Questions tagged [axioms]

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

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Math theories that allow for limits but not for infinitely big numbers

Are there maths axiomatic theories that accept finite limits (with possibly an infinite number of digits as long as we can define them by an infinite number of operations like say a mechanical ...
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50 views

Modus ponens rule

Prove for the calculus of propositions: $$ A\rightarrow B\mapsto(B\rightarrow C)\rightarrow(A\rightarrow C) $$ I had used axioms and most suitable was this one: $$ X\rightarrow Y\rightarrow(X\...
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Proof of Well-Ordering, without Induction, by the Least Upper Bound Axiom

On, Alan F. Beardon's, 'Limits: A New Approach to Real Analysis' it's asked: Show (i) by induction, and (ii) by using the Least Upper Bound Axiom, that any set of n real numbers contains a largest ...
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Is it really important to do axiomatic study of real numbers before learning Calculus? [closed]

I am currently beginning with Calculus Volume 1 by Tom M. Apostol . It has an introduction chapter divided into 4 parts namely Historical introduction Basic concept of set theory A set of axioms ...
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Non-contradictory axiom system for a binary operation

Suppose we want to define a binary operation $\otimes:\mathbb{N} \times\mathbb{N} \rightarrow \mathbb{N}$ on a ring $(\mathbb{N},+,\cdot)$ with an arbitrary system of axioms. The axioms may be given ...
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About ZFC and Partition perfect set into perfect sets

I was asking this question Another way for partition of perfect set two days ago and there was a nice discussion over there but now I need to change the question by adding more conditions. Question: ...
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Why finding an undecidable proposition doesn't imply consistency, if anything is provable in an inconsistent axiomatic system? [duplicate]

I'm reading these independent notes on the Lectures on the Geometric Anatomy of Theoretical Physics, by Dr. Frederic P. Schuller. The first chapter is a brief summary of axiomatic systems and set ...
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146 views

Another way for partition of perfect set

Let $P$ be a perfect $P\subset\Bbb R.$ Then there exists a family $\{P_{\alpha}\subset P\colon \alpha<\mathfrak c\}$ of pairwise disjoint perfect subsets such that $$P=\bigcup_{\alpha<\mathfrak ...
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Is there a prove that every unprovable proof, provable unprovable? [duplicate]

I learned today that there are some true statements that are unprovable (gödel incompleteness theorem), which is a liddle sad. But not too sad if we could at least proof that: For every unprovable ...
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56 views

How to more formally express the axioms of hyperbolic geometry?

I have copied word-for-word the axioms (minus the last one which is too long) for hyperbolic geometry as written in Cellular Automata in Hyperbolic Spaces, Volume I: Theory. Axioms of Incidence Two ...
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57 views

Is ZF equivalent to every subhierarchy coming from below being a set, over Zermelo + Ranks?

Define hierarchy as a family of sets well ordered by inclusion such that each successor is the powerset of its immediate predecessor, and the limit sets are the unions of all their predecessors. A ...
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What does the parameter $p$ refer to in this context?

I'm reading about the axioms of set theory (from Jech), and I'm having some confusion. There are various parts where it talks about well-formed formulas that include a parameter $p$. This appears here:...
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61 views

Gödel's proof: What if all axioms of a formal system are Gödel sentences

By proof, we know that Gödel's first Theorem applies to certain formal/axiomatic system, while the unprovable statement to which Gödel refers, the so-called "Gödel Sentence", is designed to ...
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21 views

What's the effect of changing the ordinal index of ranks to Scott ordinals, on the consistency strength of Zermelo + Ranks + Ordinals?

This question is about indexing the stages of the cumulative hierarchy in Zermelo set theory $``\sf Z"$ Now if we add to $\sf Z$ the following two axioms: Ranks: $\forall x \exists \alpha : x \in V_\...
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Can those definitions capture the cumulative hierarchy?

Working in first order logic with identity and membership: Definitions: $ \begin {align} hierarchy(x) \iff \forall y \in x : y=\{z \subseteq k \mid k \in x \, ( k \subsetneq y)\} \end {align}$ $level(...
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83 views

Does Gödel's incompleteness theorems imply there are infinitely many axioms in mathematics?

I'm currently reading up on Gödel's incompleteness theorems for a thesis paper but I have trouble grasping the concept and significance of such theorems as it seems to be very abstract and non-...
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complete axiomatization of reals vs we can't prove every statement about the reals vs every statement about the real numbers is either true or false

To clarify what this question's about: First I state the title and some stuff I believe, then something which contradicts that and my personal feelings about this contradiction. Then I pose my ...
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Is this replacement sentence consistent with Z? If so what's its consistency strength over Z + Ranks?

This question comes in continuation with this. Is the following replacement like statement disprovable in $\sf ZFC$? And if not, then what's the consistency strength of adding it to $\sf Z + \forall x ...
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Name of these lemmas in set theory

Lemma 1.2 If $S$ is countable and $S'\subset S$, then $S'$ is also countable Lemma 1.3 If $S'\subset S$ and $S'$ is uncountable, then so is $S$. I was wondering if there was a name for the logic/...
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What is the proof of Replacement in iterating functions over the empty set?

Define (ordinal): $\begin {align} ordinal(x) \iff & trs(x) \land \forall y \in x (trs(y)) \,\land \\& x \, \text {is} \in \text{-well-founded} \end{align}$ Where $trs(x)$ means $x$ is ...
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How to draw Axiom of Continuity : $\exists c \in\mathbb{R} :\forall a \in A, \forall b \in B \implies a \leq c \leq b$

In Real Analysis, while we are constructing the Real Numbers Axiomatically, we (in some books) define one important Axiom, Axiom of Continuity, which goes like this : "If $A, B\subseteq\mathbb{R}$...
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What does it really mean to draw a conclusion from a set of axioms? [closed]

I've started the part of my math journey that involves writing proofs, and while I think I have a good sense of when one statement implies another or not, I don't quite understand what the basis of my ...
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Examining (semi-) random structures?

Typically, much of mathematics today is ultimately based on pretty abstract axiomatisations in first order logic (or, rarely, some other logic). But all of those axiomatisations, whether for ...
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55 views

Is parameter free definability consistent with ZF?

Is it consistent to add the following axiom to $\sf ZF$? Definability: $$\forall X \exists \alpha \exists \phi: X=\{y \in V_\alpha \mid V_\alpha \models \phi(y)\}$$ Where $\phi$ is a formula in one ...
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undecidable problems in Euclidean geometry

What is (are) the most "elementary" question(s) one could ask in Euclidean geometry with all its postulates/axioms including the "5th" that is (are) known to be undecidable. The &...
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62 views

Is the existence of arbitrary definable partial functions bounded by a set equivalent to the axiom of choice?

Is the axiom of choice (125) equivalent to the following axiom (120)? If we take (120) and the companion axioms (180) and (190), can we demote specification (140) and replacement (160) to theorems? ...
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Can we have V=HPD? And is it equivalent to V=HOD?

I've lately came to know that the axiom $\sf V=HOD$ can be stated as: $$\forall X \, \exists \theta \, \exists \alpha < \theta \, \exists \varphi : X=\{y \in V_\theta\mid V_\theta\models\varphi(y,...
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Why do axiomatic systems for propositional calculus include IFF axioms?

I am reading : https://en.wikipedia.org/wiki/Propositional_calculus#Axioms, and the following three axioms seemed unnecessary to me: $$ IFF-1 : ( \phi \iff \chi ) \implies (\phi \implies \chi ) \\ ...
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Reference request: is axiom of choice motivated along type-set lines?

Language: bi-sorted first order logic with equality and its axiom and additionally the extra-logical primitives: $ ``\tau, < , \in"$, the first is a total unary function on sets denoting is the ...
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Are there any cumulative axiomatizations of the numbers?

By "cumulative axiomatization" I mean an axiomatization of the numbers where each set in the hierarchy of number types is explicitly a subset of the previous set. That is, $\mathbb{N}\subset\...
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Can full replacement follow from well ordered replacement over rest of axioms of ZF?

Would axiom schema of Replacement follow from well-ordered Replacement? What I mean is that if we restrict Replacement to only well ordered sets, then can that still prove full Replacement? Well ...
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About the axioms of a metric space. I think we don't need (D2) to show (D1).

I am reading "Topological Spaces: Examples and Exercises" by Tetsuro Kawasaki (in Japanese). There is the following exercise in this book: Definition: Let $X$ be a set. Let $d$ be a ...
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Does the following principles capture the modern standard line of set theory?

I think that the following axioms describes what modern set theory is all about (on top of mono-sorted first order logic with equality and membership) Extensionality: Two sets with the same elements ...
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Does this set have cardinality $\beth_\omega$? And if so, how does it require the Axiom of Replacement to construct?

I was reading this Wikipedia article: https://en.wikipedia.org/wiki/Von_Neumann_universe, and it mentions that the Axiom of Replacement is required to go outside of $V_{\omega+\omega}$, one of the ...
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Can an axiomatic system have uncountably many axioms?

The original question I had was vaguely "Does consistency of ZFC decide Continuum Hypothesis?", but I'm not even sure whether it is a valid question. So let me ask this question in prior to ...
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Trying to Understand the Axiom of Induction in the context of formalizing Natural Numbers

I have a question about formalizing the Natural Numbers, and the role of Peano's fifth axiom within his scheme. Let me describe what is my current understanding, and then get to some of my confusions ....
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Can a line in a projective plane have just two points?

Here is the definition of a projective plane from Stillwell's The Four Pillars of Geometry (2005, Springer): Let $\cal P$ ("points") be a set, and let $\cal L$ ("lines") be a set ...
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Can a line in a projective plane have only one point?

Here is the definition of a projective plane from Stillwell's The Four Pillars of Geometry (2005): Let $\cal P$ ("points") be a set, and let $\cal L$ ("lines") be a set of subsets ...
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Why is the Fixed-Point Axiom true?

In Epistemic Logic, the Fixed-Point Axiom says that $\varphi$ is common knowledge among the group $G$ if and only if all the members of $G$ know that $\varphi$ is true and is common knowledge: $$\...
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Why W is not a subspace of $\mathbb{R}^2$? [closed]

The set $W = \{(x,y)\in\mathbb{R}^2\mid y = x \vee y = -x\}$ is not a subspace of $V=\mathbb{R}^2$. I know $0\in W$, but I'm not sure if the others conditions are satisfied.
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Axiomatic system and Proof for axioms

So I am told by a friend that "axioms in an axiomatic system cannot be proved within the axiomatic system". I was wondering how true this is. Is there any actual mathematical theorem that ...
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Good rules / axioms for logic to be encoded into a computer program?

I'm kind of new to this whole thing so sorry if my question isn't great. My goal is to be able to encode a set of axioms into my program so that it can then infer statements from other statements. I ...
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experiment has N outcomes e1,e2,,eN. outcome e(j+1) is twice as the outcome ej for j=1,2,,N-1.Let Ek={E1,E2,…,Ek }.prove P(Ek )=(2^k-1)/(2^N-1)

PROBABILITY AND STATISICS university level. THIS IS AN ASSIGNMENT ON THE PROBABILITY OF COMPOUND EVENTS Prove that the theory is true showing all steps and examples. i don't understand how to go about ...
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Are these axioms of real number strict?

After comparing with some other textbooks about introductory real analysis, I find that many books' content about axioms of real numbers are not strict (at least for me, I think they are not strict). ...
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81 views

Proving $\{a\}$ is a set

I recently came across the following post Proving $\{a\}$ is a also set given that $a$ is a set. Introduction to Set Theory.. While I understand the answer given I am still unsure whether or not the ...
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Is it legal to define a function that gives different results for 1.0 and for 1?

In programming languages I can define such function, because in most programming languages 1.0 is not 1, because 1.0 has type "float", and 1 has type "integer". In math I don't see ...
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In a formal axiomatic system with both the law of the excluded middle and the axiom of infinity does an uncountable set have to exist?

This was something I have been thinking around when examining writings of some “philosophers” who seemingly argue against the existence of a continuation but at the same time accept the law of the ...
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Intersection of sets is indeed a set

My question is pretty straight forward - how do I use ZF axioms to prove that if $\{A\}_{i\in\Bbb{N}}$ is a family of sets, then $\bigcap_{i}A_i$ is a set? I feel like I should be using the axiom of ...
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158 views

Axioms that are not evident

How can we give a meaning to the three axioms of prepositional logic? As axioms, is not it supposed to be obvious? For example, how is $(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \...
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How to use axioms for a proof in axiomatic set theory?

I am learning set theory and find it confusing sometimes to understand how to use an axiom in order to solve an exercise. My main problem is beginning the proof and understanding how to proceed. ...

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