Axioms of set theory from TGD viewpoint
Lubos made interesting comments related to the work of set theorist set theorist Paul Cohen. Cohen proved that axiom of choice (AC) cannot be proved nor disproved from Zermelo-Fraenkel axioms for set theory. He proved same also for continuum hypothesis (CH). Checking the definitions of various notions from Wikipedia led to an intellectual adventure of Sunday morning.
Axiom of choice from physics point of view
As a physicist I tend to share the skepticism of Lubos concerning AC. Stating it more formally: given a collection of sets indexed by some index set and not containing empty set, one can select from each set single element. What AC says that Cartesian product exists as a set. For finite sets there is no problem but already for a Cartesian power of reals one has a problem. Can one really choose from each factor R single element? Physicist might regard this kind of choice as too strong an idealization since there is always a finite accuracy involved. One can fix a given point of real axis only in finite accuracy since in the real world one must always perform a cutoff on the expansion of the real number in powers of 10, 2, or prime, or any natural number larger than one..
Physicist might prefer to weaken the axiom to a choice of a finite open ball around a point. This would lead from set theory to topology. One could return to set theory by postulating discrete topology in which each point is an open set. Physicist has however motivations to believe that physical world obeys less trivial topology and metric topology with the notion of distance defining the concept of nearness is natural for physicist. One can characterize the measurement resolution in terms of the radius of the ball replacing point. In p-adic context the situation is simpler than in real context since p-adic open balls are either disjoint or nested. Also the ordering, which is un-necessary luxury below measurement resolution, is lost in p-adic context. Since we want to minimize thinking, this is is a good reason for cognition to be p-adic;-).
TGD framework the notion of finite measurement resolution (cognitive resolution, numerical accuracy) plays a key role and has very powerful implications for the proposed mathematical framework of physics. For instance, the inclusions of hyper-finite factors relating closely to quantum groups often assigned to Planck scale represent finite measurement resolution at quantum level and at space-time and imbedding space level discretization become the counterpart for the finite measurement resolution at the level of geometry.
In p-adic context total disconnectedness implies difficulties with the notion of p-adic manifold and with integration and TGD suggests a notion of p-adic manifold based on coordinate charts mapping p-adic manifold to its real counterpart so one can induce real topology and well-orderedness of reals to p-adic context. Also here finite measurement resolution plays a key role.
Infinite primes and axioms of set theory
Cardinal numbers are defined as cardinalities of sets: two sets have the same cardinality if there is a bijection between them. Ordinals in turn can be seen as ordered sets of sets: set theoretically a given ordinal is the ordered sequence of ordinals defined by the cardinalities of the sets of smaller cardinality. The successor axiom of Peano arithmetics is essential here. Finite cardinals and ordinals can be identified but the notion of infinite ordinal is more refined notion than that of cardinal: this is obvious from the fact that infinite ordinals x and x+1 are not equivalent unlike corresponding cardinals.
Power set axiom states that the set of subsets of set exists. The cardinality of the power set is larger than the cardinality of set. In the case of natural numbers the cardinality of power set would be the cardinality of continuum and CH states that there are no cardinals between cardinality of natural numbers and cardinality of continuum postulated to that for the power set of natural numbers. CH can be extended to apply to any set and its power set. Cohen suggests that CH is not true although he only proves that both CH and not-CH are consistent with ZF axioms.
The notion of infinite prime provides number theoretic notion of infinity, which does not seem to reduce to the notions of infinite ordinals and cardinals. There is infinite hierarchy of infinite primes and infinite integers have a detailed number theoretic anatomy distinguishing them from cardinals and also "ordinary" ordinals. Infinity ceases to be some kind of limit and becomes something quite concrete expressed by an explicit formula having as a basic building brick the products of all primes at levels below a given level of hierarchy expressed as formal variables.
Physically this infinite hierarchy corresponds to a repeated second quantization of an arithmetic quantum field theory. The many-particle states of previous level become elementary particle states of the new level. Simple infinite primes correspond Fock states consisting of fermions and bosons labelled by prime valued "energy". There are also not so simple infinite primes analogous to bound states, which is rather interesting from the point of view of quantum field theories. Many-sheeted space-time could serve as a natural space-time correlate for this hierarchy in the sense that even galaxy sized objects could be seen in some aspects as elementary particles and proton could be also characterized as elementary fermion at particular level of the hierarchy.
The construction of infinite primes can be easily understood in terms of repeated second quantization.
- The Dirac vacuum is what one starts from and is at the first level of the hierarchy identified as the product X1=∏i pi of all ordinary primes. At the level n one has objects Xk, k=1,..n with Xk defined as the product of all primes belonging to levels m<k.
- At the lowest level simple infinite primes are obtained by dividing X with a square free integer m and adding to the result m: X→ X/m+/- m. The physical analogy is kicking of the fermions corresponding to prime factors of m from Dirac sea to positive energy states. After this one can add bosons (pm corresponds to m particles with "energy" p). One can also multiply X/m by an arbitrary integer n having no common factors with m: its decomposition to prime powers tells the numbers of bosons in corresponding modes. It is also possible to multiply m with integer r with any integer consisting of primes dividing m: also now the powers characterize the boson numbers in various modes. By construction nX/m and rm have no common prime factors so that the infinite integer is nmX/m+/- rm represents infinite prime. Depending on the sign factor one obtains two kinds of primes and X+/- 1 represent the simplest infinite primes and obviously differ by 2.
Side remark: One can add to/subtract from X+/- 1 any integer and obtain an integer always divisible by some prime (any prime factor in the prime decomposition of X). One obtains therefore infinitely long range of infinite integers containing no primes. Analogous theorem for finite integers states that the range defined by the numbers n! +m, 0≤ m<n+1 contains no primes.
How do infinite primes and integers relate to the axioms of set theory? I am not a set theorist and can make only some observations of a dilettante. In the following I restrict the consideration to the infinite primes at the lowest level so that they have only ordinary primes below them.
The notion of infinity emerges in two different manners in the construction of infinite primes.
- How the infinite size of a given infinite prime relates to the infinite cardinals and ordinals? To me the relationship is not obvious. The number theoretical anatomy of infinite primes and the hierarchy of Dirac seas Xn defined by products of primes at various levels seem to distinguish the notion of infinite natural number from infinite ordinals and cardinals. In particular, infinite prime at a given level (first level now) has infinite number predecessors at the same level. Hence bi-directional ordering seems to replace ordinary uni-directional ordering of natural numbers. Does this mean something genuinely new or can it be reduced to set theory? Certainly it is difficult to imagine a set having number of elements given by infinite prime: infinite prime is more like energy than a number of elements.
- What is the number of infinite primes/integers/rationals at given level of hierarchy and how it relates to the notions of infinite cardinal and ordinal? The number of infinite primes at given level depends on what one allows. Consider the construction of infinite primes at the first level. The numbers n and m could be chosen to be finite. In this case the number of simple infinite primes would be the number of finite rationals n/rm. This implies denumerably of simple infinite primes (cardinality is same as for natural numbers).
If one allows n or m to have infinite number of factors, the choices of m correspond to the set of subsets of finite primes and the resulting set is not denumerable and larger than the set of natural numbers (by CH it would be same as the cardinality of reals). One can argue that since the states with infinite fermion or boson number have infinite energy as physical states, they should be excluded so that the simple infinite primes would form a denumerable set. This result might generalize also to not so simple infinite primes, and might hold true at all levels of the hierarchy. Finite energy would correspond to denumerability. One can of course, ask "What about the entire Universe": should it correspond to the limit when the hierarchy level approaches infinite or to infinite particle numbers.
- Well-ordering theorem states that every set can be well-ordered. The existence of well-ordering of course depends on axiomatics. For real numbers well-ordering is natural and generalizes to Cartesian powers of reals in obvious manner. In p-adic number fields well-ordering is not possible if one stays with the restrictions of p-adic topology. As already mentioned, canonical identification map to reals can induce well-ordering of positive reals to p-adic context.
- To my opinion the status of AC is not a mere academic question even for physicist. Banach-Tarski paradox illustrates this. If AC is true, one can decompose sphere into parts and re-arrange them by using only rotations and translations, which are volume preserving operations, so that the volume of the resulting sphere is two times larger than the original! The operation involves a construction of non-measurable sets having no well-defined volume and an infinite number of choices. This result would suggest that set theory alone is not enough for the needs of physics. The notion of measurable set is needed.
- Well-ordering theorem is implied by AC. By definition set X is well-ordered by a strict total order if ever nonempty subset of X has a least element. This seems to be essential for the induction hypothesis generalized from naturals to infinite context: in induction the first step is to show the triviality of the theorem for n=0 and then show it for arbitrary n using induction hypothesis. For infinite positive integers at given level the least element does not however exist. Consider the infinite integers at the first level of the hierarchy. Infinite primes nmX/m -rm are always positive but there is no smallest infinite prime of this kind. In fact, also the integers X-n have no lower limit just as X+n do not have any upper limit. One can have a least element only by allowing a jump to a lower level, and this limit is of course 0. Unless one is ready to make this jump, well-ordering theorem is not true and therefore also AC cannot hold true. Maybe one could replace the uni-directional induction by bi-directional induction for infinite integers.