# Questions tagged [axioms]

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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?

 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?

 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 ...
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 $s$ ...