Maybe they are chasing me, see here.
I would like to highlight a cute question in a recent paper,
Recall that W. Ackermann verified what in modern terms we call the bi-interpretability of and , where the latter is (first-order) Peano arithmetic, and the former is finite set theory, the result of replacing in the axiom of infinity with its negation (and with foundation formulated as the schema of -induction). The reference is
Die Widerspruchsfreiheit der allgemeinen Mengenlehre.
Math. Ann. 114 (1937), no. 1, 305–315.
I have written about this before. Briefly, one exhibits (definable) translations between the collection of hereditarily finite sets and and verifies that the translation extends to a definable translation of the relations, functions and constants of the language of each structure in a way that verifies that holds in the translation of and verifies that holds in the translation of . Recall that consists of those sets whose transitive closure is finite, that is, is finite, and all its elements are finite, and all the elements of its elements are finite, and so on. Using foundation, one easily verifies that , that is, it is the collection of sets resulting from iterating the power-set operation (any finite number of times) starting from the empty set.
In the direction relevant here, one defines a map by
One easily verifies, using induction on the set-theoretic rank of the sets involved, that this recursive definition makes sense and is injective (and, indeed, bijective).
Of course this argument uses foundation. In the D’Agostino-Policriti-Omodeo-Tomescu paper they consider instead the theory resulting from replacing foundation with the anti-foundation axiom, and proceed to describe a suitable replacement for that injects (codes) into the real numbers. They do quite a bit more in the paper but, for the coding itself, I highly recommend the nice review by Randall Holmes in MathSciNet, linked to above.
The anti-foundation axiom became known thanks to the work of Peter Aczel, and it is his formulation that I recall below, although it was originally introduced in work of Forti and Honsell from 1983, where they call it . Aczel’s presentation appears in the excellent book
Non-well-founded sets. With a foreword by Jon Barwise.
CSLI Lecture Notes, 14. Stanford University, Center for the Study of Language and Information, Stanford, CA, 1988. xx+137 pp.
The original paper is
Given a binary relation , its field is the union of its domain and codomain. A decoration of is a function satisfying
for all . When is and the sets in question are well-founded, the only decoration is the identity. Similarly, any well-founded relation admits a unique decoration. Define as the statement that any binary (whether well-founded or not) admits a unique decoration.
In with foundation replaced with one can prove the existence of many non-well-founded sets. One of the appealing aspects of is that the resulting univere is actually quite structured: Other anti-foundation axioms allow the existence of infinitely many Quine atoms, sets such that , for instance. Under , there is exactly one such , usually called . The axiom is sometimes described as saying that it provides solutions to many “equations” among sets. For instance, consider the system of equations and . Under the system has as its unique solution. Note that assuming , is in , as are many other non-well-founded sets.
Here is the open question from the D’Agostino-Policriti-Omodeo-Tomescu paper: Work in set theory with instead of foundation. Is there a unique, injective, function satisfying
for all ?
Note that there is a unique such on the well-founded hereditarily finite sets, and it is in fact injective. In general, existence, uniqueness and injectivity of appear to be open. The claim that there is such a function is a statement about solutions of certain equations on the reals, and the claim that is unique requires moreover uniqueness of such solutions. The expectation is that is transcendental for all non-well-founded hereditarily finite but, even assuming this, the injectivity of seems to require additional work.
For example, consider . The function must satisfy
and, indeed is the unique solution of the equation .
I would be curious to hear of any progress regarding this problem.
I had recently learned of the problem through another paper by Zamboni and a collaborator,
It is a nice result and I think it may be enjoyable to work through the argument here. Everything that follows is either straightforward, standard, or comes from these papers.
1. The problem
To make the post reasonably self-contained, I begin by recalling some conventions, not all of which we need here.
By an alphabet we simply mean a set , whose elements we refer to as letters. A word is a sequence of letters from where is a (not necessarily non-empty, not necessarily proper) initial segment of . If we denote for all , it is customary to write the word simply as
and we will follow the convention. The empty word is typically denoted by or . By we denote the collection of all finite words from , and By we denote the length of the word (that is, the size of the domain of the corresponding function).
We define concatenation of words in the obvious way, and denote by the word resulting from concatenating the words and , where . This operation is associative, and we extend it as well to infinite concatenations.
If a word can be written as the concatenation of words
we refer to the right-hand side as a factorization of . If and is non-empty, we say that is a prefix of . Similarly, if is non-empty, it is a suffix of . By for we denote the word resulting form concatenating copies of . Similarly, is the result of concatenating infinitely many copies.
By a coloring we mean here a function where is a finite set of “colors”.
Apparently the problem I want to discuss was first considered by T.C. Brown around 2006 and, independently, by Zamboni around 2010. It is a question about monochromatic factorizations of infinite words. To motivate it, let me begin with a cute observation.
Fact. Suppose is an infinite word, and is a coloring. There is then a factorization
where all the have the same color.
Proof. The proof is a straightforward application of Ramsey’s theorem: Assign to the coloring of the set of -sized subsets of given by whenever . Ramsey’s theorem ensures that there is an infinite set such that all with have the same color. We can then take and for all .
In the fact above, the word was arbitrary, and we obtained a monochromatic factorization of a suffix of . However, without additional assumptions, it is not possible to improve this to a monochromatic factorization of itself. For example, consider the word and the coloring
If nothing else, it follows that if is an infinite word that admits a monochromatic factorization for any coloring, then the first letter of must appear infinitely often. The same idea shows that each letter in must appear infinitely often.
Actually, significantly more should be true. For example, consider the word
and the coloring
This example shows that in fact any such must admit a prefixal factorization, a factorization
where each is a prefix of .
Problem. Characterize those infinite words with the property P that given any coloring, there is a monochromatic factorization of .
The above shows that any word with property P admits a prefixal factorization. But it is easy to see that this is not enough. For a simple example, consider
Consider the coloring where if is not a prefix of , 1, and otherwise. If
is a monochromatic factorization of , then so and each must be a prefix of of length at least . But it is easy to see that admits no such factorization: For any , consider the first appearance in of and note that none of the first zeros can be the beginning of an , so for some we must have and since , in fact , but this string only appears once in , so actually . Since was arbitrary, we are done.
Here is a more interesting example: The Thue-Morse word
was defined by Axel Thue in 1906 and became known through the work of Marston Morse in the 1920s. It is defined as the limit (in the natural sense) of the sequence of finite words given by and where, for , is the result of replacing each letter in with .
This word admits a prefixal factorization, namely
To see this, note that the sequence of letters of can be defined recursively by , and . To see this, note in turn that the sequence given by this recursive definition actually satisfies that is the parity of the number of s in the binary expansion of from which the recursive description above as the limit of the should be clear. The relevance of this observation is that no three consecutive letters in can be the same (since for all ), and from this it is clear that can be factored using only the words , , and .
But it is not so straightforward as in the previous example to check whether admits a factorization into prefixes of length larger than .
Instead, I recall a basic property of and use it to exhibit an explicit coloring for which admits no monochromatic factorization.
I have just posted on my papers page a preprint of a review of
Reflections—the magic, music and mathematics of Raymond Smullyan.
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. x+213 pp.
ISBN: 978-981-4644-58-7; 978-981-4663-19-9
that I have submitted to Mathematical Reviews.
Announcement of the plan to revise the Mathematics Subject Classification
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Version of 2016.07.25
Help us identify all mathematicians in this picture (click on it for a larger version). Please post comments here, on G+, or email me or Paul Larson.
The picture will appear in the book of proceedings of the Woodin conference, http://logic.harvard.edu/woodin_meeting.html. (Thanks to David Schrittesser for allowing us to use it.)
On Thursday March 10, Peter Cholak gave a beautiful talk at the Logic Seminar at the University of Michigan, on Rado’s path decomposition theorem and its effective content. I want to review here some of the results covered by Peter. Slides for another version of the talk can be found in Peter’s page. This is joint work by Peter, Greg Igusa, Ludovic Patey, and Mariya Soskova.
As usual, given a set , let denote the collection of 2-sized subsets of . If is a positive integer, an –coloring of (or simply, a coloring, if is understood) is a map (where we use ordinal notation, so ). We can think of this as a coloring using colors of the edges of the complete graph whose underlying set of vertices is . When , we have an even simpler interpretation: A 2-coloring is just a graph on .
Given an -coloring of , a path of color is a sequence of distinct elements of (which may be finite or infinite, or even empty, or of length 1) such that for all , if is defined, then . Note that this is a much weaker requirement than asking that be monochromatic (which would mean that for all ). Also, in what follows is either a finite number or . However, we do not require that the elements in the sequence be listed in their natural order: We may very well have that for some .
The starting point is the following observation:
Fact (Erdős). If is finite and is a 2-coloring of , then there are paths of color 0 and of color 1 such that every (vertex) appears in exactly one of the
In general, if is an -coloring of , we say that , , is a path decomposition of iff each is a path of color and every vertex appears in exactly one of the . Using this notion, what the fact states is that for any finite , any 2-coloring of admits a path decomposition.
Proof. Suppose the result holds for and is a 2-coloring of . We can then find paths and of color 0 and 1 respectively such that each appears in exactly one of the . We want to show that the same holds for the full coloring (which includes edges one of whose vertices is ) at the possible expense of having to modify the partial paths we currently have. If one of the is empty, this is clear. Assume then that and . The result is also clear if or . Finally, if and , consider . If this color is 0, we can let the paths be and . Similarly, if , we can let the paths be and . (This is perhaps most obvious if a picture is drawn.)
Rado’s paper is a generalization of this result and its countable version. The reference is
The paper opens indicating that Erdős sketched his proof to Rado; there does not seem to be an actual reference for Erdős’s proof. Rado proceeds to prove a more general version. I will only discuss here a particular case.
First, it should be noted that, unlike typical results in Ramsey theory where, once the case of two colors is handled, the argument easily generalizes to any number of colors, the proof above does not lift to more than two. The usual way of doing this lifting is by identifying all but one of the colors. This would result in two paths and , where along we only see color 0 and along we only see the other colors, but not 0. Let be the given coloring and be the set of vertices appearing in . If the restriction of to does not use color 0 we could indeed proceed inductively. But there is nothing to prevent 0 from being present as well, so the “easy” lifting argument actually breaks down.
The situation is indeed worse:
Theorem (Pokrovskiy). For any and any there is an and an -coloring of that does not admit a path decomposition.
The proof can be found in:
Partitioning edge-coloured complete graphs into monochromatic cycles and paths.
J. Combin. Theory Ser. B 106 (2014), 70–97.
On the other hand, we have:
Theorem (Rado). For any finite , any -coloring of admits a path decomposition.
As already mentioned, Rado’s result is more general, in particular allowing the use of countably many colors. However, the arguments that follow only apply directly to the stated version.
Before sketching the proof, note that even for , the result does not follow as usual from the finite version: Given a 2-coloring of , the standard approach would consist of letting be the paths resulting from successively applying Erdős’s theorem to the restrictions of to . But the inductive argument we presented allows the paths to be modified from one value of to the next, which means that we cannot ensure that the process will successfully identify (via initial segments) paths for the full coloring (the partial paths do not “stabilize”). Together with Pokrovskiy’s negative result just indicated, this leaves us with a curious Ramsey-theoretic statement to which the usual compactness arguments do not apply. (Its finite counterpart, Erdős’s result, is weaker in the sense that it only applies to two colors, and requires a different argument.)
Proof. Consider a nonprincipal ultrafilter on . The ultrafilter provides us with a notion of largeness. Given and a coloring , define for and the set of neighbors of in color as
Note that for any the partition and therefore there is exactly one such that is large (that is, it is in $\mathcal U$). For , define
and note that the partition .
We proceed by stages to define the paths as required. We set for all . In general, at the beginning of any given stage we have defined (finite) partial approximations to each path , say has length , with , using the convention that indicates that . For each , we will ensure that end extends (for all ), and simply set as the resulting path. Inductively, we require that each is a path of color , and that if , then .
Now, at stage , we simply consider the least not yet in any of the . There is a unique with . We set for all . If , then set . Finally, if , the point is that since and are both large, then so is their intersection (all we really need is that the intersection of sets in is nonempty). Let be a point in their intersection, and set . The induction hypothesis is preserved, and this completes stage of the construction.
It should be immediate that the so constructed indeed provide a path decomposition of , and this completes the proof.
It is interesting to note that the notation just developed allows us to give a quick proof of Ramsey’s theorem for pairs: Given a coloring , use notation as above, and note that for exactly one , the set is in . We argue that there is an infinite subset that is homogeneous for with color , that is, . Indeed, we can simply set , where the are defined recursively so that and for all .
As Peter indicated in his talk, these pretty arguments are somewhat dissatisfying in that invoking a nonprincipal ultrafilter is too strong a tool for the task at hand. He then proceeded to indicate how we can in fact do better, computationally speaking. For instance, if the coloring is computable, then we can find a path decomposition below . The key to this improvement comes from two observations.
First, we do not really need an ultrafilter to carry out the argument. It suffices to consider a set that is cohesive with respect to all the , meaning that is infinite and, for any , either or , where is the eventual containment relation: iff there is a finite subset of such that , in which case we say that is almost contained in .
The point is that we can replace all instances where we required that a set is in by the new largeness condition stating that is almost contained in . For instance, note that if are large, then so is their intersection. As before, for any there is a unique with large, and we can redefine as the set of such that
With these modifications, it is straightforward to verify that the proof above goes through. This shows that a path decomposition of is .
In more detail: Note first that these two conditions are indeed equivalent, and second, clearly the are pairwise disjoint since is infinite and, moreover, for all there is a unique such that :
Suppose that holds, and let be such that . Since is infinite, we can indeed find elements of larger than , and any such witnesses .
Conversely, if holds, then , because is cohesive and has infinite intersection with . But then holds, as wanted.
To see that any is in a unique , fix and use that is cohesive to conclude that if for all , then , which contradicts the infinitude of . It follows that for some and, since the are pairwise disjoint, this is unique. This proves that holds and therefore . Uniqueness follows from this same observation: If , then (as shown above) . But there is only one for which this is true.
The second observation is that there is an easy recursive construction of a set that is cohesive with respect to all the : Consider first . One of these sets is infinite (since their union is ), say , and let be its first element. Consider now
Their union is , so one of these sets is infinite, say . Let be its first element above . Etc. The set so constructed is as wanted. Note that this construction explicitly obtains an infinite set that, for each , is almost contained in one the , which is superficially stronger than being cohesive. However, as verified above, any set cohesive for all the must actually have this property.
Computationally, the advantage of this construction is that it makes explicit that all we need to access a cohesive set is an oracle deciding of any whether it is infinite. For computable , these are all questions.
Peter further refined this analysis in his talk via the notion of a set being over : This is any set such that for any uniformly computable sequence of pairs of sentences for such that at least one is true, there is an that predicts the true sentence of each pair in the sense that for all , if , then is true. In symbols, say that . The point of the notion is that a result of Jockusch and Stephen gives us that if then there is a cohesive set such that . The relevant paper is:
This shows that a path decomposition for a computable coloring can actually be found below (and more).
Peter concluded his talk by indicating how for special colorings the complexity can be further improved. For instance, say that a coloring is stable iff exists for all . One can check that for stable , we can use cofinite as a notion of largeness in the preceding arguments, and that a path decomposition can accordingly be found when is computable below . On the other hand, this is optimal, in that one can find a stable computable such that any path decomposition computes .