Abstract
The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, 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 at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.
Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme.
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Notes
- 1.
The set of sets that have hereditary cardinality < κ i.e. are included in a transitive set of cardinality < κ.
- 2.
Thanks to the Löwenheim–Skolem Theorem of \({L}_{{\omega }_{1}{\omega }_{1}}\).
- 3.
For example, its Hanf number can be bigger than the first measurable cardinal (Kunen 1970).
- 4.
Joint work with Moshe Vardi.
- 5.
Take the sentence which says that < is a well-order of the universe such that every initial segment is of smaller cardinality than the whole universe. Add a conjunct stating the existence of an element a and a binary relation E such that a is the least limit element of < but not the least element of < , \(\forall x\forall y(\forall z(zEy \leftrightarrow zEy) \rightarrow x = y)\) and ∀X ∃y ∀x(N(x) → (X(x) ↔ xEy)).
- 6.
We use the symbols {2 and }2 to denote the usual set formation symbols { and } in the sense of the epsilon symbol ∈ 2.
References
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Väänänen, J. (2012). Second Order Logic, Set Theory and Foundations of Mathematics. In: Dybjer, P., Lindström, S., Palmgren, E., Sundholm, G. (eds) Epistemology versus Ontology. Logic, Epistemology, and the Unity of Science, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4435-6_17
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DOI: https://doi.org/10.1007/978-94-007-4435-6_17
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