Mathematics in the Alternative Set Theory

The Future of Set Theory

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The incompleteness of ZFC means that the mathematical universe that its axioms generate will inevitably have holes.

To Settle Infinity Dispute, a New Law of Logic

Their combined results demonstrated that the continuum hypothesis is actually independent of the axioms. Something beyond ZFC is needed to prove or refute it. With the hypothesis unresolved, many other properties of cardinal numbers and infinity remain uncertain too.

In the universe of sets that results, the continuum hypothesis is true: There is no infinite set between that of the integers and the continuum. To keep this symphony of infinities, set theorists have striven for decades to find an inner model that is as pristine and analyzable as L but incorporates large cardinals. However, constructing a universe of sets that included each type of large cardinal required a unique tool kit. And that kind of makes it look hopeless because, you know, life is short.

Because there was no largest large cardinal, it seemed like there could be no ultimate L, an inner model that encompassed them all. In work that was published in , he discovered a breakaway point in the hierarchy. It provided this new hope that this approach can work.

Although it has not yet been constructed, ultimate L is the name for the hypothetical inner model that includes supercompacts and therefore all large cardinals. Woodin, who is moving from Berkeley to Harvard in January, recently completed the first part of a four-stage proof of the ultimate L conjecture and is now vetting it with a small group of colleagues.

To expand ZFC, address the continuum hypothesis and better understand infinity, advocates of forcing axioms put stock in a method called forcing, originally conceived of by Cohen. If inner models build a universe of sets from the ground up, forcing expands it outward in all directions.

In the extended universe created by forcing, there is a larger class of real numbers than in the original universe defined by ZFC. This means the real numbers of ZFC constitute a smaller infinite set than the full continuum. This is where forcing axioms shine. Work over the past few years by Todorcevic, Moore, Carlos Martinez-Ranero and others shows that they bestow many mathematical structures with nice properties that make them easier to use and understand. To Moore, these sorts of results give forcing axioms the advantage over inner models. At the recent Harvard meeting, researchers from both camps presented new work on inner models and forcing axioms and discussed their relative merits.

The back-and-forth will likely continue, they said, until one or the other candidate falls by the wayside. Ultimate L could turn out not to exist, for example. As many of the mathematicians pointed out, the debate itself reveals a lack of human intuition regarding the concept of infinity. Mathematics has a reputation for objectivity. However, the field of mathematics is known for its unity and cohesion.

Just as ZFC came to dominate alternative foundational frameworks in the early 20th century, firmly embedding actual infinity in mathematical thinking and practice, it is likely that only one new axiom to decide the fuller nature of infinity will survive. This article was reprinted on ScientificAmerican. Get highlights of the most important news delivered to your email inbox. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.

Moderators are staffed during regular business hours New York time and can only accept comments written in English. Read Later. Borel Determinacy is a deep result within ZF set theory which relies heavily on transfinitely iterated use of the replacement axiom. It does not appear to fit well with other foundational approaches. This wouldn't matter if it were only of set-theoretic interest; however, as I shall explain, it has concrete implications in the field of topological infinite graph theory.

In this talk, I shall give an overview about what is known about these questions for systems for homotopy type theory. I shall recall earlier work on the proof theory on type theory and explain how recent work on cubical type theories furthers our understanding. This approach is enabled by homotopy type theory and gives very direct ways of reasoning about homotopy types analogously to how euclidean geometry provides direct reasoning about points, lines and circles , but otherwise follows the tradition of other approaches to abstract homotopy theory.

In the workshop, we shall introduce the basic ideas through many examples, and we shall in particular focus on the use of higher inductive types to furnish us with many constructions on homotopy types. Higher inductive types also figure prominently in other uses of homotopy type theory, and we shall briefly touch on those as well.

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More details will be given during my talk on Saturday. Hence, set theory ZFC cannot give us the promised paradise. We give a pluralistic vision, accepting that different parts of mathematics need different approaches.

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The method of forcing was first used in by Paul Cohen in order to show that the Continuum Hypothesis is independent of the axioms of Zermelo-Fraenkel set theory ZFC. Since then, forcing has turned out to be a powerful tool for obtaining independence results in set theory.

In this workshop I will give a short introduction to forcing and its applications. Homotopy Type Theory HoTT is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that so construed it is a candidate for being an autonomous foundation for mathematics.

More importantly, the presentation of HoTT given in the HoTT Book is not 'autonomous' since it explicitly depends upon other fields of mathematics. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. We offer a derivation of path induction, motivated from pre-mathematical considerations, without recourse to homotopy theory.

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We use this as part of an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory but is compatible with the homotopy interpretation , and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on an interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. I will give an introduction to models of type theory from a categorical perspective.

I will discuss problems of strictness that arise and some solutions. I will also discuss the relationship between models of type theory and weak factorization systems. In conclusion, I will give some examples of models in various categories of spaces. In the Type Theoretic approach to mathematical foundations, proofs about properties of mathematical objects can have the same status as the objects themselves.

In particular, proofs about equality of objects themselves become mathematical objects that can be studied. The homotopical approach to Type Theory views such proofs of equality as paths, possibly in an abstract sense. Taking this view literally, what is required of an interval-like object I in order to give a model of Type Theory in which elements of identity types really are just functions on I?

I will discuss this question and introduce a surprisingly simple coherent theory of the interval that suffices to model Coquand's recent axiomatization of propositional identity types. Moreover, consistency strength-wise, they can be related to extensions of classical Kripke-Platek set theory.

1. Why Set Theory?

The aim of the talk is to present some of the current knowledge. While the standard notion construes a theory as a set of propositions, HoTT supports a notion of theoretical object that serves as a vehicle of both propositional and non-propositional contents. This non-standard view on theories and their models appears unusual in the context of mathematical logic of the 20th century but it has strong historical roots in some earlier logical traditions and in the current mathematical practice. Homotopy type theory is argued to have semantics in infinity-toposes.

We consider 0-types, so-called hsets, in the univalent foundation. Technically 0-types form a predicative topos. This is based on the article: Egbert Rijke and Bas Spitters. Demise of robust realism. Antos, S. Friedman, R.

Honzik, and C.

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