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

learn more… | top users | synonyms

3
votes
3answers
495 views

Using field axioms for a simple proof

Question: If $F$ is a field, and $a, b, c \in F$, then prove that if $a+b = a+c$, then $b=c$ by using the axioms for a field. Relevant information: Field Axioms (for $a, b, c \in F$): ...
0
votes
1answer
101 views

Axiom of Completeness for set of integers

If $A$ is a subset of the integers $\mathbb{Z}$, and is bounded above, then A has a supremum $\alpha$ that is an element of the integers $\mathbb{Z}$. Is this statement true?
1
vote
1answer
81 views

Doubts on the axiom of union?

I'm reading Guerino Mazzolla's Comprehensive Mathematics for Computer Scientists 1: Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set $$\{x| \text{ there exists an element } b \in ...
10
votes
2answers
218 views

When the mathematical community consider the inclusion of a new axiom?.

At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It ...
1
vote
2answers
52 views

Showing a class is a set.

The problem is to determine that a given class is or is not a set only using the basic axioms of modern set theory and some examples of classes that are or are not sets. The class in question is ...
3
votes
2answers
457 views

A set is not an element of itself.

I know that in modern set theory that for a given set $A$, $A \notin A$, specifically by the axiom of regularity. However, I'm not permitted to use this axiom in my proof. What I am permitted to use ...
4
votes
2answers
64 views

Is this a helpful way of thinking about modular arithmetic?

Could one consider the structure of $\mathbb{Z_{n}}$ under addition and multiplication to simply be that of the ordinary integers, with one additional axiom - namely that $n=0$? So we would have ...
7
votes
3answers
257 views

Why is zero important?

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators. On the computer science site I came across the following ...
6
votes
3answers
215 views

Axiom schema of specification (formula arguments)

Some sources define the formula like this: $$ \forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \varphi(x, w_1, \ldots, w_n , A) ] ) $$ Why ...
1
vote
3answers
62 views

One question about additive identity arising from Apostol's field axiom in the Mathematical Analysis 2nd edition

In the begining of this book, the field axiom 4 talks about "Given any two real numbers x and y, there exists a real number z such that x+z=y and this z is denoted by y-x." Therefore, for each x, we ...
1
vote
1answer
62 views

Multiple context available for the AC?

I am surprised at the language used in connection with the axiom of choice. From the answer to a question a made (which turned out to be duplicate) about involvement of AC in Wiles’ proof of Fermat’s ...
2
votes
2answers
90 views

Proving order in real number set

Can we prove such a statement, or is it axiomatic ? $$ \forall (x,y,z) \in \mathbb{R}^3 ∶(x \leq y)∧(y \leq z)⇒(x \leq z) $$
2
votes
2answers
182 views

Automatic theorem prover for proving simple theorems?

Is there a simple software that I could use to practice proving theorems in my course of mathematical logic? Basically what I need is ability to 1) define what axioms and laws I am allowed to use in ...
1
vote
1answer
75 views

If AC is false , is this statement about the halting problem true?

Assume AC is false. (AC = axiom of choice ) Let $n,m$ be positive integers. Let $f: \Bbb N \rightarrow \Bbb N$ and $f(m)=m$. Let $g(n,m)=1$ if the iterations $f(n),f(f(n)),...$ converges to $m$. ...
3
votes
2answers
249 views

axioms of equality

Every text I've come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other's elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in ...
1
vote
1answer
159 views

Book covering introduction to mathematical proofs

I am looking for some introductory books covering mathematical proofs, axioms, propositions, proof techniques etc in general.
4
votes
2answers
121 views

Is it necessary to use the axiom of Regularity to prove the successor function being injective?

Basically the problem is that given an inductive set $X$ we can define the successor function on $X$ such that $S:X\longrightarrow X$ and for all $x\in X$, $S(x)=x\cup \{x\}$. So, one of Peano axioms ...
11
votes
3answers
546 views

Proof that every circle has the ratio of $\pi$

We know that the ratio between the circumference's length and diameter is equal to $\pi$, but can this be proved for every circle? Or is this an axiom?
1
vote
1answer
168 views

Explain mathematical practice and axiomatization to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
2
votes
2answers
143 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
1
vote
2answers
201 views

Example of Non-axiomatic theory?

Im fairly sure that non-axiomatic theories exist (in a mathematical logic sense, aka not considering Creationism and all that...), but I cant think of any examples. Im wondering if I have a theory ...
0
votes
1answer
164 views

Problems with axioms and their potential uses in real life.

Okay, so I am looking for more examples that revolve around this topic. My topic is: "Is the assumption of basic truth needed in order to create a system of mathematics that is reliable?" Here, when ...
2
votes
1answer
130 views

Problems with axioms

I am currently exploring the idea of axiomatic truths. As of now, I have looked into axioms dealing with euclidean geometry and they are said to be self-evident truths. Each axiom in euclidean ...
5
votes
2answers
1k views

Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three ...
3
votes
1answer
154 views

Axioms based on $\leftrightarrow, \lor, \bot$ for propositional intuitionistic logic?

Propositional intuitionistic logic can be axiomatized based on $\;\to, \land, \lor, \bot\;$, with modus ponens $$ \text{from }\; \phi \;\text{ and }\; \phi \to \psi \;\text{ infer }\; \psi $$ as the ...
3
votes
0answers
173 views

Power-set in Hypercube: historical background of indexing each term like Hasse Diagram?

My instructor wants references about the indexation over the hypercube, related question here, he does not believe that I was the first who used it -- [update] thanks to a comment, the name is Hasse ...
4
votes
1answer
387 views

Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
15
votes
2answers
260 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
6
votes
1answer
139 views

Axiomatic definition of complex numbers

Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb ...
1
vote
2answers
774 views

Prove if $ac>bc$ and $c>0$, then $a>b$

How can I prove that if $ac>bc$ and $c>0$, then $a>b$ without division? Or Should I prove it by contrapositive?
4
votes
1answer
181 views

Describe a sound and complete proof system

I have a homework assignment that I am a little stumped on, the questions is: Describe a sound and complete proof system (axioms and proof rules) for proposition logic. Explain in detail why you ...
3
votes
2answers
113 views

Does the language of group theory need a constant?

I would like to verify the understanding of my claim. In Model Theory: An Introduction by David Marker Example 1.2.5 defines the language of the theory of groups as follows to be $\mathcal{L}=\{., ...
13
votes
1answer
507 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
3
votes
1answer
111 views

Deriving the Axiom of Infinity

I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ...
2
votes
2answers
217 views

Can multiplication be defined without addition?

I'm struggling to understand how to define multiplication and addition, now that I've been told that multiplication is not just repeated addition. It seems that the axioms for the two are ...
5
votes
1answer
169 views

What if we change axiom of arithmetic?

I want to experiment with some ideas about changing the axioms of arithemetic. (You know, like changing axioms of Euclidian geometry, we get non-Euclidian geometry.) What happens when we change the ...
3
votes
2answers
125 views

Help with a proof by induction.

I'm reading On Mathematical Induction by Leon Henkin in JSOTR. And well, in the first part of the article the author given us the necessary properties to be a Peano Model. A model $\langle ...
2
votes
1answer
56 views

Are forcing techniques possible to automate for mechanized reasoning?

Looking at Cohen's success at proving independence of the Axiom of Choice and the Continuum Hypothesis, I was wondering if it was possible to mechanize forcing techniques for the purpose of proving ...
5
votes
1answer
67 views

Weak classical Deontic Logics

I am writing a paper at the moment and an area of Deontic Logic has cropped up in it. I know very little about the area and I was wondering if people could give me opinions on the axiomatic system ...
3
votes
2answers
116 views

What is meant by one axiom being “weaker” than another?

If we have two axioms $A$ and $B$, what exactly is meant by axiom $A$ being weaker than axiom $B$? This question is a follow-up to A weaker Axiom of Infinity?
5
votes
1answer
180 views

A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at ...
0
votes
2answers
140 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
2
votes
5answers
1k views

Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?

Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type ...
5
votes
5answers
147 views

Help understanding $x=y\Rightarrow(x=z\Rightarrow y=z)$

I was reading a proof that opened with the integer axiom of $x=y\Rightarrow(x=z\Rightarrow y=z)$ What would be an accurate statement in English to express this? The "implies" within the first ...
2
votes
2answers
87 views

What is the absolute minimum that must be accepted/defined in order to prove 1+1=2?

As I know the "1 + 1 = 2" question has been asked before, my question (for now anyway) is more specific: Assuming as little as possible and accepting as few definitions as possible, what are the very ...
2
votes
1answer
105 views

weaker version of some axioms

I'm complementing my knowledge of set theory using other sources in addition to the book that normally I use. Working with the Jech's book (great book, by the way), in one exercise of the first ...
0
votes
1answer
86 views

Proving statements for individual and/or all sets of axioms

If something is proved for one set of axioms then you can use it for that set of axioms but wouldn't you have to prove it for another set of axioms before you could use if for the second set of ...
6
votes
4answers
429 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
11
votes
6answers
320 views

Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if ...
2
votes
4answers
246 views

Definition of $0$?

In real number axioms, it is defined that there is $0$ such that $x+0=x$ for all $x.$ I was wondering an example is there any other algebraic structure than real numbers which satisfy the real number ...