Questions tagged [axioms]
For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.
1,312
questions
3
votes
1answer
50 views
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
$$\forall z\forall w_1 \forall ...
0
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0answers
26 views
Euclidean geometry axioms
What are the best axioms to have in account before start exploring Euclidean geometry? I know that there are Birkhoff and Hilbert axioms. There are more? Which is the best?
2
votes
1answer
69 views
Probability axioms does not make sense?
Assume a unit square to be sample space (infinite points inside it being its elements). Let the points are $\{p_1, p_2, ...\}$
then, by probability axioms,
$$1 = Pr(p_1 \cup p_2 \cup \cdots ) = Pr(\{...
1
vote
1answer
43 views
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:
$1\in\mathbb{N}$
$n\in\mathbb{N} \Rightarrow \overline{n}\in\mathbb{N}$
$\...
-4
votes
0answers
42 views
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 ...
2
votes
1answer
94 views
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.
...
0
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2answers
89 views
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, \...
0
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1answer
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 ...
0
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2answers
39 views
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$$\...
3
votes
1answer
90 views
Can a regular directed graph be axiomatized in first order Logic?
Given the first order edge relation $E$ is it possible to axiomatize directed graphs where every vertex has an equal amount of incoming and outgoind edges, including graphs with infinite edges?
0
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1answer
22 views
Show equivalence with BPI: every Boolean algebra has a prime ideal
I want to prove that the following statements are equivalent:
(1) Every non-trivial Boolean algebra has a prime ideal.
(2) In every non-trivial Boolean algebra, every ideal is contained in a prime ...
0
votes
1answer
100 views
Is the following provable in Zermelo–Fraenkel Set Theory?
Zermelo–Fraenkel Set Theory is a system of axioms for describing set theory.
For example, Zermelo–Fraenkel Set Theory says things like:
For any set $x$ and any set $y$ there exists a set $z$ such ...
0
votes
2answers
58 views
What axioms are required to imply the existence of all conceivable subsets?
In ZF, the Axiom of power set implies that for any set, there exists a set that contains all sets known to be subsets of such. However, as noted by the answer to this question:
It doesn't guarantee ...
0
votes
1answer
47 views
Are abstract definitions in math still subject to our understanding of concrete objects? To what extent?
It seems like in mathematics there are two separate meanings of the word axiom:
One is like the real or natural numbers, whose axiomatization is based off on concrete objects.
The other is like ...
1
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0answers
36 views
If a set theory allowed the existence of $x\cup y$, would that imply the existence of $\cup S$? [duplicate]
Suppose the set theory we are working with implies that $\forall x \forall y \exists(x \cup y)$.
Would that implication alone further imply the existence of an arbitrary union?:
$$\forall S\exists(\...
2
votes
0answers
57 views
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 &= \...
0
votes
1answer
136 views
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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0answers
36 views
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 ...
3
votes
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) ...
1
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0answers
77 views
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 ...
11
votes
1answer
267 views
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 ...
3
votes
1answer
84 views
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 ...
0
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3answers
102 views
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, ...
1
vote
2answers
74 views
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 ...
1
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0answers
55 views
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 ...
0
votes
2answers
61 views
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?
1
vote
1answer
49 views
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 ...
1
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0answers
66 views
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, $\...
1
vote
1answer
45 views
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?
4
votes
1answer
60 views
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 ...
2
votes
3answers
102 views
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/...
2
votes
1answer
82 views
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 ...
2
votes
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 [...
2
votes
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 ...
2
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0answers
31 views
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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vote
2answers
44 views
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 ...
1
vote
1answer
58 views
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 ...
0
votes
1answer
69 views
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 ...
-1
votes
3answers
108 views
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"....
1
vote
1answer
30 views
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 ...
0
votes
1answer
75 views
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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votes
3answers
134 views
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 ...
2
votes
0answers
60 views
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 \...
1
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1answer
37 views
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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votes
2answers
69 views
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 ...
3
votes
2answers
72 views
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 ...
1
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3answers
95 views
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 ...
0
votes
1answer
86 views
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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votes
1answer
39 views
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 $s$ ...
0
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1answer
81 views
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—...
