For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.

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1answer
34 views

Is the $\varphi \to \varphi$ axiom in Hilbert calculus redundant?

When I see the Hilbert calculus in logic, I sometimes notice that $T \vdash \varphi \to \varphi$ is listed as an axiom and sometimes not. Is there some reason? Could I get it somehow from the other ...
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0answers
31 views
+50

Is there an established method for assigning a truth value $0<t<1$ to undecidable statements?

Is there an established method for assigning a truth value $0<t<1$ to undecidable statements? Classical logic assumes every proposition is either true or false, i.e., the law of excluded ...
0
votes
1answer
69 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
1
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4answers
25 views

Deriving probability of exactly one event occuring

Show that for any events A and B, the probability that exactly one of them occur is Pr(A) + Pr(B) − 2 Pr(A ∩ B). Solution: The probability that exactly one event occurs is $$ Pr ((A \cap B^c) ...
6
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0answers
130 views

What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
1
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1answer
17 views

Is it an axiom that the inequalities are translation-invariant or can we prove it?

I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.: $x < y \implies x + c < y + c ...
3
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0answers
78 views

Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
4
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1answer
105 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
8
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3answers
244 views

Is the “domain of discourse” in axiomatic set theory also a “set”?

The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of ...
3
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1answer
104 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
2
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3answers
65 views

Name of an axiom

What's the name of the following axiom: Let $A$ and $B$ be sets of real numbers, and $(\forall a\in A\text{ and }\forall b\in B)a \leq b$. Then $\exists c\in\mathbb R$ such that $(\forall a\in A\text{ ...
2
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1answer
117 views

Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?

A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), ...
3
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1answer
37 views

Prerequisites for Hartshorne: Euclid and beyond?

as the title suggests, I am looking for the prerequisites to Hartshorne's Euclid and beyond. I just found this book and I think it's wonderful, but the downside is that I only know math up to single ...
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0answers
50 views

Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
1
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3answers
235 views

Can we prove that axioms do not contradict?

We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So: Is it ...
1
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2answers
42 views

Applying the Axiom Schema of Separation for the property $x = \{x\}$

On a past exam paper in a set theory module I am taking I am asked the question: Express as a first-order sentence in the language of set theory, the instance of the Axiom Schema of Separation for ...
0
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0answers
22 views

Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...
4
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1answer
75 views

Is the existence of such a transitive model $M$ of ZFC consistent?

Questions. Q0. Does anyone know of a refutation, in ZFC, of the following statement? Q1. If not, does ZFC plus large cardinals prove its consistency? Statement. There exists a transitive model $M$ ...
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0answers
56 views

Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
3
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2answers
46 views

origin of syntax for mathematical equations

Bear with me, I don't have any formal training in mathematics. I wonder if there is something that accounts for the syntax of mathematical equations, some deeper logic or reasons why I know that ...
1
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3answers
71 views

Are there in pure mathematics ensembles of number's which not divided by them self except $0$?

In pure mathematics we know well that each number divided by him self except $0$ , the question that let me confused is: Is there a proof in pure mathematics show to us that there are others ...
0
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0answers
23 views

Axioms for ordered fields

If i have a set of numbers, that contains every positive number, including 0 , except the negative ones, can i claim it as an ordered field? I would say no, because you can't find an $y$ for any $x$ ...
2
votes
1answer
54 views

Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
3
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3answers
83 views

Axiom of infinity and empty set

The axiom of infinity is formulated as $$\exists S ( \varnothing \in S \wedge (\forall x \in S) x \cup \{x\} \in S)$$ Can someone explain why the use of $\varnothing$ in the axiom of infinity makes ...
4
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1answer
65 views

Proving $V=V_{\sf On}$ in $\sf Z+Reg$

Define the function $$V_0=\emptyset,\qquad V_{\alpha+1}={\cal P}(V_\alpha),\qquad V_{\delta}=\bigcup_{\beta<\delta}V_\beta.$$ Since we are working in $\sf Z$ (i.e. $\sf ZF$ without the axiom of ...
0
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1answer
25 views

A question about a theorem derived from a given set of postulates

Currently, I am reading the book 'Godel's proof' by Ernest Nagel and James Newman, with the forward by Douglas Hofstadter. In that book, on page 15, the authors give an example of an axiomatic system ...
0
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1answer
17 views

Formal axiomatic system(s) which has/have--or, alternatively, hasn't/haven't--produced at least one equation or model that matches observation

Hopefully I'm not wrong to suspect that the various formal axiomatic systems, which mathematicians develop, have varying amounts of empirical support (not that I generally know such systems, except by ...
0
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1answer
44 views

By proposition 3.21 is ether acute, right or obtuse

Let A, B, C be points on a line with A ∗ B ∗ C. Let D be a point not on that line such that angle ABD is obtuse. Show that angle CBD is acute. Do NOT use the Measurement Theorem. (Hint: By ...
0
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0answers
37 views

Verifying certain congruence axioms in taxicab geometry

Given: I need some help I've shown what I have so far d(A, B) = |a1 − b1| + |a2 − b2| where A = (a1, a2) and B = (b1, b2). Some people call this ...
3
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0answers
143 views

Why exactly is Bourbaki difficult? [closed]

I keep hearing people say that Bourbaki is difficult for most undergraduates but I still don't understand why. Surely if it starts from definitions/axioms then practically anyone should be able to ...
0
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1answer
8 views

Do multiclass logistic regressions obey Kolmogorov's second axiom?

Logistic regressions were taught to me using the intuition that they approximate $\mathbb{P}(Y=y|x;\theta)$. Multiclass regressions use one-vs-all classification, selecting one $y$ and classifying ...
4
votes
1answer
61 views

What *can* Euclid prove?

It is well-known that Euclid's axioms for geometry are not up to modern standards of rigor: in particular, there are a lot of times when he used "obvious" facts about the geometric objects which were ...
0
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1answer
77 views

Is the parallelogram law a theorem or an axiom?

I'm learning about inner product spaces and I am able to prove it within an inner product space. Is this a theorem or an axiom in euclidean geometry?(note: not the geometry of Descartes)
3
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0answers
37 views

Proving Boolean Logic using Axioms

I am trying to prove boolean logic formulas using axioms. I have a lot of trouble deciding what to do, which axiom to use, when to use it, etc. I've asked others on proving formulas but they have all ...
0
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0answers
44 views

Alternative definition of a matroid?

$\mathcal M\,$ is a nonempty subset of the set of all subsets of the finite set $X$, such that: $(1)\quad\; \neg\exists U,V\in\mathcal M:U\subset V\quad$ $(2)\quad\; A\subseteq B\in\mathcal M\wedge ...
2
votes
3answers
270 views

Existence of numbers such as $\pi^{-1}$

For my non-mathematics students (this particular class are computing), I would define $\displaystyle \frac{1}{n}$ for $n\in\mathbb{N}$ as the solution of the equation $$nx=1,$$ and then ...
1
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1answer
43 views

Vector space and axioms

How would i go around proving a(u+v)+ u= (a+1)u+av using axioms? I started with distributivity with respect to vector addition, associativity of addition, commutativity of addition, distributivity ...
0
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3answers
36 views

Proving $x^2 < y^2$ by means of the Ordering Axioms [closed]

How do I prove $x^2 < y^2$, if $0 \le x < y$ with the ordering axioms? thanks!
1
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1answer
78 views

The fundamental axioms of mathematics

Having known about what axioms are, I want to know whether there are some "fundamental axioms of mathematics" on which every branch of mathematics depends. If yes, what are they ? Or Do we have ...
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1answer
147 views

What are all kind of “metamath” good for? Can it help me here? [closed]

Those logical theories, which deals with questions that isn't really mathematics but reach mathematics more or less, often seems to be like textbooks full of definitions, plus some theorems of the ...
1
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1answer
71 views

Some problems from section 4 of Munkres

I'm right now covering Section 4 of Topology by James R. Munkres, 2nd edition, and am stuck with the following problems in the exercise set after Section 4: Problem 8(c): Show that given $a$ with ...
1
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1answer
40 views

How do I prove this implication using axiomatic methods?

I am trying to prove that $a >0$ if and only if $\frac{1}{a} >0$, But I am having a lot of trouble doing so. So I would have to prove both sides of the implication. That is $a>0 \implies ...
0
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2answers
37 views

Proving that a set A cannot contain another set B which contains A?

From the ZFC axiom of regularity, which states that every non-empty set contains an element disjoint from it, we can deduce that there is no set $A$ such that $A \in A$. A proof is outlined here: ...
2
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3answers
101 views

Russell's paradox and axiom of separation

I don't quite understand how the axiom of separation resolves Russell's paradox in an entirely satisfactory way (without relying on other axioms). I see that it removes the immediate contradiction ...
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1answer
70 views

Axiomatic Set Theory: Why do we need the “Axiom of Union”?

I've been reviewing the axioms of ZFC, and I'm trying to make sense of why the "Axiom of Union" was put in place. While the existence of the intersection (of two) sets seems to be a "Theorem" we can ...
3
votes
1answer
48 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
2
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2answers
86 views

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
0
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1answer
39 views

Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
1
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1answer
37 views

Whitehead's axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
0
votes
1answer
34 views

Axiomatic proof and Boolean algebra?

I'm trying to prove that: $$(c'd') + (bc') + (a'b'c) + (ab'c) = (b' + c')(b + c + d')$$ using an axiomatic proof (i.e. using only the basic axioms and theorems of Boolean algebra).However, no matter ...