Articles of axioms

Can all laws be derived from axioms?

I don’t really understand the purpose of an axiom if some laws cannot be derived from them. For example, how is one supposed to prove De Morgan’s laws with only the axioms of probability?

How do we introduce subtraction from these field axioms?

I am familiar with two different sets of field axioms. The first one is from “Mathematical Analysis” by Apostol. It has the first $3$ usual axioms, but the $4^{th}$ one is different: Axiom 1: Commutative Laws $x+y=y+x$, $xy=yx$ Axiom 2: Associative Laws $x+(y+z)=(x+y)+z$, $x(yz)=(xy)z$ Axiom 3: Distributive Law $x(y+z)=xy+xz$ Axiom 4: Given any two real […]

Prove if $x<y$ then $x+z<y+z$

This question already has an answer here: For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$ 2 answers Prove that if $x\leq y$ then $x+z\leq y+z$ 1 answer

How are primitive mathematical objects chosen?

On page of 22 Terry Tao’s Analysis I, he writes (after presenting the Peano axioms): Remark 2.1.14. Note that our definition of the natural numbers is axiomatic rather than constructive. We have not told you what the natural numbers are (so we do not address such questions as what the numbers are made of, are […]

ZF: Difference between POW and SEP?

In ZF, the Power Set Axiom (POW) says that given any set $A$, there exists a set $\mathcal{P}(A)$ such that $$ \forall a(a\in\mathcal{P}(A)\leftrightarrow a\subseteq A)\tag{1} $$ Questions: On many occasions, I’ve seen POW stated like this: “For any set $A$ there is a set consisting of all subsets of $A$.” Would a more precise formulation […]

How does one prove that proof by contradiction has completely and utterly failed?

This question already has an answer here: Is Godel's modified liar an illogical statement? 1 answer

$\exists x ~ \forall y ~ f(x, y) \iff \forall y() ~ \exists x~ f(x, y(x))$ Name? Proof?

I was looking at potential theorems, and this one came up: $$\bigg(\exists x ~ \forall y ~:~ f(x, y) \bigg) \iff \bigg(\forall y() ~ \exists x ~:~ f(x, y(x))\bigg)$$ (where the second $y$ is being quantified over all total unary functions) Does this proposition have a name? It is a generalization of $$(a_{00} \land a_{01}) […]

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the present one. According to Wikipedia(, the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least […]

Why is matrix multiplication defined a certain way?

This question already has an answer here: Intuition behind Matrix Multiplication 12 answers

Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that “every non-empty set $x$ contains a member $y$ such that $x$ and $y$ are disjoint sets.” In particular, how can $y$ be an element of a set […]