# axioms's questions - English 1answer

972 axioms questions.

### What exactly is an axiom, and can an axiom be proven?

I've come here after reading about proving SAS congruence, where the response I read was about how it was an axiom. I'm not sure, however, what exactly is an axiom. And can it be proven? And if it ...

### Why is SAS congruence an axiom?

Why is SAS congruence an axiom? What does it being an axiom even mean? Does it mean it can't be proved; if so, why? And if it can be proved, can you explain to me how SAS can be proved? Can you please ...

### 3 Is every proposition on Cartesian geometry provable on synthetic Euclidean geometry?

0 answers, 41 views geometry logic euclidean-geometry axioms
Obviously everything that is associated with coordinates can’t be analyzed within synthetic geometry. But existence, measure and incidence statements are provable; since Cartesian geometry is an ...

### 2 Choosing axiom schemes for a logical theory

In a Hilbert system, there are many ways that we can choose axiom schemes. My question is: 1- How do we know that we have defined enough schemes? What would happen if I remove a scheme from the list? ...

### Doubts about cartesian product existence.

Why I can't to show that the cartesian product between two sets exists without replacement or power set axioms?

### 5 The Deep Structure of the Real line

1 answers, 126 views logic set-theory intuition axioms forcing
For the sake of brevity, throughout this post I will identify real numbers with subsets of $\mathbb{N}$. The question that I want to ask here is more heuristic than definite; I want to understand the ...

### 1 $\mathbb{R}^2$ and the axiom of choice [duplicate]

3 answers, 69 views set-theory axiom-of-choice axioms
Is choice required to guarantee that $\mathbb R^2 := \mathbb R \times \mathbb R$ – or $\mathbb R^n := \displaystyle\prod_{k=1}^n \mathbb R$ in general – isn't the empty set $\varnothing$? If not, what'...

### Axiom of dependent choice in König's lemma

0 answers, 24 views proof-explanation axioms
Consider proof 1 from König's lemma from wikipedia: https://proofwiki.org/wiki/König%27s_Tree_Lemma At the end, they say they apply the axiom of dependent choice. I wonder on which set we apply the ...

### 1 A finite axiomatization of $\mathbb N$ and two non-standard models

In my book Complexity Theory by C. Papadimitriou he talks about first order axiomations of $\mathbb N$ and non-standard models. But what I do not get is that his examples of non-standard models did ...

### 3 The axiomatic method to real number system VS the constructive method(genetic method)

According to book Georg Cantor: His Mathematics and Philosophy of the Infinite - Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...

0 answers, 22 views euclidean-geometry axioms
1.While reading questions and answers on this forum, I read that the fact that Hilbert's axioms are built upon second-order logic is kind of disadvantage, but why ? 2.I heard that we can build ...

### 1 Is Non-Euclidean geometry really “Non”?

1 answers, 86 views geometry axioms noneuclidean-geometry
The definition of a straight line according to google. I do not understand why I call these geometries "non-Euclidean". In my view, both hyperbolic and elliptical geometry are just a dimensional ...

### 1 When to write something in terms of axioms?

3 answers, 118 views axioms article-writing
Most mathematical structures are defined according to axioms. e.g. we state: Definition. Monoid. A monoid is a tuple $(S,\cdot)$ where $\cdot$ is a binary operation $S\times S\to S$ that satisfies ...