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972 axioms questions.

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? 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 ...

In a Hilbert-style system, the axiom schemes can be written as (From Bourbaki, Book I): S1. If $A$ is a relation in $\mathscr C$, the relation $(A\text{ or }A) \Rightarrow A$ is an axiom of $\...

This the exercise 2 of section 6 of chapter 1 of "Set Theory: An Introduction to Large Cardinals", by Frank R. Drake: Show that A4 can be deduced from A3, and that A3 can be deduced from A9. Show ...

In Terence Tao's Analysis, he states The axiom is almost universally accepted by mathematicians. One reason for this confidence is a theorem due to the great logician Kurt Godel, who showed that a ...

I'm learning ZFC set theory and I'm very confused about the Axiom of Pairing. Axiom of Pairing: For any $a$ and $b$ there exists a set $\{a, b\}$ that contains exactly $a$ and $b$. It seems that ...

If two planes α, β have a point A in common, then they have at least a second point B in common. I perfectly understand the axiom, but i don't see why it's necessary, and it's kind of ...

I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$ My question is why is this an axiom? why can't we derive this from the ...

Looking up dictionary definitions of algebra geometry is pretty unsatisfying as they are usually along the lines of "the branch of mathematics in which letters and other general symbols are used ...

Markov's Principle: Let $ x \in \mathbb{R}$. Then the following holds: \begin{align*} \neg (x = 0) \Longrightarrow \vert x \vert >0. \end{align*} In constructive mathematics (no law of excluded ...

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity ...

From the ZFC axiom of regularity, which states that every non-empty set contains an element disjoint from it, we can deduce that there is no set $A$ such that $A \in A$. A proof is outlined here: ...

The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem ...

Are any mathematicians working on finding new axioms, either for ZFC or another foundational theory of math? I know that because of Godel's incompleteness theorem, it's impossible to construct a ...

EDIT: problem solved don't bother reading. In a vector space $V = (V, F)$, the zero product property states that for all $\lambda \in F$ and $\underline{v} \in V$, if $\lambda \cdot \underline{v} = \...

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 ...

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?

Is there an axiomatic characterization of the Lebesgue integral w.r.t. some finite measure $\mu:\mathcal{F}\rightarrow[0,\infty)$, for instance as the function $I$ over the set of real-valued, $\...

The system of second-order $ZFC$, presented in Shapiro, "Foundations without Foundationalism", is formulated in second-order logic and includes the usual axioms of extensionality, foundation, pairs, ...

AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a ...

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...

This question is related to but different from a previous question of mine. I have been looking for an axiomatic set theory that handles classes elegantly from a mathematical perspective. The best I ...

Ηere: Equality and its axioms, the rule of substitution of functions is placed among the axioms of equality, but at the same time, it says "is (equivalent to) a special case of the third schema", that ...

Is there a precise axiomatization of the Eudoxus theory of proportions? For example, a) (D, +, <) is a structure such that < is a strict linear order, b) + is an order-preserving ...

While I was reading the Wikipedia's article dedicated to Hilbert's problems I've known the so-called Hilbert's sixth problem from this page and the corresponding Wikipedia's entry dedicated to this ...

I am trying to self study Set Theory. While studying the ZFC axioms, I am introduced to Russell's Paradox and why the Universal Set does not exist. With the Axiom of Restricted Comprehension, Russell'...

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...

Recently I have been reading about Tarski-Grothendieck set theory, and have been impressed by its short axiomatisation, inclusion of inaccessible cardinals, and capability of supporting category ...

Is the continuum hypothesis (CH) independent of Tarski-Grothendieck set theory (TG)? Own thoughts: I have only very limited knowledge of the different axiom systems. I know that TG implies ZFC. I ...

In his Calculus book, Spivak wants to establish all basic properties of real numbers so that he can prove calculus upon it. But I thought of some properties which Spivak should have also listed. And ...

So pretty basic question http://prntscr.com/jcz64c So I am going through the answer, and I check mine, and I got the first one correct, but I cannot figure for the life in me why the second one fails ...

Generally, the structures that people study in every-day modern mathematics can be seen as defined on sets, so far as I know. (In the sense that they are collections of objects that don't give rise to ...

Let there be a set $\Bbb{N}$ defined by these 3 axioms: There exists a set $\Bbb{N}$ such that $1\in \Bbb{N}$ and a function $s:\Bbb{N}\rightarrow\Bbb{N}$ satisfying these properties: $$\not\exists ...

In the lecture notes of Bilaniuk 2003 the following axiom system is chosen: A1: $(\alpha\to(\beta\to\alpha))$, A2: $((\alpha\to(\beta \to\gamma))\to ((\alpha\to\beta)\to(\alpha\to\gamma)))$, A3: $(((...

The axiom of infinity implies that there exist infinite sets. We can construct the natural numbers without this axiom, but we cannot put them together in a set, as this would violate this axiom. The ...

My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further. Generally, the Real Number System is said ...

I'm confused about the axiom for the additive inverse and zero vector. If I define a vector space with normal addition but scalar multiplication like so: k(x, y) = (kx, 0). One axiom says: There is ...

Please let me ask a theoretical question. According to derivative definition, in accelerating motion (first) derivative at moment t1 is the limit of the function (constructed from position's function) ...

Define exponential function $\exp$ as follows: $$\exp:\Bbb{R}\rightarrow(0,\infty)$$ $$\begin{align}i) \ \forall x,y\in \Bbb{R}:\exp(x+y)=\exp(x)\cdot\exp(y)\end{align}$$ $$\begin{align}ii) \ \forall ...

So, we know that duality principle stands in geometry of projective planes. Do we know some other sets of axioms and the spaces that are generated by them, where the duality also applies? If they ...

I was trying to prove if $l=m$ and $m=n$ then $l=n$ but when doing this I had to add $-m$ to both sides of both equations.i think it is not appropriate to proceed without proving "if $a=b$ then $a+c=b+...

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 ...

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'...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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