axiomatizations love

# axiomatizations

## Definitions

• noun Plural form of axiomatization.

## Etymologies

Sorry, no etymologies found.

## Examples

• As Halmos (1956) noted, these new algebraic logics tended to focus on studying the extent to which they captured first-order logic and on their universal algebraic aspects such as axiomatizations and structure theorems, but offered little insight into the nature of the first-order logic which inspired their creation.

• It's also the conception which lay behind the 19th-century results that arithmetic and analysis can be given categorical (second-order) axiomatizations.

• On the algebraic conception of set theory, basic set-theoretic notions are characterized by looking at the model theory of first-order axiomatizations of set theory.

• It's important to emphasize that this analysis also explains why Skolem's Paradox doesn't introduce contradictions into various forms of axiomatized set theory, even when these axiomatizations are themselves understood formalistically or model-theoretically.

• Of course there will still be challenges here: we need to account for the status of the background theories in which we prove the LÃ¶wenheim-Skolem theorems, we need to explain the special significance of first-order axiomatizations of set theory, and we may need to explain how we can identify elements across various models of set theory.

• For constructivists, therefore, as for those who are willing to countenance second-order axiomatizations of set theory,

• Bayesian epistemology did not emerge as a philosophical program until the first formal axiomatizations of probability theory in the first half of the 20th century.

Bayesian Epistemology

• Condensed detachment has been used extensively to refine axiomatizations of various implicational logics, especially, in search for shorter and fewer axioms.

Combinatory Logic

• This may be due in part to the fact that in Hilbert-style axiomatizations of number theory, computation is reduced to proof in Peano Arithmetic.

Philosophy of Mathematics

• After the first forty years, the by-products of the paradoxes included axiomatizations of set theory, a systematic development of type theory, the foundations of semantics, a theory of formal systems (at least in nuce), besides the introduction of the dichotomy predicative/impredicative which was important for conceptual reasons, but also for the future of proof theoretical methods.