MOPA

In a paper published in 1990, Kossak showed that all countable models of $\Sigma_n$ collection where $\Sigma_n$ induction fails have continuum-many automorphisms. We extract from his proof a(nother) quantifier-elimination result. This gives new information about pigeonhole principles and expansions to second-order models. The work is joint with David Belanger, CT Chong, Wei Li, and Yue Yang at the National University of Singapore.

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Advances in understanding the incompleteness of PA in the 1970s and 80s built on the work of an earlier generation in the 1930s and 40s. This talk will offer historical and personal reflections on what was known, and what was not known, by both generations of logicians.

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Athena sprang forth full grown from the head of Zeus. Newton/Leibniz created Calculus. Galois created Galois Theory. Cantor created Set Theory. Boole created Boolean Algebra.

But Models of Peano Arithmetic doesn’t have a dramatic origin myth like that and took some 100 years to emerge as a discipline in itself - from Dedekind’s Second Order Axioms for Arithmetic (1863), through Frege’s Begriffsschrift (1879) and First Order Logic, through Godel’s Completeness and Incompleteness Theorems, through Skolem’s elegant construction of a non-standard model, through the War and après-guerre and on into the 1970s where the subject at last emerges as a discipline in itself. We’ll discuss the convergence of people and ideas from diverse fields like Model Theory, Set Theory, Recursion Theory, Proof Theory, Complexity Theory, … that led to the field we know and love today.

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As the history goes, and was confirmed recently [vP20], Gödel first proved his first incompleteness theorem [G31] for sound theories (that extend Principia Mathematica). Later he weakened the soundness condition to “ℵ₀-consistency”, which later evolved to “ω-consistency”. This condition was needed for irrefutability of (what is now called) Gödelian sentences; the simple consistency of a theory suffices for the unprovability of such sentences. Gödel already notes in [G31] that a necessary and sufficient condition for the independence of Gödelian sentences of T is just a bit more than the simple consistency of T: the consistency of T with Con_T, the consistency statement of T.
In this talk, we ask the following questions and attempt at answering them, at least partially.

1. Why on earth Gödel [G31] had to introduce this rather strange notion?
2. Does it have any applications in other areas of logic, arithmetical theories, or mathematics?
3. What was Gödel’s reason that ω-consistency is “much weaker” than soundness? He does prove in [G31] that consistency is weaker (if not much weaker) than ω-consistency; but never mentions a proof or even a hint as to why soundness is (much) stronger than ω-consistency!
4. Other than those historical and philosophical questions, is this a useful notion worthy of further study?

We will also review some properties of ω-consistency in the talk.
References:

• [G31]   Kurt Gödel (1931); “On formally undecidable propositions of Principia Mathematica and related systems I”, in: S. Feferman, et al. (eds.), Kurt Gödel: Collected Works, Vol. I: Publications 1929–1936, Oxford University Press, 1986, pp. 135–152.
• [vP20]   Jan von Plato (2020); Can Mathematics Be Proved Consistent? Gödel’s Shorthand Notes & Lectures on Incompleteness, Springer.
  Reviewed in the zbMATH Open at https://zbmath.org/1466.03001

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The Scott rank of a countable structure is the least ordinal $\alpha$ such that all automorphism orbits of the structure are definable by infinitary $\Sigma_{\alpha}$ formulas. Montalbán showed that the Scott rank of a structure is a robust measure of the structural and computational complexity of a structure by showing that various different measures are equivalent. For example, a structure has Scott rank $\alpha$ if and only if it has a $\Pi_{\alpha+1}$ Scott sentence if and only if it is uniformly $\pmb \Delta_\alpha^0$ categorical if and only if all its automorphism orbits are $\Sigma_\alpha$ infinitary definable.

In this talk we present results on the Scott rank of non-standard models of Peano arithmetic. We show that non-standard models of PA have Scott rank at least $\omega$, but, other than that, there are no limits to their complexity. Given a completion $T$ of $PA$ we give a reduction via bi-interpretability of the class of linear orders to the models of $T$. This allows us to exhibit models of $T$ of Scott rank $\alpha$ for every $\omega\leq \alpha\leq \omega_1$. In particular, every completion of $T$ has models of high Scott rank.

This is joint work with Antonio Montalbán.

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When properly arithmetized, Yablo's paradox results in a set of formulas which (with local disquotation in the background) turns out to be consistent, but $\omega$-inconsistent. Adding either uniform disquotation or the $\omega$-rule results in inconsistency. Since the paradox involves an infinite sequence of sentences, one might think that it doesn't arise in finitary contexts. We study whether it does. It turns out that the issue depends on how the finitistic approach is formalized. On one of them, proposed by Marcin Mostowski, all the paradoxical sentences simply fail to hold. This happens at a price: the underlying finitistic arithmetic itself is $\omega$-inconsistent. Finally, when studied in the context of a finitistic approach which preserves the truth of standard arithmetic, the paradox strikes back - it does so with double force, for now the inconsistency can be obtained without the use of uniform disquotation or the $\omega$-rule. This is joint work with Rafał Urbaniak from the University of Gdańsk.

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I will survey some well-known and some more recent undefinability results about models of Peano Arithmetic. I want to contrast first-order undefinability in the standard model with a much stronger notion of undefinability which is suitable for resplendent models, and use the results to motivate some more general questions about the nature of undefinability.

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In their technical report “Alien Intruders in Relevant Arithmetic,” Robert Meyer and Chris Mortensen explored models of relevant arithmetic including nonstandard numbers and proved an “Alien Intruder Theorem” that there are models of relevant arithmetic R# in which all rationals exist and act as natural numbers. They observed some “magical” phenomena about these models, like the fact that induction holds of these rational numbers, but did little to explain them. In this talk, I will show how techniques from ultraproduct constructions reveal some of the reasons for these “magical” features, which help demystify some of Meyer and Mortensen’s observations. This is joint work with Elisangela Ramirez at UNAM.

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In this talk we discuss important results concerning the automorphism groups of countable recursively saturated models of PA and automorphism groups of the countable boundedly recursively saturated models of PA which are short (aka short recursively saturated models). We compare and contrast and also list some open questions.

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In this talk we investigate generic automorphisms of countable models. Hodges-Hodkinson-Lascar- Shelah 93 introduces the notion of SI (small index) generic automorphisms which are used to show the small index property. Truss 92 defines the notion of Truss generic automorphisms. We study the relationship between these two types of generic automorphisms.

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In the 1970s, the logician and philosopher Robert Meyer proposed a novel response to Goedel's Incompleteness Theorems, suggesting that perhaps the results' impact could be blunted by analyzing Peano arithmetic with a weaker deductive system. Initial successes of the program of relevant arithmetic were positive. E.g., R# (the theory of Peano arithmetic under the relevant logic R) can be shown consistent in the sense of not proving 0=1 and this can be shown through arguably finitistic methods. In this talk I will discuss the rise and fall of Meyer's program, detailing the philosophical foundations, its positive development, and the context of Harvey Friedman's negative result in 1992. I'll also suggest why the program, although not necessarily successful, is nevertheless an interesting object of study.

Also note that a great deal of context—including Meyer's two long-unpublished monographs on the topic—have recently appeared in a special issue of the Australasian Journal of Logic I co-edited with Graham Priest, which can be found at https://ojs.victoria.ac.nz/ajl/issue/view/751.

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The use of nonstandard models *N of the natural numbers has recently found several applications in arithmetic Ramsey theory. The basic observation is that every infinite number in *N corresponds to an ultrafilter on N, and the algebra of ultrafilters is a really powerful tool in this field. Note that this notion also makes sense in any model of PA, where one can consider the 1-type of any infinite number.

Furthermore, nonstandard natural numbers are endowed with a natural compact topology, and one can apply the methods of topological dynamics considering the shift operator $x \mapsto x+1$ . This very peculiar dynamic has interesting characteristics.

In this talk I will also present a new result in the style of Hindman’s Theorem about the existence of infinite monochromatic configurations in any finite coloring of the natural numbers. A typical example is the following monochromatic pattern:
a, b, c, $\ldots$ , a+b+ab, a+c+ac, b+c+bc, $\ldots$ , a+b+c+ab+ac+bc+abc.

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Harvey Friedman shows that, over Peano Arithmetic as base theory, the consistency statement for a finitely axiomatised theory can be characterised as the weakest statement that, in combination with the base, interprets the given theory. Thus, we have a coordinate-free characterisation of these consistency statements modulo base-theory-provable equivalence.

Let us call a base theory that, in analogy to Peano Arithmetic, has such weakest extensions: Friedman-reflexive. We call such a weakest statement the interpreter of the finite theory. Interpreters are not always consistency statements, but they are still 'consistoids'.

Which theories are Friedman-reflexive and what more can we say about their consistoids and the associated provability-like notion? We will sketch some preliminary insights. (E.g., all complete theories are Friedman-reflexive.)

We discuss Friedman-reflexive sequential base theories. We introduce an example of an attractive very weak base theory that shares many properties with Peano Arithmetic, to wit Peano Corto. We have a look at what Friedman-reflexive sequential theories look like in general. It turns out that they may look somewhat different from Peano Arithmetic and its little brother Peano Corto.

Given an interpretation $K$ of a Friedman-reflexive base $U$ in a finitely axiomatised theory $A$, we can define an analogue of provability logic: the interpreter logic of $A$ over $U$, relative to $K$. All interpreter logics satisfy K4, aka the Löb Conditions. Two theories are irreconcilable iff they do not have finite extensions that are mutually interpretable. If $A$ and $U$ are irreconcilable, then their interpreter logic relative to $K$ contains at least Löb’s Logic. If one of $A$ or $U$ is sequential, then $A$ and $U$ are irreconcilable.

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