VII (e) 91
Abbreviations/Quine: defining abbreviations are always outside of a formal system - that's why we need to get an expression in simple notation before we examine it in relation to hierarchy.
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IX 190
System/Quine: a new system is not introduced by new definitions, but by new distinctions.
((s) Example (s): if I always have to note "n + 1" to mark the difference between real and rational numbers, I did not eliminate the real numbers, but kept the old difference. I only changed the notation, not the ontology.)
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Theory/Enlargement/Extension/System/Quine: an enlargement is not an extension!
Extension: addition of axioms, can create contradictions.
Magnification/Quine: means to relativize an added scheme to already existing axioms of a system, e.g. to "Uϑ", (s) so if something exists in "Uϑ", it must be a set.
Such a magnification never creates a contradiction.
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Theory/stronger/weaker/Quine: if a deductive system is an extension of another in the sense that its theorems include all of the other and others, then in a certain way one is stronger than the other. But this basis of comparison is weak:
1. It fails if each of the two systems has theorems that are not found in the other. (Comparability).
2. It depends on randomness of interpretation and not simply on structural properties.
Example: suppose we would have exactly "=" and "R" as primitive two-digit predicates with an ordinary identity axiom and transitivity. Now we extend the system by adding the reflexivity "x(xRx)".
The extended system is only stronger if we equate its "R" with the original "R". But if we reinterpret its "xRy" as "x = y v x R y" using the original "R", then all its theorems are provable in the non-extended system. (>
Löwenheim, >
Provability),
Example (less trivial): Russell's method ((1) to (4), Chapter 35) to ensure extensionality for classes without having to accept them for attributes.
Given is a set theory without extensionality. We could extend it by adding this axiom, and yet we could show that all theorems of the extended system could be reinterpreted with Russell's method as theorems already provable in the non-extended system.
Stronger/weaker/Quine: a better standard for the comparison of strength is the "comparison by reinterpretation": if we can reinterpret the primitive logical signs (i.e. in set theory only "e") in such a way that all theorems of this system become translations of the theorems of the other system, then the latter system is at least as strong as the first one.
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If this is not possible in the other direction, one system is stronger than the other.
Def "ordinal strength"/Quine: another meaningful sense of strength of a system is the following surprising numerical measure: the smallest transfinite ordinal number, whose existence can no longer be proven in the system.
Any normal set theory can, of course, prove the existence of infinitely many transfinite numbers, but that does not mean that you get them all.
Transfinite/Quine: what is so characteristic about it is that we then iterate the iteration further and iterate the iteration of iterations until our apparatus somehow blocks. The smallest transfinite number after blocking the apparatus then indicates how strong the apparatus was.
An axiom that can be added to a system with the visible goal of increased ordinal strength is the axiom that there is an unattainable number beyond w (omega). (End of Chapter 30).
An endless series of further axioms of this kind is possible.
Strength of systems/Ordinal Numbers/Quine: another possibility to use ordinal numbers for strength: we can extend the theory of cumulative types to transfinite types by accrediting to the x-th type for each ordinal number x, all classes whose elements all have a type below x.
So the universe of the theory of cumulative types in chapter 38, which lacks the transfinite types, is even the ω-th type.
Def "Natural Model"/Montague/Vaught/Quine: this is what they call this type, if the axioms of set theory are fulfilled, if one takes their universe as such a type.
So Zermelo's set theory without infinity axiom has the ω-th type as a natural model. (We have seen this in chapter 38). So the ordinal strength of this system is at most ω, obviously not smaller.
With infinity axiom: ω + ω.
Strength of the system of von Neumann-Bernays: one more than the first unattainable number after w.
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XII 33
Object/existence/system/Quine: systematic considerations can lead us to reject certain objects
XII 34
or to declare certain terms as non-referring.
Occurrence: also individual occurrences of terms. This is Frege's point of view: an event can refer to something on one occasion, not on another (referential position).
Example "Thomas believes that Tullius wrote the Ars Magna". In reality he confuses Tullius with Lullus.
