There Is No Standard Model of ZFC and ZFC_2 with Henkin Semantics
Article There Is No Standard Model of ZFC and ZFC2 with H...
An posible generalization of the Löb's theorem. Jaykov Foukzon*, Israel Institute of Technology, Haifa, Israel (1089-03-60)http://www.ams.org/meetings/sectional/2210_program_ss17.html
Peter Breuer ,
The link to the paper is
Article Consistency of Set Theory
That he has a publication of an empty page and that the paper in the link I provide has proof 2.1that is 1sentence long is interesting to say the least
>But we don't know the thing you assert we know anyway. Goedel proved consistency, if it is true, is unprovable. So in particular we can't prove there is a model,
I claim that such a model does not exist in principle
https://philpapers.org/archive/FOUTIN-5.pdf
https://cs.nyu.edu/pipermail/fom/2013-February/016987.html
>I would kind of agree that it is hard to imagine how we could ever physically touch such a model and therefore be sure that it exists and is not just a (false) product of our own fevered imagining about how logic and thinking may be extended into a realm beyond our immediate experience. But Goedel's formalizations avoid that problem by being about concrete formal marks on paper with agreed modes of thinking and reasoning about them
Dear Peter Breuer
Tere exists hypothesis about Actual (potential) infinity
https://en.wikipedia.org/wiki/Actual_infinity
This Turing machine should run forever unless maths (ZFC+foundation axiom) is wrong
Read more: https://www.newscientist.com/article/2087845-this-turing-machine-should-run-forever-unless-maths-is-wrong/#ixzz7GuUYPuYp
https://www.newscientist.com/article/2087845-this-turing-machine-should-run-forever-unless-maths-is-wrong/
For example this Turing Machine Shouldn't Run Forever for the case of NC that is.,naive set theory by Kantor.
>Foundation (well foundedness of elementhood?) is independent of ZFC
This statement is true iff ZFC is consistent!
Of course that is not very important which model is considered, standard or nonstandard.
ZFC+Aczel's anti-foundation axiom
the same inconsistent.
Dear Peter Breuer
If ZFC really is inconsistent, then ZFC prove any statement in his own language.
>I am hoping for you to specify the formal language in which you think such things should be phrased, and its rules of deduction.
I meant standard set-theoretical language since it presented in classical handbooks
when ZFC is inconsistent, then in ZFC provable A&~A for some A,
but A&~A trivializing the system
My claim consist that set theory ZFC +foundation (regularity) is inconsistent
Of course this asserts that ~Con(ZFC)
As I understand your question is why the axiom of regularity is added?
this is a reasonable question :-(.
You are not the first one who asked me this question.
>But your claim now is that ZFC |- not(F)? At least THAT is a clarification!
My claim consist that ZFC+F |- ~Con(ZFC+F) and therefore
ZFC+F |-~Con(ZFC).
Thus ZFC |- not(F) since ZFC is inconsistent
>[* that is anyhow very dubious because nobody is specifying the formal system in which the proofs are phrased.
Why are you so sure of this? explain in more detail.
If I understand correctly, your question, you doubt the availability of formal proof of the my claim mentioned above?
Dear Peter Breuer
I know that you believe that ZFC is consistent But you belive wrong
>ZFC+F |- ~Con(ZFC+F)" is not a well-formed formula,
No at all. Note that ~Con(ZFC+F) is expressible by ZFC language.
The formula ZFC+F |- ~Con(ZFC+F) says that there is a proof in ZFC+F the formula ~Con(ZFC+F)
The formula ZFC+F |- ~Con(ZFC+F) says that there is a proof in ZFC+F the formula ~Con(ZFC+F)
Of course such proof must be presented exactly.
Are you saying that this is impossible?
~Con(ZFC+F) mean that: there is no standard model of ZFC
Axiom M.
There is a set M and a binary relation ∈ ⊂ M X M which makes M
a model for ZF.
We emphasize that Axiom M is a single statement in ZF,
see P. Cohen pp.78-79
>If that is equivalent to "no standard model of ZFC", please explain...
1.By the following P.Cohen theorem, axiom (F) holds only in standard model of ZFC
THEOREM. (see P.Cohen handbook p.83) Let M be a model for ZF, and let xRy
denote the relation between x and y expressing that x is a member of y in the
model M. A necessary and sufficient condition for M to be isomorphic
to a standard model is that there does not exist a sequence (x_n ) of
elements in M (the sequence itself need not be in M) such that x+1_ nRx_ n.
Thus if model of ZFC+F exists such model exactly standard model
>if there were a set-sized model M of ZFC it would not be provable that it exists,
2.Godel proved that an model M exists (if ZFC+F is consistent) but by P.Cohen Theorem this model alwais standard model.
If such standard model is not exists, then theory ZFC+F is inconsistent and therefore
ZFC also is inconsistent since axiom (F) independent from ZFC if ZFC consistent.
>I presume the intuition is that if one can prove there is no such M, then it is because any attempt to posit such properties for M as it must possess leads to falsehood. But those properties are the axioms of ZFC, plus the claim that the whole thing, M, is set-sized. So I can imagine that one can think proving no such M amounts to proving ZFC inconsistent,
3.yes of course you are right
>But it seems also to assume that the model is set-sized, against which one can set several objections. Why think that set-sized does not matter?
4.set-sized does not matter since a contradiction derivable in any standard mode
>I do not know what you mean by a "standard model".
https://math.stackexchange.com/questions/310772/how-does-one-define-a-standard-model-of-zfc
>No, Goedel did not claim a model of ZFC exists
I meant this Godel theorem
THEOREM 2. Go'del Completeness Theorem. Let S he any consistent
set of statements. Then there exists a model for S whose cardinality
does not exceed the cardinality of the numher of statements in S if S
is infinite, and is countable if S is finite.
see P.Cohen p.13
>It essentially says there is a model (where? Check!) of every
1.This implicitly means the hypothesis of the Actual infinity
>.I am aware btw that measurable cardinals give rise to interior (i.e. set-sized) models of set theory and therefore can't be proved to exist, which doesn't stop people writing books about them. (A measurable cardinal number is one that is bigger than the power set cardinal of every cardinal smaller than it).
2.There is no measurable cardinals in ZFC since it form a standard model of ZFC
Article Generalized Lob's Theorem.Strong Reflection Principles and L...
>which doesn't stop people writing books about them.
3 I told them that There is no measurable cardinals in ZFC and that they are crazy, but these crazy people started scolding me ...
>If not logically consistent surely there is no model as a model would just give up, wail and die on seeing the proof that it could not exist.
This is not so because the contradictions are deeply hidden and do not lie on the surface as in naive set theory.
>One can chuck in a +C and/or a +F with toast and beans to all that, I don't care.
There is no need for this assumption.
There is a very foolish opinion that if the ZFC is inconsistent, then it would have already been discovered. There are many problems that have not been solved by anyone.
Dear Jaykov Foukzon, This is an interesting discussion but a bit over my head. The subject document seems to have been pulled and replaced with a cover sheet only.
On the general subject of Con(ZFC) and because of Godel's completeness theory, it seems very improbably that anything syntactically derived from 8 or 9 axioms of ZFC are a subset of con(ZFC).
For example, I am working on a proof of a complete model of the continuum hypothesis. There is a surprisingly simple resolution to CH, but the problem is to show that a semantic model is a form of syntactic proof.
https://www.dropbox.com/s/s360j0gqh1e90qz/CompleteModelofCH_2020_Oct_12_RevE1_Abdallah2020_Oct20_Abstract.pdf?dl=0
Without going any deeper, when I take the standard results (e.g. reference Suppes) on the smallest model of infinite set cardinalities, this minimum model would supplant about ZFC 4 or 5 axioms. When I look at the axioms I see much more latitude for semantic interpretation than the consistent set in this minimum model. In fact, there is redundancy in the axioms despite their presumed independence.
While Pinter's "A book on Set theory" is not an advanced text, nevertheless the author argues (page 219 Models), that while it is accepted the fact that a theory is consistent iff it has a model, he argues that it is impossible to conceive of a model for ZFC. So how can it be assumed that Con(ZFC)=ZFC is fact?
https://www.maa.org/press/maa-reviews/a-book-of-set-theory
There are other free online references but they seem to be blocked.
My approach in using ZFC is to construct inner models that necessarily distribute across all of con(ZFC) and therefore prove the model properties exist in ZFC. Equivalently the axioms of ZFC is a semantic interpretation of an inner model but not necessarily consistent.
This begs the question about whether this inner model applies in many particular situations (i.e. is it a global property). This does require an inner model of sets in the whole domain of set theory. This is a work in process but and is a very productive use of set theory but one based on model theory rather than the axiomatic method.
Kind regards Jim
Dear Peter
In model theory, a model is a structure of properties which for all intents and purposes is a set therefore an inner model is a model structure subset. According to downward Lowenheim- Skolem it would also be called an elementary substructure which I view as an inner model.
So for example the Propositional Calculus is a subset or inner model of ZFC where we would say that ZFC specializes PC.
If I use |= as "double turnstile" to reference semantic implication then
PC |= con(ZFC)
Here is a Base Cardinality Model of ZFC, a set of WFFs based on the standard results. PC |= M_BaseCard |= Con (ZFC)
https://www.dropbox.com/s/2x3dl5h4nrlrmoy/MetaModelofCHIndepence_2020_Oct_16_RevF1_Preprint_NonStandardBaseCardinailityModel.pdf?dl=0
It is my contention that due to the compaction theorem that an inner model distributes to all semantic interpretations. It is literally what the formal definition semantic implication as indicted by double turnstile.
Do I have something wrong here?
Regards Jim
Con(ZFC) is a statement about the existence of a model for the consistent part of ZFC. If ZFC were consistent then we could say Con(ZFC)=ZFC. That is the primary question so that my intent was to indicate Con(ZFC) as the consistent parts of ZFC. For example, Godel's Constructible Universe is an accepted inner model of ZFC.
However, when Jaykov says there is no standard model of ZFC, I assume he is speaking of the whole of ZFC and not that there is no part that is consistent.
PC is consistent and complete, but models are iff associated with consistency of a theory so I dropped the obvious Con(PC)=PC.
PC|=Con(ZFC) say that the properties of PC are true in the models of ZFC (i.e. the consistent part)
to Jim Moore
>However, when Jaykov says there is no standard model of ZFC, I assume he is speaking of the whole of ZFC and not that there is no part that is consistent.
Yes of course I speaking of the whole of ZFC.
However note that. 1.There are exist locally contradictions in ZFC. These contradictions is not completely harm the system since can be easily prevented by Quinean approach as in Quinean Set Theory NF.
There an global contradiction which completely harm the system.
to Peter Breuer
>It appears that a "standard model" means a nonempty set (class?) with a well-founded "element-of" relation, with assignments for the constants and operators of ZFC that make the axioms of ZFC true?
Yes! There are in literature set-size models and class-size models
new title
What do you think about C ́esar J. Rodrigues proof of the Con(ZFC)?
Is ZFC is inconsistent?
1.In Henkin semantics, each sort of second-order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin (1950) defined these semantics and proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first-order logic.
>Is a standard model again a nonempty set- or class-sized model (in ZF or ZFC or ZFC+F?) for the operators and constants of ZFC obeying the axioms of ZFC, with a well-founded element-of relation? Are the logical operators and constants interpreted in the model too?
2.Yes. Note that in canonical mathematics we alwais deal with ZFC+F.
By Gödel's second incompleteness theorem there is no point of looking for an exclusively formal consistency proof, at least if we are interested in recursively axiomatisable systems which interpret Peano Arithmetic. Consistency proofs must appeal in one way or another to (finitary) intuitive evidence, even if this be in the form of mechanically translating a system into another system which we hold to be more self-evidently consistent. Schoenfield's book Mathematical Logic gives us two philosophically illuminating examples of consistency proofs for Peano Arithmetic: one based on the Hilbert-Ackerman Consistency Theorem and another based on Gödel's Dialectica interpretation. For ZFC, given a finite set of instances of the axioms, we can prove (in ZFC) that they have a model. But this is not the same as proving in ZFC "every finite set of instances has a model" even though this is self-evident to us. This is sometimes called a "reflection principle".
The appeal to intuitive evidence is found also in Gödel's first incompletelenss theorem. We perceive that the sentence G which affirms its own non-provability is actually true, for Gödel showed that no proof of G can exist in the system.
>That stuff comes straight from wikipedia, as I recall it!
Dear Peter Breuer. For a detailed explanation, you should contact the late Professor Gaisi Takeuti
Proof Theory
Gaisi Takeuti
https://www.perlego.com/book/112275/proof-theory-second-edition-pdf
C.f. Einstein, "if you can't explain it then you don't understand it".
These are only Einstein's problems
Dear Peter Breuer,
Sorry for picking up this thread after a protracted absence.
From the Wikipedia oracles:
- "Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic."
- "Since the converse (soundness) also holds, it follows that T|=s if and only if T|-s and thus that syntactic and semantic consequence are equivalent for first-order logic."
https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
Therefore it is a first-order result model theory that a theory, T, is consistent iff it has a model Mt. We state this as Con(T) Mt (see the attached picture from Pinter's book.
Now in view of these putative results, you have asserted that PC |= con(ZFC) can be replaced (i.e. is equivalent to) PC |- con(ZFC). Unfortunately, if you review material Equivalence in the PC you will see that it is no defined equality. Formally we can say : {} ~= {=}
Now while you were able to deduce by a series of syntatic substitutions that your assumption was incorrect, you blamed me for your incorrect assumption.
You could have more easily deduced at the onset that your statement would lead nowhere because you violated Godeian incompleteness theorems by suggesting that PC, which can not even prove itself, could somehow prove all of con(ZFC).
Kind Regards
Jim
Dear Jim Moore
It must be emphasized that the notion of model is expressible by ZFC language!
Dear Peter Breuer,
My work has been in Model Theory at a very fundamental level because of related questions in systems engineering which has the unanswered question "Why do we use models?" .
I would pose a related question to set theorists, "With full knowledge of the completeness theorem, why would you do anything excluding models?". For obvious reasons, under putative interpretations of incompleteness, consistency is the only obtainable goal and that requires a model.
The implications of the completeness with compaction theorems is that inner models exist and distribute across the whole and can be composed minimally as per Godel's Constructible Universe.
So while Pinter's comments that he doesn't see any model for set theory unmistakably and categorially means he sees no way that set theory is consistent, it also implies he doesn't envision the consistent part of the set theory being constructed from inner models as per the Constructible Universe either.
This and the Downward Lowhenheim Skolem in mind, I have found that there is a complete model of completeness largely within the propositional calculus but also includes the simplest level of a model theory signature (including membership). My point is that using this model many of these great mysteries of formal logic are laid bare including incompleteness.
In software terms, if you are trying to solve a complex computational problem, you would try to figure out your computational stack to attack the problem. Theoretically if applied optimally or minimally this would look like an ordinal construction. This gets back to the original questions about why we use models to build solutions.
I'm working on a formal proof of why constructability always works (Upward Lowenheim Skolem) and it relates to modal collapse (when contingent and necessary truths coincide) which has an inner model in the PC and therefore distributes to all formal systems.
Kind Regards Jim
A. 0055, posted 2022/01/13
Goedel´s incompleteness theorems are only valid for monotonic proof-systems. Conrad Kuck completed Hilbert´s program. His non-monotonic proof-system is out of Goedel.
[Conrad Kuck, Non-monotonic learning automata]
`Stays proved´ was on the rules of multiplication by prefixed factors (Babylonien way).
Also on the rule of even roots on negative radicands (√-1 = i; no calculation / no result).
Following this on the Euler-equivalence (eiφ = cosφ + i sinφ).
Last but not least on the second diagonal argument (Cantor).
All this got outruled by doing the non-monotonic way of Kuck.
Jaykov Foukzon — Did you read Conrad Kuck´s paper? Would read yours.
@Peter Breuer, do you mean isometry by `a|-b => a,c|-b´? I am not familar with that way of spelling.
to Peter Kepp
I am using a similar approach
https://www.journaljamcs.com/index.php/JAMCS/article/view/30339
Article Set Theory INC # Based on Intuitionistic Logic with Restrict...
Dear Peter Breuer.
This paper beyond main discussion.
Abstract.In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
https://www.journaljamcs.com/index.php/JAMCS/article/view/30339
I cant see any which is near to the `Intutionistic set theory´ Kuck had introduced. Sorry. There are four parts / jpg of part 2.
And I am not interested to learn every decade extended and different spelling.
If English is too much ambiguous one should look for the best match for the expression.
@ Peter Breuer, maybe old but not my way to use all this types which get used by shortened spelling expressions of logic. I am not `in´ such crypto´s.
Jaykov Foukzon
As I remember parts of your work, you used equivalences between natural numbers (of amount) and limits of calculating-sequences. This is of infinitesimal calculation. And useful there for to be as correct as needed while calculating on different measures (naturals vs. π i.e.).
But for proofs in number-theory you should only use absolute identities. Not doing so proofs nothing! Sorry.
>But for proofs in number-theory you should only use absolute identities. Not doing so proofs nothing! Sorry.
Dear Peter Kepp.I don't know what you mean by absolute identity may be you mean st(x)=st(y)? For the proof by a contradiction such equivalence is not always necessarily!
Obviously you are not an expert in the field of non-standard analysis These papers use non-conservative extensions of IST.
No need for clarification if you don´t understand my simple statement.
I will not learn to use all the sybols you are using. So I can´t follow your statements. And also wont look again at your paper for citing.
Don´t worry. For me it seems that we are working at opposite direction.
It agree
>Peter Kepp added an answer
21 hours ago
No need for clarification if you don´t understand my simple statement.
I suspect that Peter Kepp rejected any proofs with infinitesimals.
Peter,
Again sorry for the delayed response. I have been having a related discussion in another forum with ISSS.org .
· But let's take the positives ... Jim, since you're an expert on Model Theory, perhaps you can explain what the Henkin stuff is all about on 2nd order (and higher?) models?
Thank you for your compliment, but I am not an expert in Model Theory. Jaykov is way above my level. I guess it remains to be seen if I have an advantage or disadvantage by not knowing what is “impossible” . As it may have allowed me to see associations/linkages (i.e. inner model inheritance) between Hahn-Banach Dual Linear Vector Space Optimization and Set Theory and even beyond to zeroth order propositional calculus.
- Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. https://en.wikipedia.org/wiki/Propositional_calculus
You can see this Wikipedia oracle suggesting that “all the machinery” which I read as “inner model” is included in all higher logics.
As a simple comprehensive guide to set theory, [Cohen 1966]”Set Theory and the Continuum Hypothesis” provides a compact introduction to well-formed formulas (wff) including a section entitled ”Universally Valid Statements”. This is apparently taken as first-order logic by the oracles https://en.wikipedia.org/wiki/First-order_logic. In particular section First-order theories, model and elementary classes
From this, I would assume that ZFC or NBG are first-order theories with axioms stated as wffs. And so I have not had to focus on higher-order logic because I’m working the other way from Hahn-Banach and started at Set theory. Generally speaking, I am interpreting Gödel’s Constructible Universe as an inner model of infinite-dimensional linear vector spaces. These are two well-established mathematical objects and what happens in between is not particularly relevant to the work of finding the most general form of HB.
I have been able to establish an inheritance of properties in a dual theory of sets.
For example, you should be able to relate the Min/Max pairs for the dual relationship (subset relation) in the figure below. I think it is straightforward t show that by the standard definitions of a set, all sets have to exist in a cumulative hierarchy (no set of a consistent theory exists outside of the constructible universe). This provides a formal basis to believe that the previously mentioned optimal solution to any **solvable computational problem is actually a constructive method, not just a theoretical result.
**There is no optimal solution to an unsolvable problem because the optimal solution is part of a solution class that does not exist.
RE: "In software terms ...." should be taken literally which does not mean theoretical computer science and specifically not complexity theory. I used the word computation because that is the original context of HB with a computational complexity metric/functional, but the example is software programs stacks or frameworks for computations.
Kind regards Jim
The way to understand is to use standard expressions. Yes, I am not origin English-speaker. And there is much to learn for me to do it in English what was thaught in German.
But those which are interested, seem to look more at the equations than on the words.
Nice to read that this problem is not exclusively mine.
Understanding should come first. After that correction of others is well done and very helpful. Also welcome for me.
Infinitisemal-calculation is doing algebra using different measures.
π never is correct (identic in value) using the construct `addition of series´.
So to use looking at slash2 (Cantor). One has to accept that by decimals no irrational number (of amount) is represented.
Even decimals and fractions are `at war´: 0.111 ... not equal (identic) 1/9 for example!
Peter,
Here is an example of a NON STANDARD global property (i.e. an inner model of infinite sets) that I think applies to higher-order logic. This can be deduced from standard results indicated by [std]
- · [std] In Suppes Theorem 39, you will see that every set is (my definition)” auto-infinite” because if a set is equal to itself it satisfies the definition of ”infinite”. In 39 let A=B the A=A is a condition for being infinite!
- · [std] In Suppes 44 we see that anything Dedekind infinite has to have a proper subset that is also equipotent (same cardinality as in 1:1 Diagonal method) infinite.
- · [std] It is accepted that both definitions are equivalent.
There are two profound NON STANDARD implications according to JM:
a. Theorem: Every set that exists in a consistent theory is part of the Constructible Universe L.
b. Sketch Proof:
i. Every set includes the null set.
ii. The cardinality of the null set #📷 implies the null set is unique.
iii. Every ordinal (and each rank or age of the ordinal) contains the unique null set.
iv. 📷 Every set only exists in an ordinal hierarchy (i.e. the Cosntructable Universe).
So this establishes the global and universal existence of optimal duality where there is a bijection between every set and a subset and it then has the optimal properties.
It is very important to read the MIN/MAX relation pair below and relate it to the diagram below that always exits as a generalization of Hahn-Banach dual vector space theory.
JIm
Peter,
"Suppes definition is incorrect."
As you seem to work in proof theory and not set theory, I can understand that these things are unusual. The most difficult thing is NOT new theories to solve old problems, but what is harder is trying to figure out how to accommodate old dogma in the context of a new non-standard theory.
If Suppes is not authoritative (putative) in ZFC then who is; the Wikipedia oracles?
"So your stuff is wrong from the get-go with [std] In Suppes Theorem 39,"
"Not apparently your fault, though. But surely you realized that's nuts! If not, I don't know what to say!"
You don't have to say anything as you are saying enough confirming what I suspected and that is prima facia evidence of the failures to the axiomatic proof methods to comprehend set theory because it does not focus on models. (something I asked you about before).
(Jim is now stepping up on the soapbox......)
More so than in any other field of mathematics, the axiomatic set theory appears to be so elusive because of emergent behaviors that are not expected and do not come from axioms that virtually can not be emergent. Axioms are an attempt to intentionally program axiomatically set-theoretic notions where the a priori nature of set theory is meant to step aside. Take the independence results of the Continuum Hypothesis as an example. The elders of the foundations of mathematics closed their ZFC doors claiming CH is independent of ZFC leaving the rest with the mouth's agape "but what does it really mean".
Since you have called me a nut there is no danger in telling you again I have solved CH using a complete model of the reals in a complete model of CH. The answer is embarrassingly simple (not for me because I figured it out). Here is the abstract and the model of the reals attached.
As I described before, it is much more difficult to explain in the context of 130 years of dogma than the solution itself which in the case of CH is both trivial and profound.
Jim
Extra credit question: Can you figure out what is profound?
https://www.dropbox.com/s/s360j0gqh1e90qz/CompleteModelofCH_2020_Oct_12_RevE1_Abdallah2020_Oct20_Abstract.pdf?dl=0
Peter,
I'm just a systems engineer; what do I know?
We just build models of the world and call them a reality because we can predict that we can hit targets, achieve specific complex behaviors, and analyze complex seemingly undefinable properties all without formal proof.
Jim.
>all without formal proof.
Dear Jim Moore. The great mathematician Vladimir Arnold rejected formal proof as nonsense.
Vladimir Arnold
https://en.wikipedia.org/wiki/Vladimir_Arnold
Dear Jaykov,
I think Peter Breuer's time is up for any extra credit on the profound nature of the solution to CH, so I will describe some of that here in relation to Vladimir Arnold's quote. ".....The great mathematician Vladimir Arnold rejected formal proof as nonsense."
The quote may a bit of an overstatement for dramatic effect but we should not let that diminish the underlying message. Formal proof can certainly lead to an interpretation that is nonsense due to what I would describe as “limited visibility of reality”.
To use an analogy, it would be like looking at the painting of the Mona Lisa through a soda straw and thinking you are looking at a sunset (the one in the upper right-hand corner). A concrete example will demonstrate this more clearly. I will use Gödel’s incompleteness as an example but start with a little-known quote by Gödel as to the reason for incompleteness.
- " As noted by Kurt Gödel, the reasons for the in-consistency in the logical operations of any usual mathematical system appears to stem from the inability of such a system to self-reference and ensure truthfulness in its conclusions [2]. '[2]"Godel, Escher, Bach: An Eternal Golden Braid"
The quote is from a paper " The imperative of self-reference in a theoretical framework for biology."By Richard Summers but he has failed to answer my request for a specific page in the 800 page book [2].
Of course, we immediately recognize what Gödel is pointing to as the issue of self-reference which has been prohibited by the ZFC Axiom of Foundation (AF). Regardless, the problem with formal proof cannot be completely attributed to dogmatic interpretations of incompleteness. Even the unfortunate dogma of incompleteness is a result of the narrow context of formal proof.
I can’t find the Wikipedia oracle’s reference at the moment, but paraphrasing it says “colloquially speaking a theory can be either complete or consistent but not both. “ This follows from the:
The first hint of a problem is that complete and consistent are not complements (incomplete and inconsistent respectively) but are supposed to be nevertheless disjoint. I will not produce a mapping of these categories, but rather explore a simple example I used to prove the Continuum Hypothesis as noted in a prior post.
If we say there is a model of the Reals in the Cantorian interpretation, Mr, then there also exists a complement of this model we will call MCr. Both models Mr and MCr are consistent by the Completeness theorem. The way to distinguish between model Mr and MCr is that there will be a statement with the membership relation in Mr and it will be negated in MCr.
Further, if there is a positive property in Mr, that property will necessarily be negated in MCr. This follows from the fact that there will need to be a set of double negatives in the primary model Mr to syntactically derive the complementary property in MCr.
So I have just described the basic properties of a Complete Theory {Mr,MCr} that contains two individually consistent complementary models (Mr and MCr) that are also derivable from each other (self-consistent) such that Mr |-MCr MCr|-Mr
So how do I reconcile this with Incompleteness? Well, it is because the caveat of AF is violated by the definition of a complement Mr MCr which is self-referential!
So the solution to CH is a complete and consistent set of models which is essentially prohibited by incompleteness owing to AF (as per Gödel’s assessment) and we wonder why nobody has ever solved CH is an accepted mathematical context.
In short, circular self-reference with complements is built into the Propositional Calculus(PC) Material Equivalence(ME) and then ZFC tries to outlaw self-reference in AF which it already has as an inner model of PC ME!
In reflection, Vladimir Arnold is correct formal proof sure does lead to nonsense but the wall of DOGMA is a far greater challenge even for such a great mathematician.
Kind regards
Jim
Dear Peter Breuer
I have asserted that a complete theory of the reals, Tr, is a Model of the reals joined with a complement of that same model such that:
Do you agree or disagree?
Regards
Jim
Discussion goes on `about the theory´ — nothing `by the theory´, not?
There are facts, done by research, which are not on crypting the theory itself. There are proofs constructed which refute expert opinion.
See the REFORM instead of naming the theory to proof anything ever and ever by other symbols (expressions)!
`Plus by plus equals plus´ is out, imaginary unit is out, cardinals are out, Euler-equivalence is out, uuncountable is out ...
ZFC: Group is out, so ring and field. If one wants to do set-theory and uses `C´ (for choice) one sorts and that´s no longer a single set the examination goes on.
So what are you talking about proofs? Does anyone of you understand how to proof? Goedel (incomplete) only is valid for monotonic proof systems. Kuck showed how to do it the non-monotonic-way. And I did.
Dear Peter Breuer
Substitute enumeration if you can not get your head around joining two complementary models of the reals. Of just read the comments explaining the standard curly brackets.
https://en.wikipedia.org/wiki/Set-builder_notation
- A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
- {\displaystyle \{7,3,15,31\}}📷 is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
JIm
Dear Pete Breuer,
Some background for motivation. As you are probably aware the Continuum Hypothesis (CH) remains a quandary within various technical circles even if it is resolved according to the ZFC high priests. See page viii by MartinDavis in his “Introduction to the Dover Edition”of Cohen’s 1966 Book “Set Theory and CH”.
The independence results strongly imply two solutions and therefore two models. Due to the perceived notions of Godelian Incompleteness, the strongest argument for a solution to CH is that the two models are complementary and circumscribe all sets within the domain or the Reals.
Whether by intuition, clairvoyance, or luck I arrived at (by simply looking for) such concrete and complementary model pair. The trivial part is that the Cantorian interpretative model treats numbers as points and the non-Cantorial treats numbers as segments of the continuum. The latter is exactly equivalent to what is implemented in fixed-point arithmetic.
Just as predicted by the model and its complement, the entire space of the real line can be modeled in these two complementary models. The results of testing CH is that CH is false in Cantor's interpretation and True in the non-Cantorian. The non-Cantorian is true because all numbers are segments of the continuum and therefore all have the same uncountable cardinality (nothing in between). It is still remarkable that this all worked out exactly as set theory would predict.
See this article that confirms the same. There is a formal proof document as well.
- Mathematicians Measure Infinities and Find They’re Equal
- Instead, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory that is already opening new frontiers of research in both fields.
- https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
>>>>Jim, are you trying to say something like "form the set consisting of the two models" by "join"?
First, recall earlier (too far back for me to quote exactly) I said something to the effect “for all intents and purposes we can treat a model as a set”.You would need to refer to your favorite source in Model theory although a can give you a free reference here http://www.personal.psu.edu/t20/notes/master.pdf
So models are sets and sets have complements.
For a Complete model of CH (probably close to a maximum model but I am using the simplest interpretation) , I only needed the minimum model which is basically Material Equivalence. I have formalized this as an inner (minimum) model of modal collapse (see attached) which would appear to be a central model that can be semantically interpreted to every model I have been working on. It appears to be an inner model of Godelian Completeness (I will defer any discussion of the relationship to Godelian incompleteness). It is also the inner model of Taski’s Truth and the definition of formal syntactic proof anywhere the results in A iff B.
After reading this I see there are different definitions of completeness and that even the Completeness in Godel's two proofs is considered different. I have deduced by Modus ponens that my definition of Completeness (A AND ~A)would appear to be stronger (more restrictive) than yours (A OR ~A) but there is no point getting into that now.
https://en.wikipedia.org/wiki/Complete_theory#:~:text=Maximal%20consistent%20sets%20are%20a,of%20only%20finitely%20many%20premises.
Hopefully, the attached figure answers most of your questions.
Regards Jim
A. 0090, posted 2022/01/21
There are at least two proofs; these on Cantor.
A. There is no cardinality
B. There is no uncountability
Dear Jim, you are at old pathes. Cantor on slash2 and cardinality got disproved!
A.
At lines:
P(0) = {Ø}
P(1) = {Ø; {1}}
P(2) = {Ø; {1}; {2}: {1,2}}
P(3) = {Ø; {1}; {2}; {3}; {1,2}; {1,3}; {2,3}; {1,2,3}
P(4) = {Ø; {1}; {2}; {3}; {4}; {1,2}; {1,3}; {1,4}; {2,3}; {2,4}; {3,4}; {1,2,3}; {1,2,4}; {1,3,4};
{2,3,4}; {1,2,3,4}}
and so on
One line (without double-naming / no surjection):
P(IN) = {Ø; {1}; {2}; {1,2}; {3}; {1,3}; {2,3}; {1,2,3}; {4}; {1,4}; {2,4}; {2,4}; {3,4}; {1,2,4};
{1,3,4}; {2,3,4}; {1,2,3,4}; …
Both could be counted by the naturals. q.e.d.
B.
Every sequence, basing on decimal places (0.1 = 1/10, 0.02 = 2/100, 0.008 = 8/1000, ...) is a rational number as long as every decimal place, whole the number depents on, is a rational number.
So Cantor never named any irrational number at his paper (slash2).
EI = (m, m, m, m, ...),
EII = (w, w, w, w, ...),
EIII = (m,w, m, w, ...),
Neither there (above) nor by the following:
E1 = (a1,1, a1,2, ... , a1,nu, ...),
E2 = (a2,1, a2,2, ... , a2,nu, ...),
...............
Emu = (a1,mu, a1,mu2, ... , amu,nu, ...),
q. e. d.
Because all fractions got proved to be in bijection to the naturals (Cantor slash1).
Extension:
Cantor did a circular argument by ruling the constructed line as to be unequal to those the construction was made from.
On one (examined) element of an infinite sequence (done the Cantor-way):
0.0, x, x, ... [consruction order is: take1 if 0 was original]
0.1, x, x, ...
[Not all could be under examination, because infinite.]
And look at the two lines — the same error; Cantor failed by mu = 2nu
00
01
10
11
Construct by two lines (mu = nu) and all are in at the four lines.
That´s how a constructed proof goes. Greetings, Peter
Peter Breuer, what kind of understanding do you have?
There is nothing which comes into your head. If a counter-example is shown whole the theory of the different statement is out. So, dear Peter, try to refute my proof! That´s how science goes. Not on beating around the bush, not on trying to blame the other by `not understanding´ and evaluating the opponent. You seemed to be a very little in doing any understanding. So what is your profession? It´s not on math!
Why not understanding the limitation of examination of infinite sequences? Why not understanding circular argument? Why not understanding mu = nu at Cantor´s paper? Why not understanding the relationship of mu and nu as mu = 2nu ?
Should I help you at a child´s niveau? [If there are two different values for each of the positions ...]
And Peter, we had it already! You wont understand. The trap is known!!!! The blame is yours.
Peter, had a look at your page. It seems you do it more by words than by constructed proofs on equations and so on.
It seems for me to talk to a `comic-mathematician´. What was really done by you? All is talking, none is doing, constructing, proving. So why do you think you are able to understand real math?
Dear Peter Kepp
- So what are you talking about proofs? Does anyone of you understand how to proof? Goedel (incomplete) only is valid for monotonic proof systems. Kuck showed how to do it the non-monotonic-way. And I did.
When you say monotonic, I assume you mean well ordered or by linear induction excluding self-reference. This would correlate with the quote I found from Godel posted earlier about "....incompleteness was due to inability fo self-reference.
I am working on a complete model of Godel's Constructible universe (as outlined earlier) to prove that all sets have Material Equivalence as an inner model of set theory and that all sets live within this self-referential model.
Has this been done before?
Regards Jim
Dear Peter Kepp,
- Dear Jim, you are at old pathes. Cantor on slash2 and cardinality got disproved!
Please provide some details or references if you have them. When solving P vs NP there are known barriers to the solution one being the Relativization barrier which means that the Diagonal Method can't be used to resolve P vs NP. I avoided DM in my proof using transfinite analysis of successor functions where I never have t o deal with the actual infinity.
This is quite ironic because computer scientists have told be that DM is the very definition of cardinality and It is very useful in various foundational CS complexity proofs, but the fact that it doesn't apply to P vs NP just means and I quote " it just doesn't do all we want it to!"
It never occurs to them that something might be wrong and when I asked this question they closed my question for being OFF topic LOL.
I also had a math post doc write me a comment that "There is nothing wrong with DM unless you try to use it!"
I have deduced the DM is OK as long as you do not use it inductively! I have given examples of the problems where subsets become larger than the super sets.
Regards Jim
Dear peter Kepp,
I know that Peter Breuer tends to pontificate even when out of his element. Beyond the absolutely stunning assertion that Suppes in a 1972 Dover reprint is wrong, Peter really seems to be over-stating his understanding of Control Systems in the paragraph to me below.
- PS. I used to lecture control engineering, so I guess you could talk to me about that. It's full of proofs ... but one leaves those out for the engineering courses, leading to engineers not knowing that mathematics is about proof. That does a lot of damage. Imagine you're a Kalman filter and modify your internal model of how things work in math until it matches what you see here!
In the last sentence, he completely dismisses the notion of a Kalman Filter with internal model parameter change.
Apparently, his control systems repertoire does not include adaptive control. There are two broad areas of linear control which are Robust and Adaptive control. Somehow I don't think he understands much of Robust control either.
Maybe he can clarify that point?
Regards Jim
Dear Peter Breuer,
>>Jim, that gothic-letter stuff is hilarious! How do you make it up?
The Fraktur font (you can access it in the MS Word equation editor under scripts) is one of the conventions used in Model theory to indicate a model. It is quite convenient to differentiate from a Con(T).
I also add the double arrow upper accent to denote a complete model.
Regards
Jim
A. 0099, posted 2022/01/22
Jim Moore
Often discussion is on misunderstanding.
`Non monotonic proof system´ means that there is a construct to disrupt the (maybe) never ending path of that `monotonic´ proof systems. Complete proven by Conrad Kuck (Non-monotonic learning automata / or at his four books on Intuitionistic Set Theory).
I thought at A. 0090 my (shortened) proof was to understand, was clear. But seemed not.
So, what I meant is to present the solution of the problem N vs P(N).
[N := all the naturals (0) 1, 2, 3, … as a so called set, but `reformed´ set => class]
[P(N) := power-set by definition; on finite sets like shown by lines, P(0) … P(4) and on the infinite set N shown by one line, P(N) = {0; {1}; {2}; {1,2}; … {1,2,3,4}; … ]
The proof, not shown direct as the one-to-one-correspondence (bijection or more done by a surjection) was on arranging the naturals to the lines (surjection) or to the single line (bijection).
Long version (full text) is available here at RG. Please see my papers on Cantor (Cantor, Proof, Cantor 3).
@Peter Breuer
When well done critical reading is out, arguing beneath the path begins.
My papers are open to read as well in English as in German!