Modifying the Field Axioms to Create Infinite and Infinitesimal Real Numbers

Motivated by the algebraic impossibility of division by zero, a new algebraic structure called an ascended field is introduced by modifying the standard field axioms.

An arithmetic on tuples containing elements from ascended fields is developed, leading to the development of a field extension based around the quotient space of an equivalence on elements in an ascended field.

An extended version of this framework is constructed based on a particular subspace of a quotient space containing tuples of elements from ascended fields.

Structure uniqueness, the ordering of elements, and the totality of the basic algebraic operations are thoroughly explored within these quotient spaces.

This culminates in the development of a unique structure called a complete s-extension of the reals that extends the standard real numbers, maintains totality of the basic arithmetic operations for all elements in the extension, and contains what are defined as infinite and infinitesimal real numbers.