c } > > undecided: { b >= c } > > > > accepted: { a < b, a < c, a < d } > > omitted: { b < c, c < d } > > undecided: { b >= c, c >= d, b >= d } > > > > accepted: { a < b, a < c, a < d, a < e } > > omitted: { b < c, c < d, d < e } > > undecided: { b >= c, c >= d, b >= d, d >= e , c >= e , b >= e } > > > > > > > > > > I know the difference between priority monism and priority > > pluralism and how that relates to the interpretation of > > classical term logic and the pointed sets of the partial > > orders discussed above, > > > > http://plato.stanford.edu/entries/monism/#PriorityMonism > > > > > > > > I know how my investigations depended on the Jordan curve > > theorem, and, I know that the Jordan curve theorem is of > > sufficient weakness to place my foundational investigation > > at the level of Hilbert's finitistic program (via Koenig's > > lemma), > > > > http://en.wikipedia.org/wiki/Reverse_mathematics#The_big_five_subsystems_of_second_order_arithmetic > > > > > > > > I know which form of the axiom of choice supports the > > way in which I introduce identity into the system of > > logic I use, > > > > https://groups.google.com/forum/#!original/sci.math/RDp0ahTBds/OF8Fsp3VG8sJ > > > > > > > > I know which organizing principles permit one to formulate > > a nonparadoxical model consistent with the finitary asymptotic > > properties of the weak weak Koenig lemma, > > > > https://groups.google.com/forum/#!original/sci.logic/CoO4TT7i_1g/vW5n9eB4LUoJ > > > > > > > > I know how to explain the construction of the real numbers > > in relation to the undefinability of truth and seemingly > > circular mathematical methods > > > > > > http://math.stackexchange.com/questions/533133/avectorspaceisasetaxiomorderivation > > > > > > > > I know that Frege turned to geometry when he rejected > > logicism and that Russell interpreted projective > > geometry as a meaningful notion reflecting Kant's > > sensible externality, > > > > > > > "The more I have thought the matter > > > over, the more convinced I have become > > > that arithmetic and geometry have > > > developed on the same basis  a > > > geometrical one in fact  so that > > > mathematics in its entirety is > > > really geometry" > > > > > > Frege > > > > > > > "..., I shall deal first with projective > > > geometry. This, I shall maintain, is > > > necessarily true of any form of > > > externality, and is, since some such > > > form is necessary to experience, > > > completely a priori." > > > > > > "We can distinguish different parts > > > of space, but all parts are qualitatively > > > similar, and are distinguished only > > > by the immediate fact that they > > > lie outside one another" > > > > > > "Analysis, being unable to find > > > any earlier haltingplace, finds > > > its elements in points, that is, > > > in zero quanta of space" > > > > > > "A point must be spatial, otherwise > > > it would not fulfill the function > > > of a spatial element; but again, > > > it must contain no space, for any > > > finite extension is capable of > > > further analysis. Points can > > > never be given in intuition, > > > which has no concern for the > > > infinitesimal." > > > > > > Russell > > > > > > > > > I know that my primitive relation, as a > > strict transitive relation, is interpretable > > as the mereological proper part relation. Hence, > > my interpretation does not start with a consideration > > of parts, but with a consideration of identity in > > relation to mereological parts, > > > > > > http://mathoverflow.net/questions/58495/whyhasntmereologysuceededasanalternativetosettheory/127222#127222 > > > > > > > > And, I know that my formal construction begins > > with a specification of logical constants > > as a projective geometry, > > > > http://mathforum.org/kb/plaintext.jspa?messageID=7933591 > > > > > > > > I also know that people do not think that I > > do mathematics. > > > > > > But, I cannot control what other people think.
"Which large cardinal axioms are most likely to be related to the structure of partition lattices on infinite sets?"
Would that not be those that complete the Borel hierarchy? Rather, it would be those.
"If I have interpreted them correctly, then any given model presumed to be consistent has pointwise definable elements."
Even countable models as they are constructible may be rather rich as defining them "pointwise".

