Articles of axioms

Do the proofes in set theory rely on the semantics of the formulas used in the axioms?

Motivation: The Axiom of separation $$\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \phi(x, w_1, \ldots, w_n, A) ] )$$ is used to guaranty the existence of subsets by constructing formulas $\phi$ which, I assume, say true or false for every […]

Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are: “To draw a straight line from any point to any point.” “To produce [extend] a finite straight line continuously in a straight line.” “To describe a circle with any centre and distance [radius].” “That all right angles are equal to one another.” “That, if a straight line falling on two […]

Axioms for sets of numbers

What is the most common axiomatic system used by modern mathematicans for the properties of the integers, rationals, reals, and complex numbers? Or does one commonly use a single axiomatic system that is agreed apon, to construct axiomatic systems for the other sets of numbers. Such as using axioms of the integers to construct axioms […]

Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?

Like any math newbie, Godel’s Incompleteness Theorems are easy to understand in general layman’s terms, but difficult to understand beyond the typical “liar’s paradox” and “barber’s paradox” type examples. But then I started thinking, are axioms examples of the truths of mathematics that can’t be proven? For example, Peano’s Postulates: a very popular “starting point” […]

Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these axioms, can I show that a contradiction cant be derived from them, such as $0=1$.. etc

Can you prove a random walk might never hit zero without the countable additivity axiom?

Can you prove a random walk might never hit 0, given a probability system that only uses the finite additivity axiom (rather than the standard countable additivity axiom)? Specifically: Imagine the standard random walk problem on the nonnegative integers: We start at integer location $i>0$. Every step we independently move left with probability $\theta$, and […]

What properties are allowed in comprehension axiom of ZFC?

I am trying to understand the axioms of ZFC. The axiom schema of specification or the comprehension axioms says: If A is a set and $\phi(x)$ formalizes a property of sets then there exists a set C whose elements are the elements of A with this property. I read Is the Subset Axiom Schema in […]

Geometric basis for the real numbers

I am aware of the standard method of summoning the real numbers into existence — by considering limits of convergent sequences of quotients. But I never actually think of real numbers in this way. I think of a real number as a vector in 1D Euclidean space, the good old number line. A signed distance. […]

Tarski-like axiomatization of spherical or elliptic geometry

Preamble Tarski’s axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive) and congruence $xy\equiv zw$ (meaning the line segment between $x$ and $y$ is the same length […]

axioms of equality

Every text I’ve come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other’s elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in y) \implies (x=y)\bigr)$$ Some will regard this as an axiom, while some regard this as the very definition of equality (replacing the implication with […]