Mathematics and Set Theory: A Critical Examination

A theory that allows for the existence of mathematical objects that are not sets, such as numbers, logical concepts, and properties.

A restricted version of ZFU (Zermelo-Fraenkel set theory with urelements) where the vocabulary includes the set-theoretic vocabulary and additional vocabulary from another theory T.

A fundamental principle stating that if a theory T is consistent, and another theory T* is a conservative extension of T, then T* is also consistent. This principle is crucial for establishing consistency results, particularly in mathematics.

A formal system or set of axioms used to prove mathematical theorems. Examples include Zermelo-Fraenkel set theory (ZF), Peano Arithmetic (PA), and set theory with urelements (ZFU).

A formal system or set of axioms that does not allow for the existence of entities outside of sets. Standard mathematical theories often fall under this umbrella.

A model of a theory is any set containing objects that fulfill all the axioms of the theory. Models are used to demonstrate the consistency of a theory.

A cardinality that is inaccessible to a set theory. It reflects the size of a set and is uncountable and unconstructible within the theory.

A method of constructing models for theories by using a set of basic elements and iteratively building new elements by taking subsets. This method is often used to prove consistency results.

The practical usefulness of mathematical entities, such as numbers and shapes, in solving real-world problems or making predictions.

Hypothetical entities in physics, like atoms or quarks, used to explain observed phenomena.

The ability of a theory to explain a wide range of observed phenomena.

The existence of alternative theories that can explain the same phenomena without relying on specific entities.

The argument that the usefulness of mathematical entities provides evidence for their actual existence.

The claim that the utility of mathematical entities is fundamentally different from the utility of theoretical entities.

The idea that alternative theories can be created by simply removing references to specific entities from existing theories.

The act of re-writing a theory using different axioms while preserving the same logical consequences.

Mathematical theories like number theory and set theory that don't allow for elements that aren't sets. These theories are not directly applicable to the physical world.

Mathematical theories that include elements that are not sets, allowing connections to the physical world through functions that map physical objects to abstract entities.

A non-set element that can be a member of a set. This allows for bridging pure and impure mathematical theories.

A mathematical theory that allows for the possibility of urelements and non-mathematical vocabulary, making it more useful for applying to the physical world.

Laws that connect mathematical and physical vocabulary to enable applying mathematical theories to reality.

A theory about subatomic particles formulated without using vocabulary that directly describes observable physical objects.

Mathematical arguments that rely on powerful tools like set theory, which are not necessarily part of the simpler theory we want to prove things about.

The belief that mathematics is essentially about discovering pre-existing truths, rather than creating them.

The view that mathematical concepts are ultimately based on physical objects or concrete experiences.

The philosophical debate about the nature of mathematical entities, whether they exist independently of human thought or are merely mental constructions.

The ability of a theory to make accurate predictions about the real world.

The argument that the practical usefulness of mathematical concepts provides evidence for their actual existence.

Mathematical theories that don't introduce new observable consequences beyond what the underlying theory already implies. This is considered a desirable property for mathematical theories.

The belief that standard mathematics is consistent (free from contradictions) and also conservative. This implies that standard mathematics doesn't introduce any new observable facts about the physical world.

The argument that the existence of multiple theories explaining the same phenomenon without referring to specific entities undermines the claim that the usefulness of entities implies their existence.

A theory that describes the physical world is not conservative because it often introduces new conclusions about observable phenomena beyond what its fundamental principles would suggest.

Mathematical theories are considered conservative because they don't create new observable conclusions about the physical world. They are designed to work internally and consistently, rather than making new claims about physical reality.

The claim that standard mathematics is conservative is supported by the fact that it hasn't led to any demonstrably false predictions or inconsistencies regarding observable phenomena.

The idea that mathematical theories, unlike physical theories, are considered conservative because they don't introduce new observable conclusions about the physical world. They focus on internal consistency and logical reasoning rather than making predictions about physical phenomena.

A mathematical theory is considered conservative if it does not yield genuinely new conclusions about non-mathematical entities when applied to the real world.

Standard set theory (ZF) is considered conservative because it can be used to represent objects in the real world without introducing entirely new conclusions. This means that its usage does not lead to new knowledge about the physical world.

ZFU (Zermelo-Fraenkel set theory with urelements) allows for the existence of objects that are not sets (urelements). This enables the theory to represent non-set-theoretic entities. However, it is crucial that the comprehension axioms only deal with set-theoretic vocabulary.

The principle of conservativeness states that if a theory T is consistent, and another theory T* is a conservative extension of T, then T* is also consistent. This principle is crucial for proving consistency results in mathematics.

The assumption of conservativeness in set theory is very close to the assumption of its consistency. This implies that proving the conservativeness of mathematics is crucial to understanding its reliability.

Inaccessible cardinals are large cardinals whose existence is not provable within standard set theory. They possess special properties related to their set-theoretic structure. Adding the axiom of inaccessible cardinals to set theory strengthens it and helps to prove the conservativeness of ordinary set theory.

Mathematical theories that don't allow for elements that aren't sets are considered pure. These theories, like number theory and set theory, are not directly applicable to the physical world.

Mathematical theories that include elements that are not sets are considered impure. This allows them to establish connections with the physical world through functions mapping physical objects to abstract entities.