# Second-Order Logic and Foundations of Mathematics

@article{Vnnen2001SecondOrderLA, title={Second-Order Logic and Foundations of Mathematics}, author={Jouko A. V{\"a}{\"a}n{\"a}nen}, journal={Bulletin of Symbolic Logic}, year={2001}, volume={7}, pages={504 - 520} }

Abstract We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically…

## 119 Citations

### Second‐Order Logic and Set Theory

- Philosophy
- 2015

This paper argues that it should be thought of first-order set theory as a very high-order logic, and compares the two approaches to second- order logic, evaluating their merits and weaknesses.

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- PhilosophyEpistemology versus Ontology
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The question, whether second order Logic is a better foundation for mathematics than set theory, is addressed and it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference.

### Higher-order Logic Reconsidered

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- 2007

This paper two arguments are given against the suitability of using second-order consequence as the consequence relation of axiomatic theories, and it is argued that, unless it is viewed as an applied branch of set theory, its justified use requires embracing some strong form of philosophical realism.

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- 2015

I defend here the view that (full) second order logic, if considered as a foundation of mathematics, is best understood as a fragment of set theory. This view is very common among set theorists, and…

### Categoricity and Consistency in Second-Order Logic

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It is argued that categorical characterisation of mathematical structures in second-order logic is meaningful and possible without assuming that the semantics of second- order logic is defined in set theory.

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This paper investigates the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses and characterize the strongest logic in each hierarchy.

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This paper examines how faithfully foundational theories can represent intended structures, and argues that this sheds light on the trade-off between expressive power and meta-theoretic properties when comparing first-order and second-order logic.

### Axiomatizations of arithmetic and the first-order/second-order divide

- PhilosophySynthese
- 2014

It is argued that the first-order versus second-order divide may be too crude to investigate what an adequate axiomatization of arithmetic should look like, and that, insofar as there are different, equally legitimate projects one may engage in when working on the foundations of mathematics, there is no such thing as the One True Logic.

### Second-order logic : ontological and epistemological problems

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In this thesis I provide a survey over different approaches to second-order logic and its interpretation, and introduce a novel approach. Of special interest are the questions whether (a particular…

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