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mpitluk
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Can someone point me to a first-order axiomatization of ZFC?
As I've mostly seen ZFC expressed in higher-order logics.
As I've mostly seen ZFC expressed in higher-order logics.
The higher-order axioms are replaced with axiom schema. e.g. the axiom schema of subsets is collection of statementsmpitluk said:Can someone point me to a first-order axiomatization of ZFC?
As I've mostly seen ZFC expressed in higher-order logics.
An actual first-order formulation of ZFC refers to a specific way of expressing the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) using only first-order logic. This means that all the axioms and theorems of ZFC can be written in a language with quantifiers (such as "for all" and "there exists") and variables, without the use of set theory or higher-order logic.
Having a first-order formulation of ZFC allows for a more precise and rigorous treatment of set theory. It also allows for easier comparisons and connections between different mathematical theories that use first-order logic.
An actual first-order formulation of ZFC differs from other formulations of ZFC in that it only uses first-order logic, while other formulations may use set theory or higher-order logic. Additionally, some formulations may have different sets of axioms or different ways of expressing the axioms, but they are all equivalent to each other.
One application of an actual first-order formulation of ZFC is in the foundation of mathematics, as it provides a formal and precise language for discussing sets and their properties. It is also used in many other areas of mathematics, such as mathematical logic, model theory, and proof theory.
While an actual first-order formulation of ZFC is a powerful and widely used tool in mathematics, it does have some limitations. For example, it cannot fully capture the concept of infinite sets, and it is not able to handle certain specialized or non-standard types of sets. Additionally, some mathematicians may prefer to use other formulations of ZFC that better suit their specific needs or interests.