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? ...

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 ...