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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 9, 2013 7:19 PM
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On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote:
> On 2/5/2013 9:32 AM, Charlie-Boo wrote:
>
> > On Feb 4, 4:26 pm, fom <fomJ...@nyms.net> wrote:
> >> On 2/4/2013 8:46 AM, Charlie-Boo wrote:
>
> >>> On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote:
> >>>> On 2/3/2013 10:19 PM, Charlie-Boo wrote:
> >>>> <snip>
>
> >>>>>>>> In PROLOG we use lowercase words for TERMS
> >>>>>>>> and uppercase words for VARIABLES
>
> >>>>>>>> ATOMIC PREDICATE
>
> >>>>>>> ATOMIC PREDICATE meaning relation?
>
> >>>>>>> C-B
>
> >>>>>> RELATION
> >>>>>> p(a, b, e)
>
> >>>>> If wffs are built on relations then { x | x ~e x } is not a wff
> >>>>> because ~e is not a relation.
>
> >>>> Well-formed formulas are built from the alphabet
> >>>> of a formal language. If the language contains
> >>>> a symbol of negation, then NOT(xex) will be a
> >>>> well-formed formula.
>
> >>> You have to define what value a symbol may have - how it is
> >>> interpreted in your definition of a wff. You need to complete B
> >>> below to see there is no paradox if you are consistent about what a
> >>> wff may contain and what values it may equal after substitution
> >>> (interpretation) if it contains variables for functions.
>
> >> First, I was not in a good mood when I posted. So, I may
> >> have been too dogmatic.
>
> >> What you seem to be objecting to is the historical development
> >> of a logical calculus along the lines of Brentano and DeMorgan.
>
> I meant Bolzano here.
>
>
>
> > The only objecting in my Set Theory proposal is perhaps objecting to
> > the fact that ZF has a dozen messy axioms, a dozen competing
> > axiomatizations, a dozen interpretations of the most popular
> > Axiomatization, and (Wikipedia), The precise meanings of the terms
> > associated with the separation axioms has varied over time. The
> > separation axioms have had a convoluted history, with many competing
> > meanings for the same term, and many competing terms for the same
> > concept.
>
> > (DeMorgan is an example of why Logic and Set Theory are the same thing
> > and should be combined - same as Math and Computer Science etc.)
>
> How do you see Logic and Set Theory as being the same?
Both are concerned with mappings to {true,false}. A propositional
calculus proposition is 0-place. A set is 1-place. A relation is any
number of places. (A relation is a set - of tuples.)
So you have the same rules of inference: Double Negative, DeMorgan
etc. apply to propositions and sets.
To prove incompleteness, Godel had to generalize wffs as expressing
propositions to expressing sets when the wff has a free variable.
C-B
> >> There is a struggle between forms that are purely syntactic
> >> and the fact that well-formedness must convey significance
> >> before interpretation.
>
> > But what is the significance of trying to apply axioms to different
> > functions than those of original interest? Who uses other than the
> > standard interpretation of + * ** ?
>
> From what I have gathered in reading historical references, various
> issues lead to that.
>
> In Aristotle, the justification for a deductive calculus with
> respect to science is epistemic. So, proofs trace a coherent
> framework between scientific assertion. And, Aristotelian
> necessity admits circular structure in the formation of initial
> formulas. But, Aristotle also introduced a notion of substance.
>
> It is in the fruitless search for "simple" substance that the
> history of logic and mathematics reduces itself to more and
> more abstract syntactic form.
>
> Leibniz has no problem following Aristotle and allowing
> circularity in axioms. But, when we get to Bolzano, the
> search for a non-circular definition of a simple substance
> begins to lead to the arguments about undefined language
> primitives (different from the symbols needed for transformation
> in the calculus).
>
> By the time we get to DeMorgan, one gets explicit discussion
> of the possibility of pure syntax subjected to random
> interpretation which, when coherent, we would call a
> model.
>
> Other developments shattered the "meaning" of mathematical
> words. The discovery of non-Euclidean geometries and the
> admissibility of complex arithmetic and quaternion arithmetic
> are two examples. It is this precursor that enables Cantor
> to argue for a transfinite arithmetic in the arena of justifying
> the differential calculus.
>
> >>> A. Na ve Set Theory
> >>> B. Formal definition of a wff including substitution for variables
> >>> (aka interpretation.)
> >>> C. Statement that x ~e x is not a relation (aka set or predicate.)
>
> >>> [The whole idea of interpretations is also not well designed. It is
> >>> an example of generalizing the wrong thing, as Productive Sets
> >>> generalize the set of true sentences - a fixed aspect of
> >>> incompleteness proofs instead of the premises which vary especially
> >>> those implicit in the carrying out of the proofs but never stated.
>
> >> Presuppositions are clearly problematic.
>
> > Just the omission of indicating they are premises.
>
> Perhaps. But, if one considers the various perspectives of
> people who disagree, the "new" theory of the detractor has its own
> presuppositions. It is like a game where the presuppositions
> are "hidden variables".
>
> >> The consequences of many of these landmarks in foundational
> >> studies are viewed as definitive epistemic limitations without
> >> considering them further. A number of years ago, it was
> >> shown that classical propositional logic was not categorical.
>
> > Who would say such a thing? Propositional calculus is a necessary
> > subset of categorical logic.
>
> Pavicic and Megill. 1999
>
> "Non-orthomodular Models for Both
> Standard Quantum Logic and Standard
> Classical Logic: Repercussions for
> Quantum Computers".
>
> >> The apparent discrepancy was identified as a presupposition
> >> concerning logical equivalence within the method of proving
> >> completeness.
>
> >> The semantics of "ideal language theory" has been being eroded
> >> by the study of pragmatics. But, mathematics has become set
> >> in its ways with regard to model theory.
>
> >>> We don t want to know all functions that satisfy Peano s Axioms. If
> >>> it is done right there is only one set of functions that + * ** can
> >>> be. What we have lots of variations of is the properties of N that
> >>> is what is to be generalized. Saying + is not addition is like
> >>> Fortran allowing you to redefine what 1 means. There s no need for
> >>> that either. It only muddies the water.
>
> >> Once again, you are diverting from classical notions of calculi.
>
> > Isn t this how they do logic now? And do you know of anyone using
> > other than the standard interpretation ?
>
> Mathematics is different from logic. So, for example, one
> can talk of permutations on a set of objects. The set of
> permutations have an arithmetical property with one another.
> Thus, they form a system. As a system, they are independent
> of the underlying set. Then with some additional terminology,
> one speaks of group actions on arbitrary sets, categories of
> groups, and group representation theory.
>
> Usually, that particular system is thought of as a multiplication. But,
> there are groups whose intrinsic property is one of
> addition.
>
> In any case, a "naive" programmer would think of this as
> beginning with some predefined data types, forming objects
> and object methods, and differentiate the object methods
> from the arithmetical relations of the predefined data types.
>
> But, mathematics does not really have pre-defined data types,
> although the drive for foundations has organized mathematics
> to now look as if it does. That is a good in many ways. Still,
> some things are lost.
>
> > (A calculus is a cross between a logic and a programming language, and
> > they are severely underutilized. Program Synthesis can be easily
> > explained as a program calculus, while researchers are clueless as to
> > how to address the problem - to the point of making blatantly
> > fraudulent claims about Martin-L f Type Theory.)
>
> See, this is how different backgrounds lead to different
> things. Some time ago I read a great book on formal
> language theory in the sense one would have emphasized in
> a computer science curriculum. It would be the kind of
> thing someone writing compilers would have to know. I am
> assuming that this applies somewhere in what you have written.
>
> But, I see that it is too much for a quick Wikipedia read.
>
> I have looked at the lambda calculus a small bit. I hope to
> take some time this year to learn enough to understand your
> paragraph. But, I also have other interests. I am just
> sick of not knowing about so much good work that came out
> of intuitionistic logic and constructive mathematics.
>
> >> Part of the reason that one speaks of "number systems" is because
> >> of the development of the complex numbers and the quaternions.
> >> With arithmetical systems different from the usual arithmetic,
> >> mathematicians were confronted with the genesis of model theory
> >> and interpretation of calculi.
>
> >>> Godel/Rosser/Smullyan incompleteness theorems include reference to a
> >>> wff being true but not provable. But then it must be true for which
> >>> interpretations? .
>
> >> Yes. But, while there may be a number of ways to introduce
> >> Goedel numbering, there is always the method that involves
>
> > This doesn t have anything to do with the choice of correspondence
> > between wff and number. It is the insistence that we use variables
> > for functions and indicate what + * and ** represent.
>
> Ok. The next time I dig around in my logic
> texts, I will look at one of the proofs more
> closely and think about what you are saying.
>
> >> prime decompositions. So, what kind of interpretation of
> >> arithmetic would alter the configuration of primes and their
> >> relation to the number system as a whole?
>
> >>> That is left out and opens up the question, why
> >>> the standard interpretation works and which ones work? But we really
> >>> don t care about using other functions for + * ** in the first place!
> >>> Certainly not in the middle of an incompleteness theorem.]
>
> >>>>> We don t need ZF - at all. All we need is Na ve Set Theory, a
> >>>>> complete formal definition of wff and recognition that x ~e x is not a
> >>>>> relation due to diagonalization on sets.
>
> >>>> The reason for ZF and other inquiries into the
> >>>> foundations of mathematics has to do with a coherent
> >>>> explanation for the utility of an otherwise incoherent
> >>>> collection of mathematical techniques. If such an
>
> >>> ZFC is one of several axiomatic systems proposed to formulate a
> >>> theory of sets without the paradoxes of naive set theory such as
> >>> Russell's paradox. - Wikipedia
>
> >>>> explanatory role is not forthcoming, such theories
> >>>> at least organize mathematical techniques into a
> >>>> science (in the sense of Aristotle) connected by
> >>>> the argumentation of proofs.
>
> >>> I am not saying to not formalize. (I have personally axiomatized at
> >>> least 5 branches of Computer Science/Logic. Every case of
> >>> incompleteness is handled by a single axiom to distinguish the sets or
> >>> relationships that cannot be characterized in the system. In fact,
> >>> that additional axiom is the only difference between the positive and
> >>> negative sides of a theory e.g. Universal Turing Machine vs. Halting
> >>> Problem in the Theory of Computation.)
>
> >>> I am saying that ZF is a lousy attempt at formalizing and I propose an
> >>> alternate formalization a simple addition to Frege s Na ve Set
> >>> Theory. Just using predicate calculus instead of a specialized
> >>> language to state the axioms makes ZF hard to communicate:
>
> >>> The precise meanings of the terms associated with the separation
> >>> axioms has varied over time. The separation axioms have had a
> >>> convoluted history, with many competing meanings for the same term,
> >>> and many competing terms for the same concept. - Wikipedia
>
> >> Yes. There is a problem with interpretations.
>
> >> My own issue lies with the axiom of extension. It can
> >> be eliminated in favor of axioms more consistent with
> >> the historical developments associated with the identity
> >> relation.
>
> >> Language is topological. The complex of a negation
> >> symbol with the Fregean "the True" and "the False" makes
> >> a formalized language representable as a minimal Hausdorff
> >> topology. This is a semiregular topology.
>
> >> Not surprisingly, the manipulations used in forcing
> >> involve a topology based on regular open sets which
> >> is also a semiregular topology.
>
> >> Forcing models manipulate the topological structure
> >> of language just like a coffee cup is made to look
> >> like a donut for classification purposes.
>
> >> That is before one even gets to separation.
>
> >>>>> Logic = Set Theory
>
> >>>> If this is true, it is not the logic of which
> >>>> you are thinking.
>
> >>>> What you are taking for granted is the structure
> >>>> of logic without a negation symbol. The negation
> >>>> you use in your programming has no reality in the
> >>>> underlying computer architectures.
>
> >>> With negation you have all levels of the Kleene Arithmetic Hierarchy,
> >>> which means any wff that can be expressed. (Each added ~exists adds a
> >>> level.) Without negation you have only Sigma-1 the recursively
> >>> enumerable sets, and the negation (complement) of some included sets
> >>> of natural numbers are not included.
>
> >>> Set Theory, axiomatic Logic used to express sets with wffs that are
> >>> true of its elements, and English all have negation and are
> >>> equivalent. Computer programs, proof in axiomatic Logic and the
> >>> various bases of computing developed during the 1930s (excluding a
> >>> couple of misfires) are Sigma-1, do not allow the complement of every
> >>> set allowed and are equivalent.
>
> >>> Is this what you re referring to?
>
> >> No. I really do need to "catch up" with some of the
> >> work computationally-literate mathematicians utilize.
>
> >> I literally mean considering the nature of logic
> >> without a negation symbol.
>
> >> My studies on the identity relation have essentially
> >> eliminated logic from the foundations of mathematics.
>
> >> A negation symbol is like the sign of a determinant.
> >> The sign of a determinant is correlated with the
> >> handedness of a coordinate system. Classical negation
> >> is correlated with geometric reflection through a
> >> line. The algebraic representation for this is
> >> the subdirectly irreducible DeMorgan algebra on
> >> four elements.
>
> >> Typically, the system of 16 basic Boolean functions
> >> is thought of in relation to a 16 element Boolean
> >> algebra. But, that algebra is simultaneously order
> >> isomorphic with the 16 element DeMorgan algebra
> >> formed as the Cartesian product with the DeMorgan
> >> algebra on four elements.
>
> >> It is DeMorgan algebra rather than Boolean algebra
> >> which is the foundational form.
>
> >> At this level, one can actually represent the
> >> structure in a finite projective geometry. The
> >> 16 elements corresponding with the truth functions
> >> (as "objects") are the affine points of that
> >> geometry. Negation, DeMorgan conjugation, and
> >> contraposition reflect geometric projectivities
> >> with the involution corresponding to negation
> >> having the line at infinity as its axis.
>
> >> Thus, negation as a "unary" connective is essentially
> >> the line at infinity.
>
> >> The exaggeration above concerning the elimination
> >> of logic can now be tempered with various
> >> classical observations from authors such as
> >> Carnap who recognize that the syntactical structure
> >> of logic is very much like a geometric structure.
> >> My studies have simply identified an explicit
> >> form for it.
>
> >>>> More formally, what you are taking for granted
> >>>> is that only 14 of the 16 basic Boolean functions
> >>>> are linearly separable switching functions. The
> >>>> two that are not are logical equivalence (LEQ) and
> >>>> exclusive disjunction (XOR).
>
> >>>> These particular connectives become problematic
> >>>> when considered in the context of classical quantificational
> >>>> logic because of the relation of identity, definition,
> >>>> and description.
>
> >>>> The standard account of identity (for example as
> >>>> discussed under "relative identity" at
> >>>> plato.stanford.edu) addresses trivial identity,
> >>>> that is, x=x, and substitutivity.
>
> >>>> What is not addressed is informative identity,
> >>>> that is, x=y.
>
> >>>> In classical model theory, however one has
> >>>> determined an object in a model and a name
> >>>> for that object has consequences for the
> >>>> satisfaction map. That is how the classical
> >>>> model theory interprets x=y.
>
> >>>> In 1971 Tarski directed his attention to the
> >>>> representation of first-order logic in the context
> >>>> of algebraic logic. In those deliberations, he
> >>>> introduced the axiom
>
> >>>> AxAy(x=y <-> Ez(x=z /\ z=y))
>
> >>>> In the formulation of these "cylindrical algebras"
> >>>> the formula
>
> >>>> x=y
>
> >>>> corresponds geometrically to a 2-dimensional subspace
> >>>> of the domain. In a 3-dimensional domain, this is
> >>>> a hyperspace separating the domain into two regions.
>
> >>>> This suggests that there is a fundamental geometric
> >>>> reason for LEQ and XOR to not be represented in the
> >>>> underlying propositional logic by linearly separable
> >>>> switching functions.
>
> >>>>> Logic = NOT AND OR EXISTS simple, easy
>
> >>>> What happens if I take NOT away?
>
> >>> Interesting question. Assuming you can express without quantifiers
> >>> all recursive sets, since all wffs can be put into prenex normal form,
> >>> you can express the same sets. Neither the quantifiers nor the
> >>> relations need the negation symbol.
>
> >> Well a "full" system of connectives has NAND and NOR. So,
> >> there should be no diminishing of what can be expressed.
>
> > I mean AND and OR only.
>
> >> My question had been directed at the complexity of determining
> >> a canonical choice between NAND and NOR for use as a negation
> >> operator.
>
> > Only a program and a few minutes of processing can answer that. First
> > question is the spec: what do you want to check for in the finite
> > world of propositional calculus?
>
> It is much more complex than that. :-)
>
> I have 4096 axioms that look like
>
> AND (IF,AND) = AND
> AND (IF,NIMP) = NIMP
> AND (IF,XOR) = NIMP
> AND (IF,IMP) = LEQ
>
> Under DeMorgan transformation, each axiom transforms
> into an axiom. So,
>
> AND (IF,NIMP) = NIMP
>
> transforms into
>
> OR (NIMP,IF) = IF
>
> This is why I no longer think about (or believe)
> Boolean algebra as a foundation for mathematics.
>
> Other transformations (i.e., negation) do not
> have that invariance.
>
>
>
> > BTW Why does NOT and EXISTS create a Kleene Arithmetic Hierarchy level
> > and not AND and OR? I recently realized the answer. I mean from a
> > high level properties of the connectives point of view? P(NOT) ^
> > P(EXISTS) ^ ~P(AND) ^ ~P(OR).
>
> Well, I do not really grasp your notation here.
>
> By definition, the Kleene hierarchy is based on quantifier
> complexity, is it not? ALL is just the DeMorgan conjugate
> of EXISTS (in the sense of "negate the arguments
> and negate the connective" applied to a single
> argument operator -- ALL=NOT(EXISTS(NOT)))
> so that NOT and EXISTS should generate the hierarchy
> as a matter of definition.
>
>
>
> >> In fact, there are four Boolean functions that negate uniform
> >> arguments. What you call P and Q I call FIX and LET. There
> >> respective negations I call FLIP and DENY. The four Boolean
> >> functions that negate uniform arguments are
>
> >> NAND, NOR, FLIP, DENY
>
> >> My solution for the complexity of making a choice was to
> >> recognize that the structure of the projective geometry
> >> could be manipulated to accommodate extensions to the
> >> propositional connectives. In effect, propositional
> >> logic is an incomplete system. The sense of a negation
> >> arises from its relation to quantifiers or operators.
>
> >> Separately, such quantifiers or operators have the
> >> four-fold structure one characterizes using negation.
> >> Organizing them into a unified system forms the
> >> lattice below, where the NOR connective is an integral
> >> part of the intersection of the component sublattices
> >> and NAND is not.
>
> >> In regard to the projective geometry, there are 5
> >> points on the line at infinity. The projectivity
> >> corresponding to negation has its center on the line
> >> at infinity. The line without that center corresponds
> >> leaves 4 points, and, that is what I am treating
> >> as a quantificational or operational complex.
>
> >> Often, mathematicians are interested in invariances.
> >> If you perform DeMorgan conjugations on all 16 Boolean
> >> functions, you will find that
>
> >> FIX, FLIP, LET, and DENY
>
> >> are invariant under DeMorgan conjugation.
>
> >> As I said, I need to catch up on the kinds of
> >> mathematics computationally-literate mathematicians
> >> use. I am almost finished with these particular
> >> geometric concerns.
>
> >>> (I did go through a period of writing software in search of minimal
> >>> bases (subsets of the 16 binary Boolean functions) for propositional
> >>> calculus.)
>
> >> With what results?
>
> > Unfortunately I must say that history does not record what happened
> > next. But given the spec and technical design I can whip up a PHP
> > script. You want the subsets of the 16 BBF (binary Boolean functions)
> > which can express all 16 and have no subset that can? Then what would
> > the technical design be? For each BBF . . . ?
>
> I cannot imagine a need for that information, although
> both memory and imagination fail me far too often in
> life.
>
> What my question really was directed toward was whether
> or not you discovered anything interesting or surprising.
>
> So, for example, I always believed that only NAND and NOR
> negated a uniform argument just because I did not immediately
> think of the semantics of the propositional connectives as
> itself being meaningful only in the context of a full system
> of representation (fancy way of saying that all sixteen basic
> Boolean functions are presupposed even if a minimal set
> of connectives is used).
>
> In the context of all sixteen functions, I realized there
> were two more.
>
> So, do you remember anything notable?
>
> >>>> There is no real way to post this picture to a newsgroup.
> >>>> It is the ortholattice which is an atomic amalgam of a
> >>>> Boolean lattice with 4 atoms (the usual 16 element lattice
> >>>> associated with basic Boolean functionality) and a
> >>>> Boolean lattice with 3 atoms.
>
> >>>> ....................................TRU....................................
> >>>> ............................./.../..//\...\.................................
> >>>> ......................../..../.../../....\...\.............................
> >>>> .................../...../..../..../.........\.....\.......................
> >>>> ............../....../...../....../...............\......\.................
> >>>> ........./......./....../......../.....................\.......\...........
> >>>> ....../......./......./........./...........................\........\.....
> >>>> .....IF......NAND.......IMP.....OR.........................ALL........NO...
> >>>> ..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
> >>>> .../...\./\......\./\....|./..\./..\...\...\...................../.........
> >>>> ../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
> >>>> .//......\.../\.../....\.|.\../....\.\......\...\......../.................
> >>>> LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
> >>>> .\\....../...\/...\..../.|./..\...././......./...\.......\.................
> >>>> ..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
> >>>> ...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
> >>>> ....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
> >>>> .....NIF......AND......NIMP.....NOR........................OTHER......SOME.
> >>>> ......\.......\.......\.........\.........................../......../.....
> >>>> .........\.......\......\........\...................../......./...........
> >>>> ..............\......\.....\......\.............../....../.................
> >>>> ...................\.....\....\....\........./...../.......................
> >>>> ........................\....\...\..\..../.../.............................
> >>>> .............................\...\..\\/../.................................
> >>>> ...................................NTRU....................................
>
> >>>> The two lattices share TRU, NTRU, OR, and NOR. The structure of
> >>>> the unfamiliar lattice has
>
> >>>> SOME=EXISTSWITH=Ex
> >>>> OTHER=EXISTSWITHOUT=Ex-
> >>>> ALL=Ax
> >>>> NO=Ax-
>
> >>>> with
>
> >>>> ALL=NOT(OTHER)
> >>>> NO=NOT(SOME)
>
> >>>> on the basis of the order relation alone.
>
> >>>> This construction, while described specifically
> >>>> for quantificational logic here, actually characterizes
> >>>> the geometric (in the sense of an atomic lattice) structure
> >>>> of any extension to propositional logic with negation.
> >>>> Modal extensions are four-fold interrelating POSSIBLE and
> >>>> NECESSARY. Deontic extensions are four-fold interrelating
> >>>> OBLIGATORY and PERMITTED.
>
> >>>> For quantificational logic, each variable has this
> >>>> four-fold structure. This corresponds with the indexing
> >>>> of quantifiers found in the cylindrical algebras of
> >>>> Tarski's later work.
>
> >>>> The negation symbol masks this four-fold structure in
> >>>> the formation rules for formulas.
>
> >>>> Yes. Logic in the absence of NOT--just like in the
> >>>> hardware of your computer systems--is not easy.
>
> >>>>> ZF Set Theory = a dozen messy axioms for which people can t even agree
> >>>>> on the specifics ??
>
> >>>> There are actually an infinity of axioms (damn those
> >>>> first-order logicists) because well-formed formulas are
> >>>> separately generated and present in the axioms of separation.
>
> >>> Yes, but this schema ranges over wffs (rather than sets) so the set
> >>> defined is aleph-0 and in fact recursively enumerable. Better to
> >>> range over sets and pull in that needed aleph-1.
>
> >> How do you mean? Any references? I am always curious why
> >> the Borel hierarchy extends to aleph_1. I am certain your
> >> statement reflects the same "need".
>
> > I am merely thinking that the reason that ZF is said to not be
> > finitely axiomatizable is simply because there are only aleph-0
> > expressions and at least aleph-1 sets (Cantor/Godel.) And the axiom
> > schemes that are blamed are not the ones that actually address or
> > cause this. The schema would have to range over sets not expessions.
> > No references.
>
> I see.
>
> In my imagination, they do.
>
> What I mean by this is that the separation and replacement
> schemas merely ensure that one cannot use grammatical methods
> to create subsets that would conflict with our basic sense
> of what a subset ought to be. If we can use language to
> delineate some heap as a well-construed collection for
> which identity is meaningful, it certainly ought to exist
> in a theory of sets as objects.
>
> And, thank you. You seem to have directed me to where
> I should be looking. If one opens a book on descriptive
> set theory, everything is aleph_1 without explanation.
> Now it makes sense. The consideration is prior and
> from sources not specifically associated with the theory.>>>>> There are a dozen set theories and a dozen interpretations of the most
> >>>>> popular set theory, and 2 or 3 versions of it (with or without Choice,
> >>>>> etc.) none of which decide any of the important questions of set
> >>>>> theory due to exhaustive work (a waste!) by Godel and Cohen.
>
> >>>> That is an odd thing to say. While I find forcing to be nonsense
> >>>> in set theory (but I reject the axiom of extension as foundational)
> >>>> it is extremely important to recursion theory, is it not? And that
> >>>> has consequences for the practical application in computational
> >>>> contexts, does it not?
>
> >>> You are referring to the advancements in prosthetics made during war.
>
> >> So, my ignorance is showing, as usual.
>
> > But it is a good idea to get that fact out of the way early.
>
> > All the best,
>
> > C-B
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