This will be a talk for a new mathematical logic seminar at the University of Warsaw in the Department of Hhilosophy, entitled Epistemic and Semantic Commitments of Foundational Theories, devoted to formal truth theories and implicit commitments of foundational theories as well as their conceptual surroundings.
Abstract. According to the math tea argument, perhaps heard at a good afternoon tea, there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions. Is it correct? In this talk, I shall discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? We shall discuss many interesting elementary examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable.
The talk will be held online via Zoom ID: 998 6013 7362.
Abstract. It is a mystery often mentioned in the foundations of mathematics that our best and strongest mathematical theories seem to be linearly ordered and indeed well-ordered by consistency strength. Given any two of the familiar large cardinal hypotheses, for example, generally one of them proves the consistency of the other. Why should this be? The phenomenon is seen as significant for the philosophy of mathematics, perhaps pointing us toward the ultimately correct mathematical theories. And yet, we know as a purely formal matter that the hierarchy of consistency strength is not well-ordered. It is ill-founded, densely ordered, and nonlinear. The statements usually used to illustrate these features are often dismissed as unnatural or as GΓΆdelian trickery. In this talk, I aim to overcome that criticismβas well as I am able toβby presenting a variety of natural hypotheses that reveal ill-foundedness in consistency strength, density in the hierarchy of consistency strength, and incomparability in consistency strength.
The talk should be generally accessible to university logic students, requiring little beyond familiarity with the incompleteness theorem and some elementary ideas from computability theory.
The best way to learn mathematics is to dive in and do it. Donβt just listen passively to a lecture or read a bookβyou have got to take hold of the mathematical ideas yourself! Mount your own mathematical analysis. Formulate your own mathematical assertions. Consider your own mathematical examples. I recommend playβadopt an attitude of playful curiosity about mathematical ideas; grasp new concepts by exploring them in particular cases; try them out; understand how the mathematical constructions from your proofs manifest in your examples; explore all facets, going beyond whatever had been expected. You will find vast new lands of imagination. Let one example generalize to a whole class of examples; have favorite examples. Ask questions about the examples or about the mathematical idea you are investigating. Formulate conjectures and test them with your examples. Try to prove the conjecturesβwhen you succeed, you will have proved a theorem. The essential mathematical activity is to make clear claims and provide sound reasons for them. Express your mathematical ideas to others, and practice the skill of stating matters well, succinctly, with accuracy and precision. Donβt be satisfied with your initial account, even when it is sound, but seek to improve it. Find alternative arguments, even when you already have a solid proof. In this way, you will come to a deeper understanding. Test the statements of others; ask for further explanation. Look into the corner cases of your results to probe the veracity of your claims. Set yourself the challenge either to prove or to refute a given statement. Aim to produce clear and correct mathematical arguments that logically establish their conclusions, with whatever insight and elegance you can muster.
This book is offered as a companion volume to my book Proof and the Art of Mathematics, which I have described as a mathematical coming-of-age book for students learning how to write mathematical proofs.
Spanning diverse topics from number theory and graph theory to game theory and real analysis, Proof and the Art shows how to prove a mathematical theorem, with advice and tips for sound mathematical habits and practice, as well as occasional reflective philosophical discussions about what it means to undertake mathematical proof. In Proof and the Art, I offer a few hundred mathematical exercises, challenges to the reader to prove a given mathematical statement, each a small puzzle to figure out; the intention is for students to develop their mathematical skills with these challenges of mathematical reasoning and proof.
Here in this companion volume, I provide fully worked-out solutions to all of the odd-numbered exercises, as well as a few of the even-numbered exercises. In many cases, the solutions here explore beyond the exercise question itself to natural extensions of the ideas. My attitude is that, once you have solved a problem, why not push the ideas harder to see what further you can prove with them? These solutions are examples of how one might write a mathematical proof. I hope that you will learn from them; let us go through them together. The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.
Welcome to Cantorβs Ice Cream Shoppe! A huge choice of flavorsβpile your cone high with as many scoops as you want!
Have two scoops, or three, four, or more! Why not infinitely many? Would you like π many scoops, or πβ 2+5 many scoops? You can have any countable ordinal number of scoops on your cone.
And furthermore, after ordering your scoops, you can order more scoops to be placed on topβall I ask is that you let me know how many such extra orders you plan to make. Letβs simply proceed transfinitely. You can announce any countable ordinal π, which will be the number of successive orders you will make; each order is a countable ordinal number of ice cream scoops to be placed on top of whatever cone is being assembled.
In fact, Iβll even let you change your mind about π as we proceed, so as to give you more orders to make a taller cone.
So the process is:
You pick a countable ordinal π, which is the number of orders you will make.
For each order, you can pick any countable ordinal number of scoops to be added to the top of your ice-cream cone.
After making your order, you can freely increase π to any larger countable ordinal, giving you the chance to make as many additional orders as you like.
At each limit stage of the ordering process, the ice cream cone you are assembling has all the scoops youβve ordered so far, and we set the current π value to the supremum of the values you had chosen so far.
If at any stage, youβve used up your π many orders, then the process has completed, and I serve you your ice cream cone. Enjoy!
Question. Can you arrange to achieve uncountably many scoops on your cone?
Although at each stage we place only countably many ice cream scoops onto the cone, nevertheless we can keep giving ourselves extra stages, as many as we want, simply by increasing π. Can you describe a systematic process of increasing the number of steps that will enable you to make uncountably many orders? This would achieve an unountable ice cream cone.
What is your solution? Give it some thought before proceeding. My solution appears below.
Alas, I claim that at Cantorβs Ice Cream Shoppe you cannot make an ice cream cone with uncountably many scoops. Specifically, I claim that there will inevitably come a countable ordinal stage at which you have used up all your orders.
Suppose that you begin by ordering π½0 many scoops, and setting a large value π0 for the number of orders you will make. You subsequently order π½1 many additional scoops, and then π½2 many on top of that, and so on. At each stage, you may also have increased the value of π0 to π1 and then π2 and so on. Probably all of these are enormous countable ordinals, making a huge ice cream cone.
At each stage πΌ, provided πΌ<ππΌ, then you can make an order of π½πΌ many scoops on top of your cone, and increase ππΌ to ππΌ+1, if desired, or keep it the same.
At a limit stage π, your cone has βπΌ<ππ½πΌ many scoops, and we update the π value to the supremum of your earlier declarations ππ=supπΌ<πβ‘ππΌ.
What I claim now is that there will inevitably come a countable stage π for which π=ππ, meaning that you have used up all your orders with no possibility to further increase π. To see this, consider the sequence π0β€ππ0β€πππ0β€β― We can define the sequence recursively by π0=π0 and ππ+1=πππ. Let π=supπ<πβ‘ππ, the limit of this sequence. This is a countable supremum of countable ordinals and hence countable. But notice that ππ=supπ<πβ‘πππ=supπ<πβ‘ππ+1=π. That is, ππ=π itself, and so your orders have run out at π, with no possibility to add more scoops or to increase π. So your order process completed at a countable stage, and you have therefore altogether only a countable ordinal number of scoops of ice cream. Iβm truly very sorry at your pitiable impoverishment.
Abstract. Recent years have seen a flurry of mathematical activity in set-theoretic and arithmetic potentialism, in which we investigate a collection of models under various natural extension concepts. These potentialist systems enable a modal perspectiveβa statement is possible in a model, if it is true in some extension, and necessary, if it is true in all extensions. We consider the models of ZFC set theory, for example, with respect to submodel extensions, rank-extensions, forcing extensions and others, and these various extension concepts exhibit different modal validities. In this talk, I shall describe the state of current developments, including the most recent tools and results.
This will be a talk for the Models of Peano Arithmetic (MOPA) seminar on 11 November 2020, 12 pm EST (5pm GMT). Kindly note the rescheduled date and time.
This will be a talk for the Barcelona Set Theory Seminar, 28 October 2020 4 pm CET (3 pm UK). Contact Joan Bagaria bagaria@ub.edu for the access link.
Abstract. The Barwise extension theorem, asserting that every countable model of ZF set theory admits an end-extension to a model of ZFC+V=L, is both a technical culmination of the pioneering methods of Barwise in admissible set theory and infinitary logic and also one of those rare mathematical theorems that is saturated with philosophical significance. In this talk, I shall describe a new proof of the theorem that omits any need for infinitary logic and relies instead only on classical methods of descriptive set theory. This proof leads directly to the universal finite sequence, a Sigma_1 definable finite sequence, which can be extended arbitrarily as desired in suitable end-extensions of the universe. The result has strong consequences for the nature of set-theoretic potentialism. This work is joint with Kameryn J. Williams.
This series of self-contained lectures on the philosophy of mathematics, offered for Oxford Michaelmas Term 2020, is intended for students preparing for philosophy exam paper 122, although all interested parties are welcome to join. The lectures will be organized loosely around mathematical themes, in such a way that brings various philosophical issues naturally to light.
Lectures will follow my new book Lectures on the Philosophy of Mathematics (MIT Press), with supplemental readings suggested each week for further tutorial work. The book is available for pre-order, to be released 2 February 2021.
Lectures will be held online via Zoom every Wednesday 11-12 am during term at the following Zoom coordinates:
All lectures will be recorded and made available at a later date.
Lecture 1. Numbers
Numbers are perhaps the essential mathematical idea, but what are numbers? There are many kinds of numbersβnatural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and moreβand these number systems provide a fruitful background for classical arguments on incommensurability and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and structuralism.
Lecture 2. Rigour
Let us consider the problem of mathematical rigour in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to the epsilon-delta limit concept, which secured a more rigourous foundation while also enlarging our conceptual vocabulary, enabling us to express more refined notions, such as uniform continuity, equicontinuity, and uniform convergence. Nonstandard analysis resurrected the infinitesimals on a more secure foundation, providing a parallel development of the subject. Meanwhile, increasing abstraction emerged in the function concept, which we shall illustrate with the Devilβs staircase, space-filling curves, and the Conway base 13 function. Finally, does the indispensability of mathematics for science ground mathematical truth? Fictionalism puts this in question.
Lecture 3. Infinity
We shall follow the allegory of Hilbertβs hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantorβs diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity; some arguments give rise to the distinction between constructive and nonconstructive proof. Zenoβs paradox highlights classical ideas on potential versus actual infinity. Furthermore, we shall count into the transfinite ordinals.
Lecture 4. Geometry
Classical Euclidean geometry is the archetype of a mathematical deductive process. Yet the impossibility of certain constructions by straightedge and compass, such as doubling the cube, trisecting the angle, or squaring the circle, hints at geometric realms beyond Euclid. The rise of non-Euclidean geometry, especially in light of scientific theories and observations suggesting that physical reality is not Euclidean, challenges previous accounts of what geometry is about. New formalizations, such as those of David Hilbert and Alfred Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarskiβs decision procedure points to a tantalizing possibility of automation in geometrical reasoning.
Lecture 5. Proof
What is proof? What is the relation between proof and truth? Is every mathematical truth true for a reason? After clarifying the distinction between syntax and semantics and discussing various views on the nature of proof, including proof-as-dialogue, we shall consider the nature of formal proof. We shall highlight the importance of soundness, completeness, and verifiability in any formal proof system, outlining the central ideas used in proving the completeness theorem. The compactness property distills the finiteness of proofs into an independent, purely semantic consequence. Computer-verified proof promises increasing significance; its role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakening the logical rules.
Lecture 6. Computability
What is computability? Kurt GΓΆdel defined a robust class of computable functions, the primitive recursive functions, and yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Alan Turingβs machine concept of computability, growing out of a careful philosophical analysis of the nature of human computability, proved robust and laid a foundation for the contemporary computer era; the widely accepted Church-Turing thesis asserts that Turing had the right notion. The distinction between computable decidability and computable enumerability, highlighted by the undecidability of the halting problem, shows that not all mathematical problems can be solved by machine, and a vast hierarchy looms in the Turing degrees, an infinitary information theory. Complexity theory refocuses the subject on the realm of feasible computation, with the still-unsolved P versus NP problem standing in the background of nearly every serious issue in theoretical computer science.
Lecture 7. Incompleteness
David Hilbert sought to secure the consistency of higher mathematics by finitary reasoning about the formalism underlying it, but his program was dashed by GΓΆdelβs incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, self-reference, and definability, showing senses in which we cannot complete mathematics. After this, we shall discuss the second incompleteness theorem, the Rosser variation, and Tarskiβs theorem on the nondefinability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength rising above every foundational mathematical theory.
Lecture 8. Set Theory
We shall discuss the emergence of set theory as a foundation of mathematics. Cantor founded the subject with key set-theoretic insights, but Fregeβs formal theory was naive, refuted by the Russell paradox. Zermeloβs set theory, in contrast, grew ultimately into the successful contemporary theory, founded upon a cumulative conception of the set-theoretic universe. Set theory was simultaneously a new mathematical subject, with its own motivating questions and tools, but it also was a new foundational theory with a capacity to represent essentially arbitrary abstract mathematical structure. Sophisticated technical developments, including in particular, the forcing method and discoveries in the large cardinal hierarchy, led to a necessary engagement with deep philosophical concerns, such as the criteria by which one adopts new mathematical axioms and set-theoretic pluralism.
Philosophical conundrums pervade mathematics, from fundamental questions of mathematical ontologyβWhat is a number? What is infinity?βto questions about the relations among truth, proof, and meaning. What is the role of figures in geometric argument? Do mathematical objects exist that we cannot construct? Can every mathematical question be solved in principle by computation? Is every truth of mathematics true for a reason? Can every mathematical truth be proved?
This book is an introduction to the philosophy of mathematics, in which we shall consider all these questions and more. I come to the subject from mathematics, and I have strived in this book for what I hope will be a fresh approach to the philosophy of mathematicsβone grounded in mathematics, motivated by mathematical inquiry or mathematical practice. I have strived to treat philosophical issues as they arise organically in mathematics. Therefore, I have organized the book by mathematical themes, such as number, infinity, geometry, and computability, and I have included some mathematical arguments and elementary proofs when they bring philosophical issues to light.
This is joint work with Wojciech Aleksander WoΕoszyn, who is about to begin as a DPhil student with me in mathematics here in Oxford. We began and undertook this work over the past year, while he was a visitor in Oxford under the Recognized Student program.
Abstract. We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement π is possible in a structure (written βπ) if π is true in some extension of that structure, and π is necessary (written β»π) if it is true in all extensions of the structure. A principal case for us will be the class Modβ‘(π) of all models of a given theory πβall graphs, all groups, all fields, or what have youβconsidered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, π-colorability, finiteness, countability, size continuum, size β΅1, β΅2, β΅π, βΆπ, first βΆ-fixed point, first βΆ-hyper-fixed-point and much more. A graph obeys the maximality principle ββ»πβ‘(π)βπβ‘(π) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs.
Follow through the arXiv for a pdf of the article.
Categorical accounts of various mathematical structures lie at the very core of structuralist mathematical practice, enabling mathematicians to refer to specific mathematical structures, not by having carefully to prepare and point at specially constructed instancesβpreserved like the one-meter iron bar locked in a case in Parisβbut instead merely by mentioning features that uniquely characterize the structure up to isomorphism.
It follows that for any two models of ZFC2, one of them is isomorphic to an initial segment of the other. These set-theoretic models ππ have now come to be known as Zermelo-Grothendieck universes, in light of Grothendieckβs use of them in category theory (a rediscovery several decades after Zermelo); they feature in the universe axiom, which asserts that every set is an element of some such ππ , or equivalently, that there are unboundedly many inaccessible cardinals.
In this article, we seek to investigate the extent to which Zermeloβs quasi-categoricity analysis can rise fully to the level of categoricity, in light of the observation that many of the ππ universes are categorically characterized by their sentences or theories.
Question. Which models of ZFC2 satisfy fully categorical theories?
If π is the smallest inaccessible cardinal, for example, then up to isomorphism ππ is the unique model of ZFC2 satisfying the first-order sentence βthere are no inaccessible cardinals.β The least inaccessible cardinal is therefore an instance of what we call a first-order sententially categorical cardinal. Similar ideas apply to the next inaccessible cardinal, and the next, and so on for quite a long way. Many of the inaccessible universes thus satisfy categorical theories extending ZFC2 by a sentence or theory, either in first or second order, and we should like to investigate these categorical extensions of ZFC2.
In addition, we shall discuss the philosophical relevance of categoricity and point particularly to the philosophical problem posed by the tension between the widespread support for categoricity in our fundamental mathematical structures with set-theoretic ideas on reflection principles, which are at heart anti-categorical.
Our main theme concerns these notions of categoricity:
Main Definition.
A cardinal π is first-order sententially categorical, if there is a first-order sentence π in the language of set theory, such that ππ is categorically characterized by ZFC2+π.
A cardinal π is first-order theory categorical, if there is a first-order theory π in the language of set theory, such that ππ is categorically characterized by ZFC2+π.
A cardinal π is second-order sententially categorical, if there is a second-order sentence π in the language of set theory, such that ππ is categorically characterized by ZFC2+π.
A cardinal π is second-order theory categorical, if there is a second-order theory π in the language of set theory, such that ππ is categorically characterized by ZFC2+π.
Follow through to the arxiv for the pdf to read more:
Iβd like to introduce and discuss the otherworldly cardinals, a large cardinal notion that frequently arises in set-theoretic analysis, but which until now doesnβt seem yet to have been given its own special name. So let us do so here.
I was put on to the topic by Jason Chen, a PhD student at UC Irvine working with Toby Meadows, who brought up the topic recently on Twitter:
Do these cardinals have special names: Ξ±'s such that there is some Ξ² with V_Ξ± being an elementary substructure of V_Ξ² (so they form a proper subset of worldly cardinals); and a stratified version: Ξ±'s such that there is some Ξ², with V_Ξ± being a Ξ£_n-elementary substructure of V_Ξ².
In response, I had suggested the otherworldly terminology, a play on the fact that the two cardinals will both be worldly, and so we have in essence two closely related worlds, looking alike. We discussed the best way to implement the terminology and its extensions. The main idea is the following:
Main Definition. An ordinal π is otherworldly if ππ βΊππ for some ordinal π>π . In this case, we say that π is otherworldly toπ.
It is an interesting exercise to see that every otherworldly cardinal π is in fact also worldly, which means ππ β§ZFC, and from this it follows that π is a strong limit cardinal and indeed a βΆ-fixed point and even a βΆ-hyperfixed point and more.
Theorem. Every otherworldly cardinal is also worldly.
Proof. Suppose that π is otherworldly, so that ππ βΊππ for some ordinal π>π . It follows that π must in fact be a cardinal, since otherwise it would be the order type of a relation on a set in ππ , which would be isomorphic to an ordinal in ππ but not in ππ . And since π is not otherworldly, we see that π must be an uncountable cardinal. Since ππ is transitive, we get now easily that ππ satisfies extensionality, regularity, union, pairing, power set, separation and infinity. The only axiom remaining is replacement. If πβ‘(π,π) obeys a functional relation in ππ for all πβπ΄, where π΄βππ , then ππ agrees with that, and also sees that the range is contained in ππ , which is a set in ππ. So ππ agrees that the range is a set. So ππ fulfills the replacement axiom. β»
Corollary. A cardinal is otherworldly if and only if it is fully correct in a worldly cardinal.
Proof. Once you know that otherworldly cardinals are worldly, this amounts to a restatement of the definition. If ππ βΊππ, then π is worldly, and ππ is correct in ππ. β»
Let me prove next that whenever you have an otherworldly cardinal, then you will also have a lot of worldly cardinals, not just these two.
Theorem. Every otherworldly cardinal π is a limit of worldly cardinals. What is more, every otherworldly cardinal is a limit of worldly cardinals having exactly the same first-order theory as ππ , and indeed, the same πΌ-order theory for any particular πΌ<π .
Proof. If ππ βΊππ, then ππ can see that π is worldly and has the theory π that it does. So ππ thinks, about π, that there is a cardinal whose rank initial segment has theory π. Thus, ππ also thinks this. And we can find arbitrarily large πΏ up to π such that ππΏ has this same theory. This argument works whether one uses the first-order theory, or the second-order theory or indeed the πΌ-order theory for any πΌ<π . β»
Theorem. If π is otherworldly, then for every ordinal πΌ<π and natural number π, there is a cardinal πΏ<π with ππΏβΊΞ£πππ and the πΌ-order theory of ππΏ is the same as ππ .
Proof. One can do the same as above, since ππ can see that ππ has the πΌ-order theory that it does, while also agreeing on Ξ£π truth with ππ, so ππ will agree that there should be such a cardinal πΏ<π . β»
Definition. We say that a cardinal is totally otherworldly, if it is otherworldly to arbitrarily large ordinals. It is otherworldly beyond π, if it is otherworldly to some ordinal larger than π. It is otherworldly up to πΏ, if it is otherworldly to ordinals cofinal in πΏ.
Theorem. Every inaccessible cardinal πΏ is a limit of otherworldly cardinals that are each otherworldly up to and to πΏ.
Proof. If πΏ is inaccessible, then a simple LΓΆwenheim-Skolem construction shows that ππ is the union of a continuous elementary chain ππ 0βΊππ 1βΊβ―βΊππ πΌβΊβ―βΊππ Each of the cardinals π πΌ arising on this chain is otherworldly up to and to πΏ. β»
Theorem. Every totally otherworldly cardinal is Ξ£2 correct, meaning ππ βΊΞ£2π. Consequently, every totally otherworldly cardinal is larger than the least measurable cardinal, if it exists, and larger than the least superstrong cardinal, if it exists, and larger than the least huge cardinal, if it exists.
Proof. Every Ξ£2 assertion is locally verifiable in the ππΌ hierarchy, in that it is equivalent to an assertion of the form βπβ’ππβ§π (for more information, see my post about Local properties in set theory). Thus, every true Ξ£2 assertion is revealed inside any sufficiently large ππ, and so if ππ βΊππ for arbitrarily large π, then ππ will agree on those truths. β»
I was a little confused at first about how two totally otherwordly cardinals interact, but now everything is clear with this next result. (Thanks to Hanul Jeon for his helpful comment below.)
Theorem. If π <πΏ are both totally otherworldly, then π is otherworldly up to πΏ, and hence totally otherworldly in ππΏ.
Proof. Since πΏ is totally otherworldly, it is Ξ£2 correct. Since for every πΌ<πΏ the cardinal π is otherworldly beyond πΌ, meaning ππ βΊππ for some π>πΌ, then since this is a Ξ£2 feature of π , it must already be true inside ππΏ. So such a π can be found below πΏ, and so π is otherworldly up to πΏ. β»
Theorem. If π is totally otherworldly, then π is a limit of otherworldly cardinals, and indeed, a limit of otherworldly cardinals having the same theory as ππ .
Proof. Assume π is totally otherworldly, let π be the theory of ππ , and consider any πΌ<π . Since there is an otherworldly cardinal above πΌ with theory π, namely π , and because this is a Ξ£2 fact about πΌ and π, it follows that there must be such a cardinal above πΌ inside ππ . So π is a limit of otherworldly cardinals with the same theory as ππ . β»
The results above show that the consistency strength of the hypotheses are ordered as follows, with strict increases in consistency strength as you go up (assuming consistency):
ZFC + there is an inaccessible cardinal
ZFC + there is a proper class of totally otherworldly cardinals
ZFC + there is a totally otherworldly cardinal
ZFC + there is a proper class of otherworldly cardinals
ZFC + there is an otherworldly cardinal
ZFC + there is a proper class of worldly cardinals
ZFC + there is a worldly cardinal
ZFC + there is a transitive model of ZFC
ZFC + Con(ZFC)
ZFC
We might consider the natural strengthenings of otherworldliness, where one wants ππ βΊππ where π is itself otherworldly. That is, π is the beginning of an elementary chain of three models, not just two. This is different from having merely that ππ βΊππ and ππ βΊππ for some π>π, because perhaps ππ is not elementary in ππ, even though ππ is. Extending successively is a more demanding requirement.
One then naturally wants longer and longer chains, and ultimately we find ourselves considering various notions of rank in the rank elementary forest, which is the relation π βͺ―πβΊππ βΊππ. The otherworldly cardinals are simply the non-maximal nodes in this order, while it will be interesting to consider the nodes that can be extended to longer elementary chains.
This will be a talk for the Oslo potentialism workshop, Varieties of Potentialism, to be held online via Zoom on 23 September 2020, from noon to 18:40 CEST (11am to 17:40 UK time). My talk is scheduled for 13:10 CEST (12:10 UK time). Further details about access and registration are availavle on the conference web page.
Abstract. I shall introduce and describe the subject of modal model theory, in which one studies a mathematical structure within a class of similar structures under an extension concept, giving rise to mathematically natural notions of possibility and necessity, a form of mathematical potentialism. We study the class of all graphs, or all groups, all fields, all orders, or what have you; a natural case is the class Modβ‘(π) of all models of a fixed first-order theory π. In this talk, I shall describe some of the resulting elementary theory, such as the fact that the L theory of a structure determines a robust fragment of its modal theory, but not all of it. The class of graphs illustrates the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, colorability, finiteness, countability, size continuum, size β΅1, β΅2, β΅π, βΆπ, first βΆ-fixed point, first βΆ-hyper-fixed-point and much more. When augmented with the actuality operator @, modal graph theory becomes fully bi-interpretable with truth in the set-theoretic universe. This is joint work with Wojciech WoΕoszyn.
Dr. Corey Bacal Switzer successfully defended his PhD dissertation, entitled βAlternative CichoΕ Diagrams and Forcing Axioms Compatible with CH,β on 31 July 2020, for the degree of PhD from The Graduate Center of the City University of New York. The dissertation was supervised jointly by myself and Gunter Fuchs.
Corey has now accepted a three-year post-doctoral research position at the University of Vienna, where he will be working with Vera Fischer.
Abstract. This dissertation surveys several topics in the general areas of iterated forcing, inο¬nite combinatorics and set theory of the reals. There are four largely independent chapters, the ο¬rst two of which consider alternative versions of the CichoΕ diagram and the latter two consider forcing axioms compatible with CH . In the ο¬rst chapter, I begin by introducing the notion of a reduction concept , generalizing various notions of reduction in the literature and show that for each such reduction there is a CichoΕ diagram for eο¬ective cardinal characteristics relativized to that reduction. As an application I investigate in detail the CichoΕ diagram for degrees of constructibility relative to a ο¬xed inner model πβ§ZFC.
In the second chapter, I study the space of functions π:ππβππ and introduce 18 new higher cardinal characteristics associated with this space. I prove that these can be organized into two diagrams of 6 and 12 cardinals respecitvely analogous to the CichoΕ diagram on π. I then investigate their relation to cardinal invariants on Ο and introduce several new forcing notions for proving consistent separations between the cardinals. The third chapter concerns Jensenβs subcomplete and subproper forcing. I generalize these notions to the (seemingly) larger classes of β-subcomplete and β-subproper. I show that both classes are (apparently) much more nicely behaved structurally than their non-β-counterparts and iteration theorems are proved for both classes using Miyamotoβs nice iterations. Several preservation theorems are then presented. This includes the preservation of Souslin trees, the Sacks property, the Laver property, the property of being ππ-bounding and the property of not adding branches to a given π1-tree along nice iterations of β-subproper forcing notions. As an application of these methods I produce many new models of the subcomplete forcing axiom, proving that it is consistent with a wide variety of behaviors on the reals and at the level of π1.
The ο¬nal chapter contrasts the ο¬exibility of SCFA with Shelahβs dee-complete forcing and its associated axiom DCFA . Extending a well known result of Shelah, I show that if a tree of height π1 with no branch can be embedded into an π1-tree, possibly with branches, then it can be specialized without adding reals. As a consequence I show that DCFA implies there are no Kurepa trees, even if CH fails.
Abstract. We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of ZF connected by a binary accessibility relation, achieving a potentialist account of the full background set-theoretic universe π. The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory S5, both for assertions in the language of ZF and for assertions in the full potentialist language.
