# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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. ...