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

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8answers
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What is an axiom in layman's terms?

I tried to explain to my kids what is an axiom in very nontechnical terminology (also known as layman's term) but I could not find anything that they could relate to easily. Do anyone have a good ...
17
votes
4answers
4k views

Has there ever been an application of dividing by zero?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
0
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0answers
8 views

Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
-1
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1answer
23 views

Prove the following using only axioms of propositional logic and the deduction theorem? [see description] [closed]

$\vdash((\alpha\implies\beta)\implies(¬\beta\implies¬\alpha))$ Give a proof for the above theorem using only the three axioms of propositional logic (below), modus ponens, and the deduction theorem. ...
0
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0answers
13 views

How to derive the five-segment axiom of Tarski's geometry from Hilbert's axioms?

We are trying to prove that in an arbitrary Hilbert plane (assuming Hilbert's axioms of Group I:Incidence, II:Order and III:Congruence https://en.wikipedia.org/wiki/Hilbert%27s_axioms), Tarski's ...
4
votes
1answer
65 views

Necessity of being rigorous

Disclamer I am no serious mathematician, just curious Context I recently discovered some set theories, ZFC and IZF in particular. It made me realize that I've studied math a whole year without ...
2
votes
3answers
56 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
2
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3answers
31 views

Inner Product proof, axiom 1

I hate to argue that my text book is wrong. That being said, I am going to try and do just that. The text book says that this IS a valid inner product, I disagree. The vectors u and v are defined ...
1
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1answer
40 views

Prove axiom of infinity from ZF$-$Infinity +Zermelo's version of infinity

Assume ZF-infinity and assume that there is a set $X$ such that $\varnothing\in X$ and for all $t\in X$, $\{t\}\in X$. How to deduce axiom of infinity from these new axioms?
2
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1answer
34 views

Requirements for Diagonal Lemma

What are the axioms required for a formal system to be able to state the Diagonal Lemma or Fixed Point Theorem? If possible, could you please also relate with the following systems, if it applies to ...
0
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2answers
55 views

Choice function without AC in special case [duplicate]

I read the Jech, Set theory, and saw following proposition. (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice. I tried to prove ...
1
vote
2answers
81 views

Proof using only field axioms

Prove if $x, y ∈ R$ and $xy > 0$ then either $x > 0$ and $y > 0$, or, $x < 0$ and $y < 0$ using only the field axioms. These include the Field axioms for addition, multiplication, ...
1
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1answer
88 views

What book on Set Theory is best to understand motivation for axiomatization?

I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing ...
0
votes
1answer
25 views

Equivalence of Taylor series and its corresponding function and Axiom for infinite summation

Given a function $f(x)$ with a taylor series expansion, is it valid to say that $$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(a)(x-a)^n$$ for all values of x irrespective of whether the taylor series ...
0
votes
1answer
35 views

Real analysis using field axioms to prove

Prove that (-x)(-y)=xy using field axiom theorems My attempt i know that -(xy) is the additional inverse for xy, and i know xy + -(xy) = 0 (equation 1) i assumed that (-x)(-y)=xy and substitute ...
0
votes
1answer
43 views

Simple proof using only field axioms

I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both ...
0
votes
0answers
102 views

What kind of Properties are Allowed in the Schema of Comprehension?

Studying an introduction to set theory, we were presented with several axioms as the Axiom Schema of Comprehension and the Axiom of Infinity. This last aximom allows the definition of the inductive ...
38
votes
3answers
4k views

Can proof by contradiction 'fail'?

I am familiar with the mechanism of proof by contradiction: we want to prove $P$, so we assume $¬P$ and prove that this is false; hence $P$ must be true. I have the following devil's advocate ...
1
vote
3answers
91 views

In relatively simple words: What could be a model of $\sf {ZFC}$?

An $\text{interpretation}$ of a theory consists of A domain of discourse $\mathcal U$, usually required to be non-empty. For every constant symbol, an element of $\mathcal U$ as its ...
1
vote
2answers
42 views

Does defining a type of mathematical object require defining a binary relation of “equality”?

I'm trying to determine whether defining a type of mathematical object requires us to know what we mean by another object being "equal" to it. For example, when we define a type of object like set, ...
0
votes
1answer
43 views

What axiom in Math says “similar inputs should yield similar outputs”?

It is easy to take for granted the simple idea that similar input $x$ to a function $f(x)$ should yield similar outputs - such that if the difference between $x$ is arbitrary small, then we should ...
14
votes
8answers
1k views

Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the ...
0
votes
1answer
29 views

How to verify that $P_n(F)$ is a vector space over field $F$?

I assume $p(x)=a_nx^n+...+a_0$ where $a_i\in F$ and $x$ is variable. To verify it as vector space, I think we need to check axioms. i.e. $\forall x, y,z\in V,\ (x+y)+z=x+(y+z)$ and $\forall x, y\in ...
1
vote
0answers
62 views

Consequences of the Fact Every Uncountable Set of Reals Has Perfect Set Property

I have recently been working through the consequences assuming the Axiom of Determinacy has and came across the fact that this axiom implies that every uncountable set of reals has the perfect set ...
4
votes
4answers
122 views

Why is the infinite set from the axiom of infinity the natural numbers?

Why is the infinite set from the axiom of infinity the natural numbers? Is there any reason such set was chosen? Couldn't the axiom yield a set that looks like $\Bbb R$ for example?
1
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1answer
51 views

Axioms of noncommutative rings [duplicate]

Is there an abelian group $R$ with multiplication operator with this properties ? (i) $a(bc)=(ab)c$ (ii) $a(b+c)=ab+ac$ , $(b+c)a=ba+ca$ And a unique element $e$ s.t (iii) $ea=a\quad ...
5
votes
0answers
85 views

How can I learn Math intuitively? [closed]

I am currently a Junior in High School. I am in an Intermediate Algebra class, but my teacher does not always explain things in a way I can understand. I like to learn Math intuitively, but my teacher ...
0
votes
0answers
36 views

I'm stuck on a proof involving the of Axiom for multiplicative inverses and modular arithmetic.

I am trying to show that the axiom of multiplicative inverses holds on sets of integers modulo P when P is prime. i just need to show that for any non zero integer, n less than P there is a unique ...
0
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0answers
22 views

Can you characterize dimension in terms of the touching axioms?

One alternative axiomatization of topology is the touching axioms. It's basically the same as the closure axioms, and is in fact equivalent to the open set definition. Can we define dimension in a ...
4
votes
1answer
141 views

Two congruent segments does have the same length?

The answer to the question in the title seems an obvious ''Yes by definition !''. And this really is the definition from Wikipedia: Two line segments are congruent if they have the same length. ...
0
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0answers
38 views

Probability without second axiom (unit measure)

I'm working with functions (namely, representing incoherent degrees of belief) which resemble probabilities, but are actually, say, quasi-probabilites: their values on atomic events (here: atomic ...
0
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0answers
38 views

How do I prove the pairing axiom by the other ZFC axioms? [duplicate]

To prove the pairing axiom I thought about using the replacement axiom but I do not really know how. Can someone show me how it is done?
4
votes
2answers
55 views

Axioms of Trigonometry

On Wikipedia it gives a picture of all trigonometric functions of an angle laid atop the unit circle, 1. Obviously there are other trigonometric identities, but what I'm wondering is, does ...
1
vote
3answers
32 views

What's a model in axiomatic theory?

In the first answer here: Kolmogorov's probability axioms It is wondered whether there is only one model of the axioms (up to an isomorphism). Could somebody explain this concept? What's a ...
4
votes
1answer
88 views

A possible alternative to the Axioms of Pair, Union, Infinity and Replacement

In this question we assume that all formulae are in the language of $\sf ZFC$ and that $\sf ZFC$ is consistent. Recall that we say that a formula $\varphi(x,y)$ represents a set-like class relation ...
1
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1answer
21 views

To uniquely axiomatize a theory … meaning?

What does the following mean, exactly? When axiomatizing a theory, it is interesting to ask whether a set of axioms uniquely axiomatize the theory - $\textbf{that is, is it only this theory which ...
5
votes
2answers
116 views

Proving the Powerset Axiom for hereditarily finite sets

Consider $\mathsf{ZF}$, and relace the Axiom of Infinity with its negation. This gives us the theory of hereditarily finite sets. Its universe is $V_\omega$. Intuitively, I feel that I can construct ...
0
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1answer
35 views

Axioms of motion (Redei version)

I try to understand the axioms of motion in Redei's "Foundationd of Euclidean and Non-Euclidean Geometries, according to F. Klein" 1968 Redei gives as Axioms: Any motion is a one to one mapping of ...
5
votes
1answer
77 views

ordinal isomorphism theorem

The only proof of the theorem stating that every well-order is order-isomorphic to a unique ordinal of which I am aware relies on the Axiom of Replacement. Is this necessary?
1
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0answers
75 views

Incompleteness theorems in encoding schemes other than Gödel numbering

Gödel's proof of his incompleteness theorems makes use of Gödel numbering, which is a device that allows a theory of arithmetic $S$ (e.g. PRA) to express and reason about metamathematical statements ...
1
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1answer
90 views

Why do we call “comprehension” and “regularity” to the axiom schemas in Set Theory?

I have several Set Theory books in my "shelve" but I have found the justification for the names in none. Usually all of them just state something like: "The axiom schema of ...
0
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1answer
23 views

How to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$ based on the following assumption?

Let $W=$Span$\{1, x\}$, $W_1=$Span$\{x^2, x^3\}$ and $W_2=$Span$\{1+x+x^2+x^3, 1+x+x^2-x^3\}$, how to show $P_3(R)=W\oplus W_1$ and $P_3(R)=W\oplus W_2$? $P_3(R)=W+ W_1$ because Span$\{1, ...
1
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1answer
41 views

If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
0
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1answer
26 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
1
vote
1answer
19 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
2
votes
2answers
122 views

Different axiomatizations of set equality

I've seen two definitions (or axioms?) of set equality: $a=b \Leftrightarrow (\forall x : x \in a \Leftrightarrow x \in b)$ $a=b \Leftrightarrow (\forall x : a \in x \Leftrightarrow b \in x)$ That ...
0
votes
1answer
69 views

Is it possible to create a software to find formal proofs?

Let's say I have a Hilbert style system, with a few axioms and rules of inference, and I want to find a proof for some formula $\varphi$, is it possible to create an algorithm that would find a proof ...
0
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0answers
17 views

Peano's 3rd axiom -explain

Peano's axioms for Natural numbers. 3rd axiom from Introduction to topology by Bert Mendelson - There is one and only one object in $\mathbb{N}$ denoted by ...
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2answers
48 views

'Multiplying' by 0 in a field, field axiom proofs

The question says: The solution set was posted and there are a few things I don't quite understand from it. For the first one, I'm not entirely sure what's happening. It appears to be using the ...
3
votes
1answer
56 views

How to show that the axiom for vector space hold for the following operation?

So the operation is sum defined by $f+'g=f\circ g$ (composite of functions) and usual scalar multiplication. First, for $(x+y)+z=x+(y+z)$ property, $(f+'g)+'h=(f \circ g)\circ h\ne f \circ (g\circ ...