On 11/21/2020 07:42 PM, Ross A. Finlayson wrote:
> On Saturday, November 21, 2020 at 6:55:12 PM UTC-8, Ross A. Finlayson wrote:
>> On Saturday, November 21, 2020 at 5:59:54 PM UTC-8, Ross A. Finlayson wrote:
>>> On Tuesday, November 17, 2020 at 12:28:41 AM UTC-8, Mostowski Collapse wrote:
>>>> Well I guess you are a hopeless case.
>>>> You still dont know the term sigma-algebra
>>>> and cannot answer the question.
>>>>
>>>> Who has not thought so? A random word
>>>> robot cannot answer a question, what
>>>> else was to expect?
>>>>
>>>> LoL
>>>> Ross A. Finlayson schrieb am Dienstag, 17. November 2020 um 03:21:34 UTC+1:
>>>>> On Monday, November 16, 2020 at 9:52:13 AM UTC-8, Mostowski Collapse wrote:
>>>>>> No meassure µ : Ω -> R+ and no metric d : X x X -> R+
>>>>>> involved in the sigma-Algebra question.
>>>>>>
>>>>>> What are you smoking?
>>>>>> Ross A. Finlayson schrieb am Montag, 16. November 2020 um 16:56:49 UTC+1:
>>>>>>> The metrizing ultrafilter construction isn't so different a
>>>>>>> ... gibberish ...
>>>>> ... gibberish ...
>>> "Finlayson's paradox: none"
>>>
>>> "
>>> 63. Cantor's Theorem
>>> HOL Light, John Harrison: statement
>>> Isabelle, Larry Paulson: statement
>>> Metamath, Norman Megill: statement
>>> Coq, Jorik Mandemaker: statement
>>> Mizar, Grzegorz Bancerek: statement
>>> Lean, mathlib: statement
>>> ProofPower, Rob Arthan: statement
>>> NuPRL, Jim Caldwell
>>>
>>> https://www.cs.ru.nl/~freek/100/
>>> "
>>>
>>>
>>> Which sigma algebra?
>>>
>>> Algebra 1, Algebra 2. or the integers'?
>>>
>>> The measurable subsets making sigma
>>> algebras is in their spaces, here of the
>>> sets of only two adjacent elements for
>>> example, but usually a setting more or
>>> less always being the same and parameterized
>>> as "length", the disjoint contiguous segments
>>> their union.
>>>
>>> I.e., building those that way, from all the
>>> separate classes of continuous segments
>>> that each of those is a different partitioning
>>> of the integral into so-many measurable pieces,
>>> then what _aren'_ measurable subsets is in their
>>> usual intensional and direct results in set-intersection,
>>> that the closing the algebra to that would make it
>>> so there were singletons or otherwise not "measurable
>>> subsets", that basically redefines the intersections of
>>> those sets as the closure, for example that it's one or
>>> another of the neighbors instead, still closing the algebra
>>> with regards to that singletons aren't measurable, and
>>> that only sequences as segments are measurable, makes
>>> for a measurable space where the sigma algebra relation
>>> holds just fine under the pairwise disjoint expectation of
>>> the sets, as what close under dense instead of sparse.
>>>
>>> Then, there's about why there were built the length adapters
>>> besides, for that being "Algebra 3" and the above "Algebra 1"
>>> and "Algebra 2" besides "any default sigma algebra of the integers",
>>> and why under length assignment and symmetry there are the
>>> five or six prototypes of a sigma algebra or placement for one
>>> under isomorphism for algebra and reduction - these "measurable
>>> sets" aren't for the rest of set theory only this "measure machinery",
>>> as for whether or not they're the same or "isomorphic" in all relevant
>>> models and here as "the" model of time-continuous terms.
>>>
>>> Then, looking also for other sigma algebras of that are the same
>>> or lesser cardinality than their model the reals, for countable additivity,
>>> is for what sigma-algebra adapters, there are for
>>> line reals
>>> field reals
>>> signal reals
>>> here that for the signal reals, which is not ran(EF) but instead an
>>> altogether another definition itself of continuity, that for the sigma-algebras,
>>> of the sets, from the signal domain, that for example it's a model that's not
>>> one of sigma-algebras of the line reals or field reals already, yet still for how
>>> that under adapters or the pair-wise it's its own organization, carrying about
>>> the signal or the critical moments what perfectly reconstruct the continuous
>>> functions of the field domain. (Or are not noise.)
>>>
>>> So, algebra 1 and 2 or "the defined "measurable subsets" from the definition
>>> of the "measure theory" ", making for length assignment whatever are
>>> sigma-algebras of those, then there's besides of course just handed the
>>> example of the sigma-algebra of the integers under omega, the "any default
>>> sigma-algebra of the integers' ".
>>>
>>> Yes, I am still keeping to that for the measurable subsets there _are_
>>> sigma-algebras. These are called "Algebra 1 and 2 for length assignment"
>>> where there's the complement terms from the other side of the _space_
>>> as a usual enough "non-Archimedean" definition of the term, for the
>>> algebra being "under the space" instead of vice-versa, that for the
>>> elements of the space they construct the other meaning of "measure theory"
>>> besides, line-drawing.
>>>
>>> It's not necessarily obvious why those things are that way here that
>>> probably though I just read in the properties ascribed to character the
>>> algebra, thus getting "thus that there are sigma-algebras, for, this
>>> measurable space and its measurable subsets", here now that for
>>> not being closed, I had to contrive some meaning under which the
>>> "measurable subsets close measure", and for how as "sets" here's
>>> nothing else for them to do, or it's shown constructively that there
>>> are its own and there are others', the "sigma algebras" in the
>>> universal space of functions of function theory.
>>>
>>> (Which are closed and complete under their terms as sets, here for
>>> example keeping them besides the real-valued values .)
>>>
>>> This is for that these "duck sigma-algebras" where they're not closed
>>> under the arithmetic, they're still complete in the space terms, and
>>> also besides how they _are_ closed in the arithmetic.
>>>
>>> This defines the sigma-algebra for "what it is: the set" besides as
>>> "what it does: the algebra", for example how it's consistent that
>>> such a thing exists anyways by itself for example axiomatically.
>>>
>>> There is then that these are "duck sigma algebras" besides for
>>> something as just being example's of "not Galois' algebras in
>>> set theory, but still having their own completeness results in
>>> spaces and here for term reduction and cancellation in algebra".
>>>
>>> Then, that there are "sigma algebras" but just that the complements
>>> only exist as on either from zero or from one the integer index, ...
>>> what results is that the sets, with their natural operations, see
>>> defined differently something like "complement exists" instead
>>> that in space terms there's the "closure exists, complement is
>>> defined to not exist so must be a set of equivalence classes under
>>> a relation that according to set theory the complement _does_
>>> exist, that though it's entirely reversed in image and makes sense
>>> the scale relation is preserved".
>>>
>>> Then, "implementing the operations of a sigma-algebra as what's
>>> used as a model of countable additivity", here is that for the integer
>>> assignment, the "measurable subsets" have a stronger condition than
>>> "subsets", that it's not really sets of integers so much as pairs or more.
>>>
>>> The "measurable subsets" here are basically "each partitioning of the
>>> integers into two parts". Then, closing those to disjoint, which otherwise
>>> the point being here is let _not_ exist, then there's next "each partitioning
>>> of [0,1] into three parts."
>>>
>>> These are then having "Sigma Algebra 1" and "Sigma Algebra 2", as for
>>> what structure in whatever space of words satisfies the result of there
>>> existing measurable subsets for extent 1.0. (For measure 1.0.)
>>>
>>> Then that the results for measure theory are about the _disjoint_
>>> measurable sets, here makes it so that for the partitioning as above,
>>> there is only countably-many, and the disjoint makes for that as sigma-algebras,
>>> the parts are hold together throughout the duck Sigma Algebra as for its model
>>> of a role in arithmetic.
>>>
>>> So, the duck Sigma Algebra _is_ a model of a sigma algebra if only though
>>> on the arithmetic of elements of an integer domain to continuous range.
>>>
>>> Otherwise it's just an example that the measurable subsets suffice to
>>> represent the analytical character of the real numbers with respect to
>>> measure, for analysis. (I.e., if "measure theories have a sigma algebra",
>>> this one does too, then as above besides the default sigma algebra on
>>> the integers, there are these more relevant "length assignment" for
>>> measure the "measurable subsets", adapters to sigma algebras with
>>> rather restricted transfer in naive set theoretic machinery, ..., here there's
>>> that for _these_ measurable subsets, that their union about the pairwise
>>> disjoint, is that the union only exists when both sides share exactly one
>>> endpoint with the other.
>>>
>>> That _those_ are the segments that are each measurable, measurable
>>> subsets containing n>1 -many consecutive integers each sharing only
>>> one endpoint, that thus the closure of the union is only defined that
>>> way, and, for those measurable subsets, when they only share one
>>> endpoint, then their intersection is defined as empty set, that for
>>> the terms of the measure theory, they're the only measurable subsets
>>> what are "disjoint" and "measurable".
>>>
>>> This is where, the union exists, but the model results what:
>>>
>>> Each lhs, rhs is some measurable subset {a, ..., a<b, b} with a < b, and usually a = 0.
>>>
>>> lhs, rhs have only _one_ partitioning into two of N ['Algebra 1'] or into three of [0,1] ['Algebra 2'].
>>>
>>> U ( lhs, rhs) = { min(lhs, rhs), ..., max(lhs, rhs) }
>>> <- still always a single partition
>>> <- undefined if lhs_b < rhs_a or lhs_a > rhs_b
>>>
>>> Int ( lhs, rhs) = { max(lhs_a, rhs_a), ..., min(lhs_b, rhs_b) } <- undefined if a singleton
>>>
>>> Basically, what it says then "undefined" is that then the "duck Sigma Algebra"
>>> is contrived and implements the otherwise usual treatment of the "sets",
>>> that are a model of partitioning of integers by containing either all the
>>> numbers greater or all the numbers lower than a given integer, or here
>>> splitting [0,1] into three with two arbitrarily large integers, is for that the
>>> measurable subsets that topology basically effects that the naive operations
>>> on the measurable subsets, are not naive, in terms of the object still being
>>> a model in the set theory, of the closure of operations on the measurable
>>> subsets (with regards to countable additivity and its import to measure theory).
>>>
>>> It's so: that these "measurable subsets" are parameterized by the scale of
>>> the integer lattice of the spaces where they're relevant, and that with usual
>>> closure or set theory, don't satisfy the definition of that "the default composition
>>> of sets defines the sigma-algebra", here instead that these "measurable subsets"
>>> instead to have a "sigma-algebra" must have some alternate definition of how
>>> the _closure_ of the space to marking all the lengths by marking a segment,
>>> that the _closure_ maintains the result as that otherwise naive set theory
>>> "maintains the result and in first order and as a pure set etc", maintains the result.
>>>
>>> Then as about what it's still a sigma-algebra, so construed in this framework,
>>> has not so much the naive sets expected to exist like singletons, instead from
>>> these objects so existing, the measures of measurable subsets for their union
>>> and intersection still _close_ the space of the sigma-algebra, in terms of that
>>> functionally with the measures those are some neatest definitions of as
>>> real value in an integer lattice being described by a real value as in an integer lattice.
>>> (And its integer part.)
>>>
>>> For basically that the union of all the measurable subsets has measure 1.0,
>>> and the intersection of all the measurable subsets has measure 0.0,
>>> keep in mind instead it's that "there are _disjoint_ measurable subsets
>>> with measure a and b that sum to a+b", that here they must share exactly
>>> and only one member (instead of being disjoint). I.e., it's a way for extensionality,
>>> that: doesn't require any changes to the "measurable subsets" as sets,
>>> to close them out in terms of their operations, instead that "the sigma algebra"
>>> includes operations on the sets in their closure what make them so.
>>>
>>> (And that the complement is variously omitted or constructed as from
>>> the other side, in partitioning.)
>>>
>>> Really this is a lot of weaseling for the definition "... for the measurable
>>> subsets: ..., what make sigma-algebras for them... [or whatever lesser
>>> structure so results in fulfilling all their meaning to topology, calling
>>> those also models of sigma-algebras]".
>>>
>>> It's fair for you to say "a model of an algebra in a set theory has a set-theoretic
>>> model", not necessarily though "... that models set theory".
>>>
>>>
>>>
>>> This way then still the field is Banach.
>>>
>>> Here then this is for writing a model of Banach space ,
>>> writing it in this measure theory here this integer-lattice
>>> theory.
>>>
>>>
>>>
>>> The gibberish....
>> Then, these "measurable subsets", besides, have simply
>> what machinery as establishes their disjoints and under
>> intersections interms of disjoints and intersections
>> (that here the partititions make two subsets each containing
>> the value under the partition the index.) (With all "sets"
>> in the modular.)
>>
>> I.e., here the product rule makes for the subsets, how they're
>> associated or so, that they intersect under terms, under these
>> closure rules above, by applying intersection, and union,
>> that the sets so model a sigma-algebra this way.
>>
>> As above then in the Algebra 1, with the n-sets and a = 0,
>> that those are partitions of the integers by one value, each.
>>
>> I.e. the product has both subsets that those fall into an equivalence class,
>> each, in terms of what descriptively of course and for what sets are,
>> that here like those that converge its these that are dual.
>>
>> Then, because they're always finite numbers, there have to
>> be maintained as different and complementary under sets,
>> both those from zero to one, and, those from one to zero.
>>
>> (This is called Equivalency Function with f(0) = 0
>> and Reverse Equivalency Function, with f(0) = 1,
>> as mostly though maintaining complement, as about
>> what bounds aren't maintained all in terms of the one side,
>> that they maintained on the other side separately then for
>> as they do in the middle, the partitions that are all centered
>> around zero, with the partition of the middle that is effectively
>> "infinite" to each, side in the middle.)
>>
>> As above then the products what results in unions, to be having
>> the factors, are from values as sets of course that, extensionally,
>> these sets are to be _implemented_ intensionally, that the
>> functional arithmetization is: implementing the product space
>> here of the values the sigma-algebra, as what necessary for
>> example structures it guarantees, and guarantees don't,
>> exist and don't exist.
>>
>> Here then it's that the various results in inequalities are framed in
>> these terms, including for where it's a usual model of arithmetic
>> and arithmetic in the linear. (That the "existence" of the "sets" of
>> the models of the arithmetic and arithmetization suffices to
>> prove their existence.)
>>
>> Then, maybe these sets besides having a value or assignment
>> like the length assignment here for these kinds of non-cuts
>> in the continuous, for the length cut, here it is the "cut/cut".
>>
>> I.e., for letting the integer lattice a real lattice this way,
>> it seems best the measurable subsets are a sort of result
>> what for that being the _assignment_, then that the internals
>> of the set's contents, result in what closures of rules there are,
>> that for example there are lots of sets in the models what have
>> lots of content or structure work out null, that the arithmetic
>> basically closes the algebra with regards to at least the _forward_
>> inductive useful part of sigma algebras in measure theory, these
>> are sets as so result with for example strong or weak associativity.
>>
>> (With other laws of arithmetic.)
>>
>> (All of which model arithmetic, but not any of which models arithmetic.)
>>
>> Then, that other sigma-algebras don't maintain measure,
>> under usual products with THESE sigma algebras, it's that
>> length assignment doesn't hold up underneath, or that what
>> otherwise usually the singletons (and thus all their products)
>> are measurable subsets, here that _no_ singletons are measurable
>> subsets, except the empty set with 0 (the additive identity).
>
> It seems with "start with all, thus no singletons" for partitions
> works out about the same way as "start with none, so no completed
> infinities" does usually.
>
> Here then the usual expectation that a closure, exists, is that the
> "partition assignment with an integer", also brings its history:
> how the set arrived as a product what in terms of all the other
> sets the are products, works out from their definition on the
> existential quantifier, that simply according to their state the
> assignment under closures, that none of the intersections of
> the "measurable sets" have singletons what result after for
> keeping mutual endpoints their partitions besides, that,
> this way, there is fundamentally the filter what almost universally
> defines measure (by composition) that it's what are the equivalency
> classes of the maintenance of the quantifier conclusion that are
> the "sets" of these Algebra 1, Algebra 2, ..., in an arithmetization,
> of the associative elements of the domain.
>
> What this means for usual frameworks what have "according to
> operations what you wouldn't take in preserving measure" that
> when they're isomorphic, the models under the sigma -algebras,
> that they're equivalent, which enough they are, but, when these
> ones aren't, as sets their length assignment simply is off into
> the field where these go to the flat instead of the non-linear.
> (Except to the spiral.)
>
>
> Basically there's that all these have no differences in their history
> that result what they would have singletons as their intersection,
> and that unions are maintained also by sharing endpoints, thus
> that only "dis-joint, that compose back together" means that both
> sets have the endpoint and one set is otherwise lesser and the
> other set is otherwise greater, those being the only "disjoint"
> measurable sets for the guarantees of the sigma algebra anyways,
> that all results simply have to be built in terms of those instead of
> that any two values between zero and infinity are different or same,
> and if difference that inequality is transitive, from both small and
> large numbers.
>
>
> Thanks though, I can definitely see how that might otherwise seem
> all left out or wrong.
>
>

The measure-theoretic aspects of iota-values, here
just happens to be in a longer discussion about quantifier
disambiguation, with a usual sort of mention of Feferman
into quantifier disambiguation or Nik Weaver and the FOM
list's impredicativity crowd.

The idea that quantifier disambiguation makes for a
first-class organization within the universal quantifier,

for-any the first, i.e. the next, ...
for-each for-any, for-any, ...

for-every a completion
for-all a completion and furthermore collection

has that this is a simplistic approach employing
the usual coverage of the usual meaning of the
distinction of these usual terms otherwise
reflecting the same thing, usually in terms
of quantifier comprehension, that here it's
a first-class thing about disambiguating quantifiers.