# Questions tagged [axioms]

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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### Showing Euclid's Proposition 30 ("Lines parallel to the same line are parallel to each other") is equivalent to the 5th Postulate.

I am trying to show that the 30th Euclid's proposition, "Straight lines parallel to the same straight line are also parallel to one another." is equivalent to the 5th Postulate: "If ...
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### Is this theory trying to capture the theory of the minimal model of ZFC correctly formalized?

I'm trying to capture theory $T_0$ written by Noah's answer to a prior posting of mine. First we add a constant symbol $\mathcal M$ to the language of set theory. Now we add all axioms of $\sf ID$ and ...
1 vote
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### Is it valid/useful to prove statement $X$ by finding $Y$ such that $Y\to X$ and $\lnot Y\to X$?

Background: Suppose I want to prove theorem X. Typically, I'd have to use a set of axioms $A = \{A_1,A_2, \ldots ,A_n\}$ or previously proved theorems $T=\{T_1,T_2, \ldots,T_n\}$ and consider all of ...
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### Meaning of notation S(x) in set theory

I am a beginner to set theory. axion of specification: To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those element x of A for which S(x) holds. What ...
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### Is $-\vec{v}$ in vector space axioms mean $-1$ multiplied with $\vec{v}$?

Is $-\mathbf{\vec{v}}$ in vector space axioms mean $-1$ multiplied with $\mathbf{\vec{v}}$? I got this doubt while i am solving the question below. Let $V$ be the set of positive real numbers ...
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### Relationship between Axiomatic geometry and Hyperbolic geometry

I have done quite a bit of hyperbolic geometry and euclidean geometry, but one thing remained obscure to me throughout: the connection between axioms and the metrical definition of geometry. As I ...
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### What is the rationale for selecting the value "one" as the probability of sample space?

Among the three axioms of the probability theory, the following is said to be the normalization axiom $$p(\Omega) = 1$$ It states that the probability of the entire sample space ($\Omega$) is equal to ...
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### All possible proofs within an axiom system [duplicate]

I firmly believe people have asked this question, but I didn't manage to find it. So I post it again. I'm wondering, couldn't we make ever true statement within an axiom system that can be proven by ...
59 views

### How is circumscribing a triangle equivalent to the Parallel Postulate?

This question states that one of the statements equivalent to the parallel postulate (Euclid 5) is "Every triangle can be circumscribed". The Wikipedia page on Tarski's Axioms lists three ...
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### Equality in category theory seems poorly defined to me

apologize if this doesn't make much sense, I am self-taught and often I am thinking about things completely wrong, but I am very lost right now. When we consider some generalization of an idea[^1], ...
1 vote
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### Why Hilbert's 24th Problem is unsolvable?

Hilbert's famous 24th problems handles the problem of the simplest possible proof of a mathematical statement, in a nutshell. It is said that there are a few problems with this problem. First of all, ...
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### How to justify the necessity of the Axioms?

I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms. In Tao's book Analysis I, I found very helpful when he points out what ...
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### Mathematical science being a tautology inquiry by Poincare

I was reading a quote from Poincare from his "On the Nature of Mathematical Reasoning" that states: THE very possibility of mathematical science seems an insoluble contradiction. If this ...
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### Consistent and independence of ZFC

My question is very simple but I want to make sure. when we say the statement is independent of ZFC means, we can not this statement true or false by using only the axioms of ZFC. Is that right? The ...
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### Axiomatic derivation - what does instancing an axiom practically entail?

I'm stunned by the chapter in my coursebook (which is in Dutch, so please advice if I am mistranslating any of the terms) about deriving from a system of axioms and derivation rules. The exercise is ...
1 vote
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### If A is a Subset of B, then the closure of A is a Subset of the closure of B.

Conditions Hello, I am trying to prove that $A \subseteq B \implies CL(A) \subseteq CL(B)$. I know how to prove when you define closure of set E as the Intersection of all sets that are closed and ...
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### Prove all 4 axioms of "less than" are necessary (for real numbers)

One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies: For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds. For all $x,y,z \in F$, if $x<y$ and ...
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### Real Numbers Cannot be Constructed: Question about Constructive Mathematics

I got into a discussion with someone stemming from the set of uncomputatble numbers and how they claimed that such numbers like $\pi$ (not uncomputable but you'll see in a second) don't exist. I was ...
1 vote
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### Must axioms be proven to be compatible with other axioms?

As we know, an axiom cannot be proven. It is an assumption that certain properties exist, and then we can learn more interesting properties (theorems) from anything that obeys these axioms. So ...
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### Could we use predicates instead of propositions in the definition of axiomatic system?

An axiomatic system is a finite sequence of propositions a_1,a_2..,a_N which are called axioms 56:23 In the whole lectures, two kind of logics are introduced: Proposition: A variable which is either ...
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### Why is the Axiom of Pairing required?

I first heard of the ZF Aziom of Pairing watching this. I don't get why it is necessary to have an axiom which states that a set exists. Doesn't a set exist simply by virtue of the fact that it has a ...
### Formal proof that $\vdash (\varphi \rightarrow \psi) \rightarrow (\neg \varphi \lor \psi)$.
I have to prove the statement $$\vdash (\varphi \rightarrow \psi) \rightarrow (\neg \varphi \lor \psi)$$ only using first order logical axioms (similar to the ones in the Hilbert System), modus ponens ...