Questions tagged [axioms]

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

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Would unrestricted comprehension and no regularity avoid Russel's paradox with this modification?

Suppose the Set theory in question followed all axioms of ZF or ZFC, except for the axiom of regularity. Additionally the axiom schema of specification was altered from $$\forall z\forall w_1 \forall ...
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Euclidean geometry axioms

What are the best axioms to have in account before start exploring Euclidean geometry? I know that there are Birkhoff and Hilbert axioms. There are more? Which is the best?
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Probability axioms does not make sense?

Assume a unit square to be sample space (infinite points inside it being its elements). Let the points are $\{p_1, p_2, ...\}$ then, by probability axioms, $$1 = Pr(p_1 \cup p_2 \cup \cdots ) = Pr(\{...
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Proving that $\mathbb{Z_{-}} \cap \mathbb{N}=\emptyset$

In assumption that $\mathbb{N}$ and successor function ($\overline{x}$) over $\mathbb{N}$ is defined by 5 Peano axioms: $1\in\mathbb{N}$ $n\in\mathbb{N} \Rightarrow \overline{n}\in\mathbb{N}$ $\...
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Philosophy of maths [closed]

It is generally accepted that the validity of a theory in mathematics depends upon the initial assumptions (axioms) and the logical framework which is applied upon it. But if a new-born baby were to ...
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Without the ZF axiom of regularity can any infinite sets be constructed?

Update with Direct Question Based on Asaf's comments, here is a related question: Prove that the mapping $n \mapsto n \cup \{n\}$ on the set $\Bbb N$ is injective without the axiom of foundation. ...
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Group theory with one more axiom [closed]

I am looking for the answer to the following question. How many models (and which ones) have (accurate to isomorphism) the group theory with an additional axiom? $\forall a, \forall b, \forall c, \...
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59 views

What constitutes a rigorous proof, and can intuitive explanations be made rigorous?

It seems that there is a marked discrepancy between how one might explain a concept in mathematics, and how one might prove it. Obviously, there are some exceptions, and some particularly elegant ...
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Are $X$ and $Y$ the same in these two equivalent formulations of the axiom of choice?

I'm having troubles understanding how the following definition is a possible version of the axiom of choice: Let $X,Y$ be sets and $f:$ $X$$\,\to\,$$Y$ a surjective function, then a function $g:$ $Y$$\...
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Can a regular directed graph be axiomatized in first order Logic?

Given the first order edge relation $E$ is it possible to axiomatize directed graphs where every vertex has an equal amount of incoming and outgoind edges, including graphs with infinite edges?
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Show equivalence with BPI: every Boolean algebra has a prime ideal

I want to prove that the following statements are equivalent: (1) Every non-trivial Boolean algebra has a prime ideal. (2) In every non-trivial Boolean algebra, every ideal is contained in a prime ...
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Is the following provable in Zermelo–Fraenkel Set Theory?

Zermelo–Fraenkel Set Theory is a system of axioms for describing set theory. For example, Zermelo–Fraenkel Set Theory says things like: For any set $x$ and any set $y$ there exists a set $z$ such ...
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What axioms are required to imply the existence of all conceivable subsets?

In ZF, the Axiom of power set implies that for any set, there exists a set that contains all sets known to be subsets of such. However, as noted by the answer to this question: It doesn't guarantee ...
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47 views

Are abstract definitions in math still subject to our understanding of concrete objects? To what extent?

It seems like in mathematics there are two separate meanings of the word axiom: One is like the real or natural numbers, whose axiomatization is based off on concrete objects. The other is like ...
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If a set theory allowed the existence of $x\cup y$, would that imply the existence of $\cup S$? [duplicate]

Suppose the set theory we are working with implies that $\forall x \forall y \exists(x \cup y)$. Would that implication alone further imply the existence of an arbitrary union?: $$\forall S\exists(\...
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Theory of the von Neumann hierarchy

The sets $V_\alpha$ in the cumulative von Neumann hierarchy, defined by transfinite induction: $\begin{align} V_0 &= \varnothing \\ V_{\alpha+1} &= \mathcal{P}(V_\alpha) \\ V_\lambda &= \...
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1answer
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I am losing my motivation to study math, what should I do? (philosophy of axioms and formal math) [closed]

TLDR; To summarize, the emphasis in most real math courses seems to be on proof-based things about seemingly esoteric or abstract structures which is, sadly, making me lose my love for math and ...
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Are Real Numbers a Formal System?

I don't know a lot of mathematics but I have noticed that every branch of Mathematics has the same structure: some axioms (For example in Geometry might be Euclid's Axioms, in Probability might be ...
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2answers
42 views

Transforming Hilbert-style Axiom Systems for Classical Propositional Logic and Retaining Soundness and Completeness

First off, I will use ~ for negation, & for conjunction, V for disjunction, -> for implication, and <-> for bi-conditional. To the question: The axioms of classical propositional logic (CPL) ...
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Meta-logic of Hilbert-style propositional calculus

I am trying to study the foundations of mathematics from the bottom-up (propositional logic then predicate logic then the axioms of set theory.) Currently, I'm considering the following Hilbert-style ...
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Could we ever hope to integrate all functions?

The Riemann integral has a weakness, in that it cannot integrate many functions of interest, such as Dirichlet's function $\boldsymbol{1}_\mathbb{Q}$. The Lebesgue and Henstock-Kurzweil integrals ...
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1answer
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Extending the Axioms of Set Theory, and the Continuum Hypothesis

I've been listening to some talks about the continuum hypothesis and I have some questions regarding how we are working on this problem. A particular talk of significance is this one. Here, Woodin ...
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Why do we even need axiom schema?

It has already been shown that a parameter-free form of ZFC is as strong as ZFC with parameters (hence it is not necessary to have a separate axiom for every list of parameters). Even so, ...
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2answers
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Why does multiplication always have the associative and commutative property?

We all know that Commutative property and Associative property of multiplication is always saved for the real and complex numbers. I know that if I recalculate it a million times, the result will be ...
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Has anyone ever proposed a relational mathematical universe?

We know from Gödel's incompleteness theorems that any significantly powerful axiomatic system is either inconsistent or incomplete, and we even have a few examples where the ZFC comes short. As you ...
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If $P(A)$ is a set, $A$ is a set? [closed]

I know from a ZFC axiom that if $A$ is a set then $P(A)$ (the powerset of A) is a set. Is it possible to prove the converse? Or is it an independence property?
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1answer
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Is multiplication of denominators in fractions a consequence of the associativity law of multiplication?

$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$. If at first we divide only by b, and then their result by d, It is clear to me. I do not understand why if we divide directly by bd, it will give the same ...
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In $\mathsf{ReS}$, prove that if $x$ admits a double well-ordering, then $x$ is in bijection with some member in $\omega$

I'm reading the paper "Weak Systems of Gandy, Jensen and Devlin" by Mathias. The $\mathsf{ReS}$ consists of this: Extensionality, Null set, Pairing, Union, Set difference, $\Delta_0$ Separation, $\...
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Why is modus ponens not an axiom?

Yes, it is a rule of inference and not an axiom. But how does it not qualify as an axiom? What specifically is the disqualifying feature?
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For any statement independent of $\mathsf{ZFC}$, can we prove it is independent of $\mathsf{ZFC}$? [duplicate]

Gödel's famous incompleteness theorem implies, in particular, that there are statements unprovable in $\mathsf{ZFC}$. This implies that we could never hope to settle the truth of every mathematical ...
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Are mathematical operations axioms?

Are mathematical operations axioms? I will give an example of multiplication, but this also applies to division, subtraction and addition. Idea of multiplication was invented by people to increase/...
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What motivated the use of $\sigma$-algebras in measure theory (or probability)?

In the definition of a measure space (or a probability space), the measurable sets are required to form a $\sigma$-algebra. That is, they must be closed under complements and countable unions (and ...
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1answer
337 views

Is there a system of mathematics where $4>2$ is false?

A recent question on propositional logic posted on Philosophy Stack Exchange yielded an answer which states, in part, that, The fact that $4$ is greater than $2$ is not a "logical fact" but and [...
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1answer
100 views

How do we know there isn't a Russell-like paradox in ZFC under Classical Logic?

From what I've understood, Naive Set Theory does not mix very well with classical logic because it eventually boils down every ...
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Why some mathematicians axiomatized number sets?

There was sense of natural numbers before Peano. Likewise why did Peona axiomatized Natural Numbers? Or later, why some mathematicians did axiomatize the Real Numbers or Complex Numbers? And before ...
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Show that the decidability of a statement is undecidable.

In this system of axioms, it can be shown that A = 5 is undecidable Axioms: A is a prime number, A is greater than 4, A is less than 8 This is because both A = 5 and A = 7 are consistent with these ...
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finite axiomatizability of first order theories?

If we consider a single schema as a single axiom, so ZFC for example would be finitely axiomatizable after this kind of counting axioms. By schema its meant a syntactical expression (string of ...
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Proving a statement is true by proving it is undecidable.

https://www.youtube.com/watch?v=O4ndIDcDSGc I was watching this numberphile video in which Professor Du Sautoy mentions that if the Riemann hypothesis is undecidable, it must be true. Let's say we ...
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Are the axioms of ZFC set theory considered to be synthetic?

Pretty straight forward questions. I assume that ZFC would be considered a logicist program if its founding axioms were logical and analytic in nature, yet ZFC is often referred to as "extra-logical"....
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Well-foundedness axiom alternate characterisation:

Show that the following two statements are equivalent: (1) $\forall x \neq \emptyset:\forall X \subseteq x: (X \neq \emptyset \implies \exists y \in X:\forall z \in X: z \notin y)$ (I.e. $\in$ is a ...
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How is self-reference usually handled in math?

Define statements A and B: A: “this statement” is Y B: “this statement” is X Assume I may combine statements using the logical operator “and”. Define statement C: C: A and B. (Meaning “C asserts ...
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Proof of the existence of $\varnothing$ and $\{\varnothing\}$ [closed]

I would like to know a proof of the following statements from pure ZF axioms: $\exists x\forall y (\lnot y\in x)$ $\exists x\forall y [y\in x \Leftrightarrow\forall z (\lnot z\in y)]$ The reason ...
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Please check my proof of 'Show that the third axiom cannot be deduced from the other two.'

I have been trying to answer the question "Show that the third axiom cannot be deduced from the other two.", where the first axiom is $p \Rightarrow(q \Rightarrow p)$ , the second axiom is $(p \...
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How can we show that if $|x| \le 1/n$ for all natural numbers, n, then $x = 0$?

I was thinking about how to define the real number system axiomatically, and can't find anywhere a proof that $$\left[\forall n \in \mathbb{N}\left(|x| \le \frac{1}{n}\right)\right] \Rightarrow [x = 0]...
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Is every math statement exponential in worst-case to prove?

[1] All math axioms can be expressed as grammar rules like the following: A -> B (directed) A <-> BC (undirected) See ...
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On the meaning of the Axiom Of Regularity in terms of “ repeatedly taking the union” .

"whenever I try to chase down a chain of members, it must stop at some finite stage. You can think of it in this way. We have a string of sets $x_1,x_2,x_3...$ where each is a member of the preceding ...
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Why is the simplest set satisfying The Axiom of Infinity the von Neumann ordinals?

The Axiom of infinity is colloquially defined as: There exists a set X having infinitely many members (see Wikipedia) In the language of first-order logic, it's convention that the following ...
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Can all math be done with axioms of length 3 or less?

[1] Assume math can be done with unrestricted grammars (See Are axioms in math equivalent to production rules in unrestricted grammars?). Can the grammar rules be rewritten to contain at most 3 ...
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Axiomatic Independence and Truth

So let the sentence $s$ be independent from an axiomatic system $A$ (like PA or ZFC), i.e. neither $s$ nor $\lnot s$ is provable there. Now I read that $s$ is true in case $s$ = $\forall x Px$ and $s$ ...
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Are axioms in math equivalent to production rules in unrestricted grammars?

In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—...

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