Questions tagged [axioms]
For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.
1,890 questions
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Does definition of empty set use universal specification
In chapter 3 of Analysis I by Terence Tao, the following definition of empty set is given:
(Empty set). There exists a set $\phi$, known as the empty set, which contains no elements, i.e., for every ...
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Shouldn't all possible sets be covered by ZFC axioms? [closed]
Are there models of ZFC where Continuum Hypothesis (CH) fails? The answer should be yes as if CH were true in all models then it should be provable from ZFC axiom but we know that it is independent of ...
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Axiomatization similarities between Naturals and Magnitudes
Given certain Peano-like axiomatizations of the naturals (but not all of them), the axioms of Well-ordering and Induction are equivalent in that you could use either one and get the same system.
Well-...
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Completeness of the Euclidean Postulates
In The Elements, Euclid states the postulates of geometry. The Euclidean postulates originally state that:
A line is defined by two points as the points whose difference in distances to the two ...
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Proving arbitrary products exist in a Mac Lane universe
I am going through some leftover exercises from Mac Lane's Category Theory for the Working Mathematician. I am currently dealing with the foundational chapter, in which the concepts of small and large ...
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Proof verification and axioms of the integers and order
Axioms of integers:
(ZE) The set of integers is non-empty.
(AE) The binary operation of integer addition is well-defined and exists and it tells us that the set of integers under addition is closed.
(...
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What real numbers correspond to second order logic definable Dekekind cuts?
This question spurred from a thought I had: does every (lower) Dedekind cut have a (finite) second order logic formula that defines it?
Fix the usual setting: the domain is $\mathbb{Q}$ with the order ...
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Is the the nested interval property equivalent to Cauchy criterion without Archimedean property?
Recall the following properties of the reals.
Nested interval property. For every sequence $I_1, I_2, I_3,\ldots$ of closed and bounded intervals satisfying
$$I_1 \supseteq I_2 \supseteq I_3\supseteq \...
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Proofs using real number field axioms
Questions :
Let $(\mathbb{R}, +, \cdot]$ be a field.
Using only the field axioms, prove the following:
$(-1)(-1) = 1$
For all $a,b \in \mathbb{R}$:
(2.A) $-(a+b) = (-a) + (-b)$
(2.B) $(-a)(-b) = ab$
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With ZFC foundations, if all math objects are sets, where do these sets "live"?
Let's say we're doing ordinary mathematics, and we want ZFC to be our foundations, such that all of our mathematical objects are sets.
I have long had the idea in my head that these sets that ...
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Are these versions of Axiom of Infinity Equivalent? [duplicate]
Does this statement:
Axiom of Infinity 1: $$\exists X \Big(\emptyset\in X \land \forall x\in X (\{x\}\in X)\Big)$$
imply this statement
this Axiom:
Axiom of Infinity 2: $$\exists X \Big(\emptyset\...
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Kunen Set Theory; Importance of $\text{rank}(x)$ being independent from Power Set
In Kunen's Set Theory, he defines $\text{rank}(x)$ by using the transitive closure of the set $x$. This definition, he claims, is important because it demonstrates that the rank function does not rely ...
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Why does an arbitrary plane contain three non-collinear points by the Hilbert's axoims?
I'm working through Hilbert's The Foundations of Geometry and I'm stuck on what seems to be a fundamental proof regarding the axioms of incidence.
From Axiom I.3
"There exist at least two points ...
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Is the syntax for sequent calculus axioms a pure convention or is there some justification for it?
I am currently studying sequent calculus from Mancosu et al. 2021, chapter 5. The syntax for axioms is given in definition 5.2:
$$ A \Rightarrow A $$
Is there some justification for this syntactic ...
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Does "Identity of Indiscernibles" replace "Axiom of Extensionality"? [duplicate]
The axiom of Extensionality states that "sets that contain exactly the same elements are equal, i.e. the same set".
My question is: if there would be no such axiom, would sets with exactly ...
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