One can however argue that algebraic numbers do not allow finite representation as do rational numbers. A weaker condition is that the coding of information about algorithm producing the cognitively representable number contains a finite amount of information although it might take an infinite time to run the algorithm (say containing infinite loops). Furthermore, cognitive representations in TGD sense are also sensory representations allowing to represent algebraic numbers geometrically (2^{1/2}) as the diameter of unit square). Stern-Brocot tree associated with partial fractions indeed allows to identify rationals as finite paths connecting the root of S-B tree to the rational in question. Algebraic numbers can be identified as infinite periodic paths so that finite amount of information specifies the path. Transcendental numbers would correspond to infinite non-periodic paths. A very close analogy with chaos theory suggests itself.

See the article What does cognitive representability really mean?

For a summary of earlier postings see Latest progress in TGD.

## 20 comments:

Empirical science can postulate and study cognitive mathematics naturally as time-dependent observation events - e.g. physical notion of 'number' or most fundamental mathematical relation as relational spatiotemporal bivectors < and > (observation event of 'more' and 'less'; number and antinumber). Quadratic area <> ("planar dot product") thus suggests itself as the most basic element of cognitive mathematics which can be also interpreted as a planar projection of geometric observation event that self-contains the three point minimum of area as well as "observer point".

Notion of time-independent mathematics - especially in the sense of metalanguage of timespace - is by it's nature non-empirical, metaphysical and theological. By definition there is no empirical way to show that e.g. non-demonstrable real numbers can perform basic arithmetic operations and form a field, such claim must be axiomatically postulated as pure matter of faith, as belief that there is divine mind beyond capabilities of human cognition where real number arithmetics can and do actualize. We should, however, make here very careful distinction between pure mathematics and applied mathematics of approximate measurements, and a) theories containing to applied mathematics that don't presuppose platonistic theology and use e.g. notion of infinite set only heuristically and instrumentally, and b) theories that give ontological interpretations to divine aspects of time-independent theories of mathematics.

This fundamental distinction does not aim to exclude theological notions of mathematics from more general philosophical or more narrow discussion of empirical science, but to clarify basic concepts and categories and make e.g. discussions about nature of causality and causal power of mathematics more fruitful and communicative.

[SS] Empirical science can postulate and study cognitive mathematics naturally as time-dependent observation events - e.g. physical notion of 'number' or most fundamental mathematical relation as relational spatiotemporal bivectors < and > (observation event of 'more' and 'less'; number and antinumber). Quadratic area <> ("planar dot product") thus suggests itself as the most basic element of cognitive mathematics which can be also interpreted as a planar projection of geometric observation event that self-contains the three point minimum of area as well as "observer point".

[MP] Cathegory theory based thinking based on objects and arrows is very powerful bureaucratic tool in modern mathematics but to my narrow understanding the notion of number, which is fundamental for all mathematics and as it seems also for physics, does not reduce to category theory. One must have also fundamental objects. I have the feeling that < and > alone are two limited notions to serve as the basis of mathematics- in particular, if they already involve the notion of spatiotemporal. I do not like fundamentalism in either mathematics or physics.

[SS] Notion of time-independent mathematics - especially in the sense of metalanguage of timespace - is by it's nature non-empirical, metaphysical and theological. By definition there is no empirical way to show that e.g. non-demonstrable real numbers can perform basic arithmetic operations and form a field, such claim must be axiomatically postulated as pure matter of faith, as belief that there is divine mind beyond capabilities of human cognition where real number arithmetics can and do actualize.

[MP] When speaking of time-dependence of mathematics I want to keep in mind the fact that there are a two times. Experienced time: with respect to this time mathematics evolves, new axioms emerge mathematical consciousness evolves. A concrete realization for this vision would be continula extension of algebraic extension of rationals defining the adele behind cognition. More and more complex structure become representable as the number of space-time points in cognitive representations increases and quantum states in WCW become algebraically more complex.

As far as geometric time is involved, mathematics is time-independent, outside space-time and time. Universe becomes quantum jump by quantum jump more conscious about it.

[SS] We should, however, make here very careful distinction between pure mathematics and applied mathematics of approximate measurements, and a) theories containing to applied mathematics that don't presuppose platonistic theology and use e.g. notion of infinite set only heuristically and instrumentally, and b) theories that give ontological interpretations to divine aspects of time-independent theories of mathematics.

[MP] Finite and discrete mathematics is what cognitive representations necessarily use. The cognitizable but cognitively non-representable part of mathematics is also there. This is the Patonia and one can call it divine. The reason why I believe this is that the option in which one has only the discrete mathematics does not work. This would reduce mathematics to theorem deduction from given axioms by computers. I think that that the successes in this respect have been rather meager - combinatorial explosion.

Generally, it would be prudent to stay open to possibility that the mathematical presuppositions of TGD do not necessarily imply tautological reduction of cognition to the presupposed axiomatic construction of Platonic ontology, but that we could interprete the presupposed theory of mathematics also as heuristic tool, one possibility among many. Spinoza's theology is what also adelic approach seems to lead to, and let's keep in mind that Spinoza's Absolut is undivided whole which has all attributes - TGD and all others. In Western tradition we do injustice to serious theology if we don't take also Spinoza's contribution seriously.

Back to physics and time-dependent mathematics, in this phenomenal world we share and experience, the most fundamental empirical mathematical relations 'more' and 'less' are not discrete but prediscrete. There is no need for any theory of discrete measurement to know that a cup of coffee is getting closer to mouth, after a sip there is less coffee in the mug and desire for coffee is getting satisfied. Perhaps the most valuable contribution of the theory of Surreal Numbers is to explicate that the basic notion of 'equality' - which notion of discrete mathematical objects to my understanding need to presuppose - is not axiomatic but naturally derived as 'not more, not less' -relation. And perhaps it is also divine "coincidence" that very simple derivation of a time-dependent discrete and chirally symmetric number theory has quadratic character also visually and more than enough room for even infinitely rich internal analytical (vector etc.) structure of number: <<> <> <>> for 'number', 'identity' and 'antinumber'.

The simple beauty of this approach is incomparable to e.g. the awful mess of Peano Axioms, and as there is no presupposition of closure by existential quantification at foundational level, it remains even more open and creative than the subsumed theory of Surreal Numbers closed by artificial postulation of 'empty set' and theory of games defined only as win-lose games. If we accept evolution as common ground ethical axiom, and mathematical etc. scientific (meta)narratives as potentially adaptive strategies for scaling up cooperation of human populations, it is time that our time dependent mathematical etc. scientific languages start playing win-win games better. More win with less axioms.

As I have said many times, the idea about axiomatic construction of Platonic ontology is impossible. Mathematical consciousness evolves all the subjective time and new discoveries representable as axsiom emerge. This process is accumulative. For instance, the notion of number field is here to stay although the formulations of this notion evolve.

About Peano Axioms in contrast to surreals I have no strong views. I am physicist and physics serves for me as a tool for becoming aware of possible new mathematical structures. I cannot think of starting from mathematics as given and then proceeding to do physics. It would lead nowhere.

Foundations of mathematics remain rather remote to what I am doing. Actually I believe that physics could give a lot for people working with foundations of mathematics - infinite primes provide a good example in TGD framework. I do not have time and resources to have battles about detailed realizations of real numbers and do not want to take strong attitudes about their existence or non-existence. I trust on Platonic intuitions and accept that cognizable is not always cognitively representable.

The view that physics can exist outside language and communication leads nowhere - or at best, to solipsism. And I do contest the view that any process is only accumulative piling up of axioms on axioms. We don't play cooperative games by just making up new rules over old rules, that kind of game sounds like bureaucratic nightmare of Kafka. The very hard work of finding new meaningful generalizations and simplifications by questioning and deconstructing axiomatic presuppositions of prevailing views is at the heart of paradigmatic revolutions of evolution. Quantum jumps, if you like, but in this context quantum drops towards the center of gravity would be much more accurate description.

There is no battle here, in terms of Spinozan theology. The new purely relational foundational theory of mathematics that starts from most natural codependent relation < and > is all about doing more with less, as Buckminster Fuller describes the promise of mathematics and science as adaptive evolutionary strategy. It is not a hostile rival of constructions based on Peano axioms or Conway games, it is new beautiful generalization that subsumes both, in which sense it is accumulative, but more importantly it can open up new mathematical and physical territory to explore and map. A more physical - time dependent - foundation that can be as you say, a "tool for becoming aware of possible new mathematical structures." Revealing of the liminal space of "more" that Badiou seeks in the following quote, with most natural formal language to accompany and explore that territory:

BADIOU: "In my view, this contrast is of the greatest philosophical importance. The prevailing idea is that what happens 'at the limit' is more complex, and also more obscure, than that which is in play in a succession, or in a simple 'one more step'. For a long time philosophical speculation has fostered a sacralisation of the limit. What I have called elsewhere [Manifeste pour la philosophie, 1989] the 'suture' of philosophy to the poem rests largely upon this sacralisation… . Every true test for thought originates in the localisable necessity of an additional step, of an unbroachable beginning, which is neither fused through the infinite replenishment of that which precedes it, nor identical to its dissemination … The empty space of the successor is more redoubtable, it is truly profound. There is nothing more to think in the limit than in that which precedes it [e.g. ω has no last element]. But in the successor [e.g. ω + 1] there is a crossing. The audacity of thought is not to repeat 'to the limit' that which is already entirely retained within the situation which the limit limits; the audacity of thought consists in crossing a space where nothing is given." Quote is from very illuminative review of Badiou's 'Number and Numbers': http://ndpr.nd.edu/news/number-and-numbers/

The language of French philosophy a là Badiou is horror to me, but perhaps with benevolent interpretation some meaning can be deciphered. In response to Badiou the event of revealing and explicating the liminal space (between discreet number and completed infinite process) as the most foundational relation 'more', '<', which already contains successor function in itself, is the unification of analytical and poetic language and comprehension that Badiou is seeking. Common communicative language to translate the monk latin of the authoritarian theology of ivory tower logicism into. The Lutheran revolution was about translating Bible into common language, and the meaning and promise of this revolution is both more and less.

It doesn't matter whether one takes platonist, constructivist or fictionalist view on metaphysics, <> has been nailed on the church door creating an opening, invitation and challenge for all who are willing to think and explore What Else?

Have been reading a history of "cold fusion", which actually began for century ago. I have had nuclear tables at hand to interpret the findings which were strange at that time. Some of them are still strange. This is so enjoyable and concrete. And it relates to the key assumptions behind our world view testable in lab. This is so different from sterile philosophical verbosity about whether this or that definition of real number is the correct one. For me philosophy and mathematics starts to breath only when there is direct connection with what we observe.

That is the question here. How do we observe number, what kind of observation event is number. It is curious how frightening that question can be. What is afraid of it?

I accept as a fact that cognition does not have only the representable part. It has also the part, ideas, which can be only conceptualized. The attemps to reduce everything to rationals or something else would throw this part of cognition away. In physics numbers are not observables: they represent the values of observables. Both physics and mathematics. Not only numbers.

I am not a mathematician and should perhaps avoid saying anything about fundamentals of mathematics. On the other, hand, I have nothing to lose and since physics inspired thoughts about basic mathematics kept me awake the whole night, it is better to recall why I could not sleep and hope that this makes next night more pleasant.

a) From the point of quantum physics set theory and the notion of number based on set theory look somewhat artificial constructs. Nonempty set is a natural concept but empty set and set having empty set as element used as basic building brick in the construction of natural numbers looks weird to me.

b) From TGD point of view it would seem that number theory plus some basic pieces of quantum theory might be more fundamental than set theory. Could set theory emerge as a classical correlate for quantum number theory and could quantal set theory make sense?

c) Could one define or at least represent the notion of number using the notions of quantum physics? A natural starting point is hierarchy of extensions of rationals defining hierarchy of adeles. Could one obtain rationals and their extensions from simplest possible quantum theory in which one just constructs many particle states by adding or removing particles using creation and annihilation operators?

Rationals and their extensions are fundamental in TGD. Can one have quantal construction for them?

a) One must construct rationals first. Suppose one starts from the notion of finite prime as something God-given. At the first step one constructs infinite primes as analogs for many-particle states in super-symmetric arithmetic quantum field theory. Ordinary primes label states of fermions and bosons. Infinite primes as the analogs of free many-particle states obtained correspond to rationals in a natural manner.

b) One obtains also analogs of bound states which are mappable to irreducible polynomials, whose roots define algebraic numbers. This would give hierarchy of algebraic extensions of rationals. At higher levels of the hierarchy one obtains also analogs of prime polynomials with number of variables larger than 1. One might say that algebraic geometry has quantal representation. This might be very relevant for the physical representability of basic mathematical structures.

How to obtain the analogs higher-D spaces?

a) One can obtain n-dimensional spaces (in algebraic sense) with integer valued coordinates from n-D extensions of rationals. Now the n-tuples defining numbers of extension and differing by permutations are not equivalent so that one obtains n-D space rather than n-D space divided by permutation group S_n. This is enough at the level of cognitive representations and could explain why we are able to imagine spaces of arbitrary dimension although we cannot represent them cognitively.

b) One obtains also Galois group and orbits of Galois group as G(A), A a set. One obtains also discrete coset spaces G/H and alike. These do not have any direct analog in the set theory. The hierarchy of Galois groups would bring in discrete group theory automatically. The basic machinery of quantum theory emerges elegantly from number theoretic vision.

c) In octonionic approach to quantum TGD one obtains also hierarchy of extensions of rationals since space-time surface correspond zero loci for RE or IM for octonionic polynomials obtained by algebraic continuation from real polynomials with coefficients in extension of rationals.

How to obtain quantum set theory?

Suppose that number theory- quantum theory connection really works and we have really obtained number theory. What about set theory? Or perhaps its quantum counterpart having ordinary set theory as a classical correlate?

a) A purely quantal input to the notion of set would be replacement of points delocalized states in the set. A generic single particle quantum state as analog of element of set would not be localized to a single element of set. The condition that the state has finite norm implies in the case of continuous set like reals that one cannot have completely localized states. This would give quantal limitation to the axiom of choice. One can have any discrete basis of state functions in the set but one cannot pick up just one point since this state would have infinite norm.

The idea about allowing only say rationals is not needed since there is infinite number of different choices of basis. Finite measurement resolution is however unvoidable. An alternative option is restriction of the domains of wave functions to a discrete set of points. This set can be chosen in very many manners and points with coordinates in extension of rationals are very natural and would define cognitive representation.

b) One can construct also the analogs of subsets as many-particle states. The basic operation would be addition/removal of a particle from quantum state represented by the action of creation/annihilation operator.

Bosonic states would be invariant under permutations of single particle states just like set is the equivalence class for a collection of elements (a_1,...,a_n) such that any two permutations are equivalent. Quantum set theory would however bring in something new: the possibility of fermionic statistics. Permutation would change the state by phase factor -1. One would have fermionic and bosonic sets. Even braid statistics are possible. The phase factor in permutation could be complex. Even non-commutative statistics can be considered.

Many particle states formed from particles, which are not identical are also possible and now the different particle types can be ordered. On obtains n-ples decomposing to ordered K-ple of n_k-ples, which are consist of identical particles and are quantum sets. One could talk about K-sets as a generalization of set as analogs of classical sets with K-colored elements. Group theory would enter into the picture via permutation groups and braid groups would bring in braid statistics. Braids strands would have K colors.

How to obtain classical set theory?

a) Many-particle states represented algebraically are detected in lab as sets: this is quantum classical correspondence. This remains to me one of the really mysterious looking aspects in the interpretation of quantum field theory. For some reason it is usually not mentioned at all in popularizations. The reason is probably that popularization deals typically with wave mechanics but not quantum field theory unless it is about Higgs mechanism, which is the weakest part of quantum field theory!

b) From the point of quantum theory empty set would correspond to vacuum. It is not observable as such. Could the situation change in the presence of second state representing the environment? Could the fundamental sets be always *non-empty * and correspond to states with non-vanishing particle number. Natural numbers would correspond to eigenvalues of an observable telling the cardinality of set. Could representable sets be like natural numbers?

c) Usually integers are identified as pairs of natural numbers (m,n) such that integer corresponds to m-n. Could the set theoretic analog of integer be a pair (A,B) of sets such that A is subset of B or vice versa? Note that this does not allow pairs with disjoint members. (A,A) would correspond to empty set. This would give rise to sets (A,B) and their "antisets" (B,A) as analogs of positive and negative integers.

One can argue that antisets are not physically realizable. Sets and antisets would have as analogs two quantizations in which the roles of oscillator operators and their hermitian conjugates are changed. The operators annihilating the ground state are called annilation operators. Only either of these realization is possible but not both simultaneously.

In ZEO one can ask whether these two options correspond to positive and negative energy parts of zero energy states or to the states with state function reduction at either boundary of CD identified as correlates for conscious entities with opposite arrows of geometric time (generalized Zeno effect).

d) The cardinality of set, the number of elements in the set, could correspond to eigenvalue of observable measuring particle number. Many-particle states consisting of bosons or fermions would be analogs for sets since the ordering does not matter. Also braid statistics would be possible.

What about cardinality as a p-adic integer? In p-adic context one can assign to integer m, integer -m as m(p-1)(1+p+p^2+...). This is infinite as real integer but finite as p-adic integer. Could one say that the antiset of m-element as analog of negative integer has cardinality -m= m(p-1)(1+p+p^2+..). This number does not have cognitive representation since it is not finite as real number but is cognizable.

Could one argue that negative numbers are cognizable but not cognitively representable as cardinality of set? This representation must be distinguished from cognitive representations as a point of imbedding space with coordinates in extension of rationals. Could one say that antisets and empty set as its own antiset can be cognized but cannot be cognitively represented?

In my line of thought the notion of quantum set is primary to existential quantification and discreet number theory. Could it be possible to think of <> as formal representation of quantum set? The main difference is that the classical set has closed boundary, but <> is open and unbounded, more and more and less and less continue beyond horizon of observability, ie. contains "successor function" already in the relational operators on both sides. But as this notion set has no closed boundary and unintuitive notion of infinite set with closed boundaries is not assumed, it avoids naturally the Russel's Paradox and much of Gödelian problematics created by postulation of finitely reductionistic completeness axiom. On the other hand, "well ordered" ordinality is inherent property of this suggestion for quantum set and thus contains in itself the basic structure on which quantified number theories etc can be formed: the basic number, identity and antinumber scheme <<> <> <>> can be given many interpretations, including integers <<1> <> <-1>> and state vector < >.

Also, the suggested formulation of quantum set <> can be interpreted as some sort of analogue to Feynman diagram. Defining the area - or more generally content - of <> as constant (cf. basic idea of cardinality) we can imagine the <> as mean of stretching it into vertical and horizontal lines, which as time-dependent continuous movement describes also wave form.

As side note, whether relevant or not, observation of the form <+> has mysterious beauty to me, as the planar shadow projections of tetrahedron looked simultaneously from both orthogonal sides. Isosceles triangle does not exist on number theoretical plane with rational parametrization, but becomes beautifully - rationally - observable inside cube.

Annihilation or merger can be seen as quantum set version of "no cloning theorem", the basis of arithmetic functions: <><> (which I have named palindromically 'testset' or in Finnish 'itsetesti" ;)) merges into <>. Thus <> + <>> is not more not less than <>>; and <<> + <>> merges into <>.

[SS] In my line of thought the notion of quantum set is primary to existential quantification and discreet number theory.

1. More about what quantum set could mean.

"Quantal existential quantification"! What I have is only an intuitive proposal what might be the basic concepts behind quantum set theory. Element of set would be replaced with wave function in the set - single particle state as physicist would say. Subset with many particle state in quantum sense.

Nasty mathematician would immediately ask whether I can really start from Hilbert space of state functions and deduce from this the underlying set. The elements of set itself should emerge from this as analogs of completely localized single particle states labelled by points of set. In the case of finite-dimensional Hilbert space this seems trivial. The number of points in the set would be equal to the dimension of Hilbert space. In the case of infinite-D Hilbert space the set would have infinite number of points.

Here one has two views about infinite set. One has both separable (infinite-D in discrete sense: particle in box with discrete momentum spectrum) and non-separable (infinite-D in real sense: free particle with continuous momentum spectrum) Hilbert spaces. In the latter case the completely localized single particle states would be represented by delta functions divided by infinite normalization factors. They are routinely used in Dirac's bra-ket (<>) formalism but problems emerge in quantum field theory.

A possible solution is that one weakens axiom of choice and accepts that only discrete points set (possibly finite) are cognitively representable and one has wave functions localized to discrete set of points. A stronger assumption is that these points have coordinates in extension of rationals so that one obtains number theoretical universality and adeles.

This is TGD view and conforms also with the identification of hyper-finite factors of type II_1 as basic algebraic objects in TGD based quantum theory as opposed to wave mechanics (type I) and quantum field theory (type III). They are infinite-D but allow excellent approximation as finite-D objects.

This picture could relate to the notion of non-commutative geometry, where set emerges as spectrum of algebra: if I recall correctly, the points of spectrum label the ideals of the integer elements of algebra.

2. About the emergence of number theory from the notion of Hilbert space.

The notions of prime and divisibility and actually basic arithmetics would emerge already from the tensor product and direct sum for Hilbert spaces. Hilbert spaces with prime dimension do not decompose to tensor products of lower-dimensional Hilbert spaces. One can even perform a formal generalization of the dimension of Hilbert space so that it becomes rationals and even algebraic number.

For some years ago I indeed played with this thought but at that time I did not have in mind reduction of number theory to that of Hilbert space. If this really makes sense, numbers could be replaced by Hilbert spaces with product and sum identified as tensor product and direct sum!

p-Adic arithmetics is however required. In p-adic arithmetics negative finite integer dimension n corresponds to -n= n*(p-1)*(1+p+p^2+....) infinite and without number theoretic anatomy as real integer so that Hilbert space of negative dimension corresponds to p-adic infinite-dimension Hilberg space defined by replcing * by tensor product, ^n by tensor power and + by direct sum in the above formula. This infinite-dimensional Hilbert space has precise number theoretic anatomy unlike infinite-D Hilbert space in real sense.

Also division of Hilber spaces becomes possible in p-adic arithmetics. m/n can be expressed as power series of p and analogous representation applies. I do not remember whether algebraic numbers can define dimension of Hilbert space in this sense. What a root of a polynomial of Hilber space could mean!?

[SS] Could it be possible to think of <> as formal representation of quantum set? Also, the suggested formulation of quantum set <> can be interpreted as some sort of analogue to Feynman diagram. Defining the area - or more generally content - of <> as constant (cf. basic idea of cardinality) we can imagine the <> as mean of stretching it into vertical and horizontal lines, which as time-dependent continuous movement describes also wave form.

[MP] I do not know how you define < and >. Does <> refer to open interval or does it have something to do with Dirac's scalar product of states , which is essentially Hilbert space scalar product?

[SS] The main difference is that the classical set has closed boundary, but <> is open and unbounded, more and more and less and less continue beyond horizon of observability, ie. contains "successor function" already in the relational operators on both sides.

[MP] To speak about boundary and closed/open sets, one must specify topology. In p-adic topologies open sets are simultaneously also closed and there are no boundaries: this makes them and - more generally Stone spaces - ideal for realizing Boolean algebra set theoretically.

In real topology open sets are not closed and complement of open set is closed. Hence Boolean algebra cannot be realized as open sets.

*If one replaces open sets with closures of open sets (closure of open ball includes also its boundary) and closed complements of open sets, the analog of Boolean algebra would consist of closed sets. One obtains a strange situation reminding of Russell's paradox but in geometric form.

Closed ball and the closure of its open complement - stament and its negation - share the surface of the ball, which is sphere. Statement and its negation would be simultaneously true at this sphere.

*If one replaces the closed complement of open set with its open interior, one has only open sets. Now the sphere would represent statement about which one cannot say whether it is true or false. This would look like Goedelian sentence but represented geometrically.

Already familiar conclusion: p-Adic topology is natural for geometric correlates of cognition, in particular Boolean cognition. Real topology for geometric correlates of sensory experience.

In algebraic geometry closed sets correspond to algebraic surfaces of various dimensions. Open sets are their complements and of same dimension as the imbedding space. Also now one encounters asymmetry. Could one say that algebraic surfaces characterize "representable" (="geometrically provable"?) statements as elements of Boolean algebra and their complements the non-representable ones? 4-D space-time (as possibly associative/co-associative ) algebraic variety in 8-D octonionic space would be example of representable statement. Finite unions and intersections of algebraic surfaces would form the set of representable statements. This new-to-me notion of representability is somehow analogous to provability or demonstrability.

[SS] But as this notion set has no closed boundary and unintuitive notion of infinite set with closed boundaries is not assumed, it avoids naturally the Russel's Paradox and much of Gödelian problematics created by postulation of finitely reductionistic completeness axiom.

[MP] Reals form infinite set and have no boundaries. Line of finite length could be seen as completion of reals has bounary as itse ends. Same can be said about natural numbers with discrete topology (every point is open and closed set simultaneously). The above argument applies to this.

Goedelian problematics is encountered already for arithmetics, that is natural numbers although naturals have no boundary in the discrete topology: this topology does not however allow ordering of natural numbers. In the induced real topology one can orderthem and can speak of boundaries of subsets of naturals. The ordering of natural numbers reflects the ordering of reals: this shows how difficult it is to think about discrete without implicitly bringing in the continuum.

For p-adic integers the induced topology is p-adic and Goedelian problematics should be absent in p-adic Boolean logic since set and its complement are both open and closed. If this view is correct, p-adic integers could replace naturals in the axiomatics of arithmetics. The new element would be that most p-adic integers are of infinite size in real sense. One has natural division of them to cognitively representable ones finite also in real sense and non-representable ones infinite in real sense.

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[SS] On the other hand, "well ordered" ordinality is inherent property of this suggestion for quantum set and thus contains in itself the basic structure on which quantified number theories etc can be formed: the basic number, identity and antinumber scheme <<> <> <>> can be given many interpretations, including integers <<1> <> <-1>> and state vector < >.

[MP] I would be extremely cautious with "well ordered", which according to the above argument derives from topology of naturals induced from real topology. In p-adic topology well-ordering is lost but one at least gets rid set from the set theoretic representation of Goedel's problematics.

What does <> say about itself? That it consists of relational operators and forms an area between something less and something more than itself. Further definitions is open for questioning and thinking.

Hilberts spaces are defined by the notion of completed metric space, a very complicated definition of 'cauchy space' with problematic presuppositions and which primarily concerns lines. We can, however, naturally compare <> with the notion of Dedekind cut: <> as prequantified and squared "cut".

Attempts to define <> based on lines and quantification would be trivial and pointless, as <> is invitation to think differently in new open spacetime for creative intelligence. Hence, standard notions of 'norm', 'topology' etc. don't yet apply - but can of course be derived and defined from some quantified theory based on <>.

Instead of metaphysics of existential quantification ("there is an accounting unit") the phenomenological rebeginning by '<' states that "more becomes".

Here the hope is that, as Buckminster Fuller recommends, we could be able to do more with less. Suggestion to consider <> as quantum set asks, could it be possible to form more beautiful, more communicative formulation with less arbitrary presuppositions, of e.g. quantum theory starting from the notion of prequantified, prediscrete area between less and more? How could we proceed to do so, if we wanted the give a try? What are the actual physical characteristics of quantum theory that do not necessarily depend from quantified number theory, if there are such? What are the most essential phenomenal more and less relations in quantum physics?

“their principles state that information should be localized in space and time, that systems should be able to encode information about each other, and that every process should in principle be reversible, so that information is conserved.”

https://www.wired.com/story/physicists-want-to-rebuild-quantum-theory-from-scratch/

Bingo. It seems that the "one sentence" could be that at large, quantum theory is just a mirror that reflects back the theory of measuring being used. And a theory of measurement is a theory of mathematics - as Wittgenstein said: "Mathematics as such is always a measure, not the thing measured."

As probability theories basically just measure and describe what is more and/or less likely, we go back to the axioms of relational operators <>, which as codependent reversible relations encode each other. Localized area <> between something less and something more. And which already in their own form can be seen as consisting of spatiotemporal (base) vectors.

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