Has anyone ever met the constraint of scotts infinite theorem; is proof by contradiction the methodology by which its constrains can be shown to hold. Ie can presumably cannot so much as use proof by induction with regard to it.
Does deriving these conditions; imply that one 'in a sense' derived the applicability of an archimedean condition; or is that implicitely assumed within the theorem itself, (I presume one does not explicitely have to assume or somehow quasi-derive the compatibility of one systems with an archimedean system before using said axiom (scotts infinite)- I presume the compatibility of ones system with continuum and infinite spaces is implicit within those constraints, and thus meeting them would already entail that such archimedean compatibility (Constrains, as opposed to necessarily derived) are already met, or is such a condition presumed within the theorem itself.
Is that its advantage of it, over and above the conditions say listed by Jaffray and Chautenauff (1984) which are apparently sufficent and necessary to a representable probability measure on an infinite space (and likewise if you want, countable additivity as well) .
What would be the consequences of its, with relata to the uniqueness of the representation; I presume that it allow for a compatible finitely additive measure (on an infinite sample space, or a continuum algebra, concatenated or otherwise, by which I mean each individual space may be infinite, or finite, but they are concatenated with infinitely many other spaces with the same number of finite outcomes) without said algebra having to being montonely continuous or a sigma algebra, or without an intermediate,or almost agreeing measure in the case when its not totally connected (total pre-order, although In my case it the space is). see https://www.researchgate.net/publication/222611452_Archimedean_qualitative_probabilities
I presume it would entail the compatibility of one's measure with the stronger archimedean (jaffrays H and other stronger conditions, perfect separability, and other conditions) conditions that need to be imposed, when monotone continuity and the sigma algebra assumption is dispensed with. See attached
Has such a (scotts infinite, in domotors 1969) representation theorem or any other infinite representation theorem ever been applied
Secondly, has one ever applied something of its ilk to gleason result in QM, ie the born rule/function measure being the only probability measure, function), compatible with the probability calculus (given non-contextuality, continuity, frame function etc)
I know other representation theorems have been applied within particular frameworks.However, I am talking about the standard gleason theorem itself. Likewise I am not talking about whether the born rule itself is a compatible with a probability measure or is representable (by its numerically identical self) or is only possibly uniquely given by its numerical identical self. It presumably just is, in some sense, by construction, necessarily a probability function.
What I mean instead, is quantum probability itself is compatible with a probability measure, by which I mean if one were, to somehow rank the frame function, using some qualitative probability methods (ie translate those numerical inequalities back into qualitaitve inequalities, and use scotts infinite theorem). Perhaps using a PCS bisymmetric.binary difference, additive difference kind of structure, using the same methods by which Gleasons compared spaces, events within spaces and between spaces (on different bases); a combination it appears, of the conditional logic structure of QM, counterfactual monotonicty, where some event is in common between non commuting spaces, the entanglement like structure between commutting spaces, orthogonal triples, and non-contextuality, to compare common events which appear in any given set of non commuting bases, and thus individually commute with the other two events, although the entire triples do not)).
Or would have use the thomson hexagon kind of condition instead (although I presume this provides a stronger form of representation, and a somewhat unique measure as well).
Incidentally, what is the born rule measure (not with relata to its being unique with regard to being the only measure compatible with being a probability measure; its presumably just unique in that sense), but rather the homogeneity of the probabilistic axioms themselves, in qm, unique up (ie interval scale etc? log interval (i doubt it), positive affine transformations of the unique probability measure). As in which other numerical representations or transformations are ruled out; that is how (quasi) uniquely probabilistic would the numerical function, produced by the ranking system be? or would just be unique some partial sums and nothing more.
For example if one were to able somehow qualitatively order the events in accordance with however, it was, that he derived the unique-ness theorem; or is this precisely why its not constructively provable. There appears to be two slightly distinct form of archimedean assumptions or derivations; one appears to be the ability to neglect infinitesimally improbable events (which may follow) given the continuous nature of the space, but then I believe that (whilst perhaps a necessary condition) is slightly distinct from what is required for countably additive representation (although I might be wrong; for example in infinite spaces).
I presume Gleason used continuity in the former sense to compare these distinct spaces, as if they were all there in exist, each being a subset of the other; as opposed to extending to dimensions greater then 3 (or in particular in infinite spaces where countable additvity is required). I know there was an issue with rational valued probabilities or amplitude norm squared (or is just when they are merely rational) which is part of the issue behind why one give a uniqueness pvm proof for dim =2 in the conventional way (and probably also because there arent not all these commuting bases (which all exist) in some sense) that one can compare, via non-contextuality over spaces?)
Article Archimedean qualitative probabilities
I'm ready to help. I just ask you to inform me of the source you are relying on for your reading.
see also https://www.researchgate.net/publication/222611452_Archimedean_qualitative_probabilities
Scotts infinite (is in domotor relational structures above, the other nec and suffient or sufccient conditions are in villegas, or in the research link attached above and in Jaffray 'ON THE EXISTENCE OF A PROBABILITY M...RE on infinite boolean algebra.pdf
-Villegas 1964,1967 qualitative probabilities on a sigma algebra, qualitaitve probabilties
Fine; Revised basis of Quantum mechanics
Koplov 'subjective probability on small domains' 2007, and his 2010
FIshurne 1986;
Narens 1985 'abstract theory of measurement (and his later books which are quite recent 'probability logic, and another on lattices just published this year I thinks), T Fine 1973, 'theories of probability' Suppes, Krantz et all Vol1-3, foundations of measurement. Kreps (1988), F Roberts
Bolker 1964 (student of A.M Gleason, who came up with a similar approach to villegas's atomlessness- constraint), Functions resembling Quotients of measures1967A Simultaneous Axiomatization of Utility and Subjective Probability
Also Alper 1987
Uniqueness and Homogeneity of
Ordered Relational Structures Luce 1986, Measurement scales on the continuum Luce 1987 (there is some mention of Gleason in these kinds of PCS style or conjoint system papers);
T. AM. Alper u. Math. Pgchol. 29, 73 (1985)), Classification of All Order- Preserving
Homeomorphism Groups of the Reals, Alper 1987
'classification of all order preserving groups hat Satisfy Finite Uniqueness' (alper,mentions an unpublished result of gleason, with relata to uniqueness and homogeneity/invariance of representations, and representation)
Wakker;'Additive representations on rank-ordered
sets', Pivato. 'Additive representation of separable preferences
over infinite products'2014
Also see https://arxiv.org/abs/quant-ph/0503066;;qualtiative representation/uniqueness approach vis a vis qualitative probability, or attempt somewhat in line with gleasons kind of approach. BY contrast to totally distinct methods (such as in everett or other approaches to qm)
s. I presume that is the issue with n=2, not only are there no non trivial disjunctions by which one can resolve inequalities through additivity, and the standard (quantum issues) commuting bases- via which these admixtures of conditional logic, entanglement like relations and non contextuality can be applied, and/or thus enough equi-probables to resolve the system (get a unique probability measure compatible with the axioms of the probability and said qualitative ranking), but that also if one were to do it, counter-possibly speaking, would have to derive a much stronger claim (as one would not terribly much of the structure of probabilism to you- ie one would have derived more than a unique represent-able probability measure, but that it would be also quasi-uniquely probabilistic in the other sense of in-variance (the usual up to 'BUT NOT INCLUDING", this that and the other, transformations of the represented numerical structuring, ie the numerical representations that still respect the rank ordering, but which may differ with relata to their values insofar as, in this case, they are not probability functions, are some transformation of the intended, probabilistc (often unique) measure). That is to say, and I may be wrong I think, that one would have to also have derived or otherwise assumed could some non-neglible part of probabilism itself (as opposed mere compatibly and uniqueness) just to get the unique compatible probability measure.
I am not sure which word Invariance/homogeneity and/or unique-ness applies to each of these distinct forms; they are often conflated and used in the wrong sense. Nonetheless, whilst one should not confuse the two, although they are often, connected with each other, to some degree, due to what one has to build into the system to get a standard unique probability measure result (particularly with finitely many outcomes, on each 'individual' space)
IF interested also see Narens 1985 'abstract theory of measurement (and his later books which are quite recent 'probability logic, and another on lattices just published this year I think), T Fine 1973, 'theories of probability' Suppes, Krantz et all Vol1-3, foundations of measurement. Kreps (1988), F Roberts 'measurement theory'
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Article Archimedean qualitative probabilities
In particular jaffray condition and chautenauff distinct conditions are mentioned in
Jaffray, J.Y., 1974, On the extension of additive utilities to infinite sets, Journal of Mathematical
Psychology 11, no. 4, Nov.;
as well as inON THE EXISTENCE OF A PROBABILITY M...RE on infinite boolean algebaea.pdf attached, and the fishburne reference1986 as well as the chautenauff attched, and the link to archimedean qualitative probabilities
there is also the conventional books on 'applications ofgleasons theorem 1993', Quantum measure theory, by Hamhalter', as well Gudder 1979, 'Stochastic as well the suppes 2002 'invariance of scientific structures'; and chrismans Nicholas R. Chrisman ,
"Rethinking Levels of Measurement for Cartography. THere are many other scales that sort of fit within these, such as hyper-ordinal and fractional interval scales as well.Louis Narens-Theories of Meaningfulness (Volume in the Scientific Psychology Series)-Lawrence Erlbaum Associates (2001), Gudder Quantum probability and Gudder 1979 'udder S 1979 Stochastic Methods in Quantum Mechanics'
I presume that gleason did not actually have to explicitely split up the events, or even so much as fuse them together as if the pure state on distinct non commuttng spaces on spin 1 systems and above actually act as if they were orthogonal outcomes, on the same space.
This may be just to round off a few loopholes, with regard to getting the measure precise (maybe irrationals, or due issues with finite additivity or other kinds of quantum states), You can somewhat see immediately that the combination of equi-probables on other spaces that are somehow counter-factually, logically connected, or otherwise via non-contextuality plus a small use of non trivial addivity will immediately resolve some outcome spaces; perhaps this is just a way to compute (you cant compute them all by hand, so in sense, its rather that uniform or absolute continuity is a consequence of the unique probabilities and not so much the other way around; at least arguably). I am not sure which form of continuity was actually required, although the issue is more with uniform continuitty and not the derivation of continuity (or perhaps they mean uniform equi-continuity, strong continuity, or absolute continuity).
Although to be honest I dont know. I am particular sure as to what strong continuity as in rao and rao, translates into, within the moduli of continuity (some form of weak uniform continuity, equi-continuity, micro-continuity or none of the above, and likewise for strong non-atomicty of convex rangedness, which is either uniform continuity or something stronger, although not quite absolute continuity)
(3).pdf 'rao rao, theory of charges on finitely additive measures 1983'
Likewise i am unclear as to whether (in an interval scale) whether the arbitrary constatnt is held fixed within (but not between) any given other transformation, and if so, is the arbitrariness, just to indicate that it may be negative or positive within any given other representing numerical structure.
I cant see how that follows. It would entail that one have merely have to fix the zero probability event, and the unit event, P(X)=1 and maybe not even 0
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