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Questions tagged [axioms]

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

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What is the status of the Axiom of limitation of size? (adrift for almost a century now)

On reviewing the wiki article Axiom of limitation of size I come away with the impression that there are issues surrounding this 'maybe too powerful' principle/heuristic/doctrine/axiom that haven't ...
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How to describe sigma algebra for both conditional and unconditional probabilities

This question is not about random variables and stochastic processes, I'm talking about 101 Kolmagorov Probability definitions. I have a problem with determining a correct probability space for this ...
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3answers
41 views

proof that weak axiom of pairing and axiom schema of specification imply axiom of pairing

as the title says I am trying to give a (nearly, but not fully formal) proof that the weak axiom of pairing (i.e. $\forall x \forall y \exists p: x \in p \wedge y \in p$) together with a suitable ...
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1answer
34 views

Why is “If ψ ∈ Γ then the sequent (Γ ⊢ ψ) is correct” true?

In Chiswell and Hodges Mathematical Logic the authors define a sequent as such "A sequent is an expression (Γ ⊢ ψ) (or Γ ⊢ ψ when there is no ambiguity) where ψ is a statement (the conclusion ...
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1answer
91 views

Reining in the Axiom of Power Set in ZF

Given the powerset operator $\mathit P$, we have the following mapping $\tag 1 \mathcal \Phi: \mathbb N \to \Phi(\mathbb N) $ $\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \...
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2answers
75 views

Example of a set of real numbers that is Dedekind-finite but not finite

Without assuming $AC$, can we find an explicit example of a subset of $\mathbb{R}$ such that it is not finite but it is Dedekind-finite?
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The Next Step (Plane Geometry) After Defining the Family of Distance Measuring Rays in the Cartesian Plane?

We start with $(M,0,+) \left( \; = (\mathbb R^{\ge 0},0,+) \; \right)$, a system of magnitudes with an additive identity. Consider linear isometric mappings $\rho: M \to \mathbb R^2$, where the image ...
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1answer
73 views

finding structures where the singleton axiom, the axiom of extensionality and the axiom of empty set do hold

Hey friends of maths, I am trying to find an example that fulfills the requirements mentioned above, more precisely: I'd appreciate your help with finding two (or better more than 200) non-...
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3answers
70 views

When Can You State that a Given Class is a Set?

Just wondering if the following is of any interest (I am an amateur in these areas, so this might be so much malarkey). Let $\mathcal X$ be a class in ZFC set theory. Let $\mathbb \Phi:\mathcal X \to ...
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1answer
190 views

Excess axioms in the definition of a metric

First, I am aware of how to show the following, my question concerns the reasoning of the standard metric definition. Traditionally, the axioms defining a metric $\rho:X\times X \rightarrow \mathbb{R}...
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1answer
69 views

intuition for PFA

Kunen (Set theory 2011) says on the page 307: The Proper Forcing Axiom (PFA) is the assertion that $MA_{\mathbb P}(\aleph_1)$ holds for all proper $\mathbb P$. My question is, what kind of axiom ...
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63 views

Reduced homology of a point is trivial from axioms

In the section Axioms for homology from Hatcher's Algebraic Topology (page 161) he says: Note that $\tilde{h}_n(x_0) = 0$ for all $n$, as can be seen by looking at the long exact sequence of ...
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1answer
48 views

Taking an unproven but “seemingly true” statement as an axiom

I'm sorry for this uneducated question, but I've been thinking of this for a few hours and I couldn't find anything on the topic. Perhaps it is just a failure on my part and a limitation of my ...
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0answers
33 views

Show that first order Peano's axioms capture the natural numbers regarding satisfiablity

Denote be $\mathcal P_{MO}$ the set of the monadic second order axioms of Peano. Then as shown by Dedekind any two model are isomorphic to $\mathbb N$. Hence for a monadic second order sentence we ...
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5answers
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Basic division and multiplication

We know that 2 x 2 /2 can be solved by removing '2' from the denominator and numerator, we can't do the same if the operation was addition, These "rules" have been established based on the ...
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1answer
34 views

Definition of rational numbers

We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some ...
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1answer
57 views

On independency of ZFC of statements in math problem solving

To be honest, this is a question that has been bothering me (provided that I understand this correctly), which probably means I'm not quite sure about what I ask, but I'm looking for some ...
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1answer
65 views

Alternative axioms for NBG or MK

While I was thinking about NBG and MK I had the idea for two alternative axioms. As usual $V$ is the class of sets. The first one: For a boolean function $f : \{T,F\}^n \to \{T,F\}$ let $\varphi_f(...
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2answers
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How badly does foundation fail in NF(etc.)?

The strongest antifoundation axiom I know is due to Boffa. Roughly, it asserts that every graph which could represent a set, does. For example, considering a graph consisting of (a root connected to) ...
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43 views

Can we state the existence of infinite set without infinity axiom? [duplicate]

I have a question about infinity axiom in ZF and maybe, it has nonsense. So I apologize in advance if it is the case. In ZF, the infiny axiom can be state as $\exists X(X\neq\emptyset\wedge\forall x\...
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1answer
52 views

What are the base assumptions we make in mathematics?

In any proof, we establish (or attempt to establish) a fact based on previously established facts. Each of these previously established facts in turn must have been established using a set of then pre-...
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Categorical Axiomatization of the theory of Dedekind complete ordered fields.

I am looking for a proof of the fact that the axiom system of ordered fields extended by the Dedekind completeness axiom is categorical. It should be a book or journal which I can cite.
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$\neg (\neg A) \rightarrow A$ part of the axioms of propositional logic? [duplicate]

When talking with a mathematics teacher the other day, we discussed these axioms in the context of proving tautologies with semantic tableaux: $(p\to(q\to p))$ $((p\to(q\to r))\to((p\to q)\to(p\to r))...
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Associativity axiom expressed in first-order logic with predicates only

I've been doing work with axioms, in particular representing common axioms using predicates only, where the predicates work like this: $$*(x,y,z) \ \text{true}\ \Leftrightarrow x * y = z$$ Many ...
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$\text {dom}(R)$ and $\text {ran}(R)$ exists for any definition of order pair.

Suppose that we define, for any sets $x,y$, a set $(x,y)$ with the propertie that $$(x,y)=(x',y')\rightarrow x=x'\wedge y=y'(*)$$ Let $R$ be a set. I want to show that the classes $\{x:\exists y((x,y)...
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Difference between axiomatizations of FOL in Blok and Pigozzi and in Andréka et al.

In the axiomatization of FOL given by Andréka, Németi, and Sain in their "Algebraic Logic" (2001, p. 224), they give the following axioms (with $i, j, k$ ordinals $< \alpha$ and $\phi$ and $\psi$ ...
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63 views

Creating a 'sequence' of subsets using the subset axiom, then using induction to prove they are all equal

Setup: $(N,\sigma)$ satisfies Peano axioms, i.e. $N \equiv ω$. Using the subset axiom we define, for each $k \in N$, $F_k = \{x \in N \; | \; \phi_k(x) \}$. Next we define $D = \{d \in N \; | \; ...
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1answer
51 views

How to study Euclidean geometry from axioms?

I want to know if there's a good book or any other type of guide to study Euclidean geometry by only the 5 axioms in plane geometry and prove every other theorems from them?
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An introduction book for analysis with the axioms of Tarski

Briefly: Is there an introduction book for analysis with The axioms of Tarski as basis? (I found them very elegant. And the most important thing for me is that the axioms are "obvious", not like the ...
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1answer
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vector spaces: verify axioms for 1, random variables and 2, power set of a set over {0,1}

Verify the axioms of a vector space for: The set of all real-valued random variables on a fixed sample space over $R$ The power set of a set $\Omega$ forms a vector spaces over $F=${$0,1$}...
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2answers
38 views

Transform a totally ordered set to a structure that is isomorphic to (R,+,.,≤)

So let $(M,\le_M)$ be a totally ordered set. Can we define $+$ and $.$ to make $M$ isomorphic to $(\mathbb{R},+,.,\le)$? I mean the well known axioms. To let this possible: $M$ is not bounded above ...
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Prove that a universal class $V$ is a proper class [closed]

Prove that $V$ is a proper class, where $V$ is universal class. Try to use these theorems: (1)if $a$ is a set and $b\subseteq a$, then $b$ is a set. (2) Russel class.
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Is there correspondence between vector spaces and independent theories?

I'm not a mathematician, but I have this idea. A vector space is defined by a basis, that is a set of linearly independent vectors. On the other hand, an independent theory is defined by a set of ...
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1answer
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Model of ordered plane with the negation of Pasch's axiom

I am interested in finding the model of particular set of geometry axioms in which Pasch's axiom fails. First I'll give the definitions. By ordered line I mean the set $L$ (line) with one three-...
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Axiomatisation of Infinite Series

I have seen a ridiculous "proof" claiming that $$\displaystyle\sum_{n=1}^{\infty} n=-1/12.$$ The starting point is a false statement that $$\sum_{n=1}^{\infty} (-1)^{n+1}=1/2.$$ Since we know that ...
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1answer
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Proof of $(-a)b=(-b)a$ starting from axioms

Prove from axioms that $ \forall_{a,b\in\mathbb{R}}:(-a)b=(-b)a$ , but $a0=0$ is allowed. My attempt: $LHS=(-a)b=(-a)b+0=(-a)b+(ba+(-ba))=((-a)b+ba)+(-ba)=b(-a+a)+(-ba)=b0+(-ba)=0+(-ba)=-ba$ So, ...
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1answer
104 views

Almost universal class

I so stuck with a problem of set theory. But first a recursive definition: Define $R_0=\emptyset$ If $R_\alpha$ is defined, then $R_{\alpha+1}=\mathcal{P}(R_\alpha)$ (the power set). For a limit ...
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1answer
60 views

Axiom Problems (Intro to Computer Logic)

"Show that—or prove that—$ \Gamma \vdash A $" means "write a $ \Gamma $-proof that establishes $ A $". The proof can be Equational or Hilbert-style. Show that $ A \equiv C \vdash A \rightarrow (B ...
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284 views

Translating Tarski's Axiomatization/Logic of $\mathbb R$ to the Theory of Magnitudes

Update: This has become a project, but I need help. All answers will now be definitions, propositions, theorems, etc. that build on the theory. I will marks some of my own answers as community wiki so ...
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2answers
112 views

Derive simple logical laws in a structure with not and implies

We can define an abstract system with the following three axiom schemes that define $\to$ and $\lnot$ as follows: ax1. $P\to(Q\to P)$ ax2. $(\lnot Q \to \lnot P)\to(P\to Q)$ ax3. $(P\to(Q\to R))\to(...
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1answer
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Redunduncy of Pasch's Axiom of Hilbert's Foundations of Geometry

I am baffled by Hilbert's Foundation of Geometry's Axioms because the relations are not defined. They are only described, not defined. For example, the notions of 'betweeness' or 'congruency' are any ...
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A theorem derived from Axiom Schema of Replacement

My textbook use this theorem implicitly. To make sure that it's indeed valid to use it, i try to give it a shot. Does my proof look fine or contain flaws? Many thanks for you! Axiom Schema of ...
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What does Hilbert's second axiom of connection mean? [duplicate]

Here are the first 2 axioms of connection from Hilbert's Book: I, 1. Two distinct points $A$ and $B$ always completely determine a straight line $a$. We write $AB = a$ or $BA = a$. Instead ...
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2answers
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Deriving least upper bound property from one particular axiom of completeness

Let's have this axiom of completeness for the reals: Axiom: Let $(I_n)$ be a sequence of closed intervals in $\mathbb{R}$ such that $\forall n\in\mathbb{N}:I_{n+1}\subset I_n$. Also let $\lim_{n\to\...
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1answer
67 views

Do we need AC to have a least upper bound property?

In my analysis course, we are considering $(\mathbb{R},+,\cdot,\leq)$ as axiomatically constructed ordered field. Now, together with that, we added a completness axiom stated as follows: Axiom: Let ...
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1answer
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Why do we need the following set to have an infinite set?

By Dedekind's definition: Definition: We say that some set $S$ is infinite if there exists an injection $f:S\rightarrow S$ such that $\operatorname{Im}(f)\neq S$ (or $f(S)\neq S)$. Now, the book ...
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Question about the distributivity axiom of vector spaces

Let $V$ be an arbitrary vector space over $K$. I'm asked to prove that $$a(v_1 + v_2 + ... + v_n) = av_1 + av_2 + ... + av_n$$ $$(a_1 + a_2 + .... + a_n)v = a_1v + a_2v + ... + a_nv$$ for all $a, a_i ...
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1answer
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axiom of extensionality word definition

I was reading about the axiom of extensionality and in words it reads "If A and B are sets such that for every element x, x is a member of A if and only if x is a member of B, then A is equal to B" am ...
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114 views

The most difficult way of proving that a countable union of countable sets is countable

In ZFC I want to prove the following result: Proposition 1: Let $A$ be a set and let ${(G_k)}_{k \in \mathbb N}$ be a (countable) family of countable nonempty subsets of $A$. Then there exist a ...
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What does this ZF+[AOC-Lite] Look Like?

We use the notation $[n] = \{0,1,2,3,\cdots ,n-1 \}$. Remove AOC completly from ZFC and then replace it with Axiom asdf: Let $X$ be any nonempty set such that $\;\text{For every injective } f:[n]\to ...