But why think those axioms are true?
Peano axioms, also known as Peano’s postulates, in number theory, five axioms introduced in 1889 by Italian mathematician Giuseppe Peano.
Using the above rules and axioms Lemmon defines four systems.
Any such axioms would be at best redundant and could be inconsistent.
In Gödel’s view, we have compelling intrinsic evidence for the truth of these axioms.
These desiderata (fewer, shorter, and more perspicuous axioms) often pull in different directions.
These axioms are apparently weaker than the usual axioms for conditional probabilities.
Large cardinal axioms are taken (by those who study them) to have certain degrees of intrinsic plausibility.
These axioms would also be “effectively complete” and compatible with all large cardinal axioms.
For this reason, axioms of this form have never been considered as plausible candidates for new axioms.
Given the truth of such axioms, it follows that propositions exist and have the features attributed to them by our axioms.
The second approach to new axioms—which ran in parallel to the approach using definable determinacy—was to invoke large cardinal axioms.
The typical axioms with which one wishes theorems to compare are the fan principle and the bar principle, Kripke’s schema and the continuity axioms.
All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms.
For the rest of this entry we will embrace non-pluralism concerning large cardinal axioms and axioms of definable determinacy and focus on the question of CH.
In this way the two approaches to new axioms—via axioms of definable determinacy and via large cardinal axioms—were shown to be closely related.
There were two main candidates for new axioms that underwent intensive investigation in the 1970s and 1980s: Axioms of definable determinacy and large cardinal axioms.
The close connections between preference axioms and choice axioms can also be employed to construct a preference ordering from a choice function that satisfies certain axioms.
In particular, Hilbert demonstrates the consistency of various sub-groups of the axioms, the independence of a number of axioms from others, and various relations of provability and of independence of important theorems from specific sub-groups of the axioms.
In 1946 he proposed as new axioms large cardinal axioms—axioms of infinity that assert that there are very large levels of the hierarchy of types—and he went so far as to entertain a generalized completeness theorem for such axioms, according to which all statements of set theory could be settled by such axioms (Gödel 1946, 151).
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In 1946 he proposed as new axioms large cardinal axioms—axioms of infinity that assert that there are very large levels of the hierarchy of types—and he went so far as to entertain a generalized completeness theorem for such axioms according to which all statements of set theory could be settled by such axioms Gödel 1946 151