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

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11
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3answers
474 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
161 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
129 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
167 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
148 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
124 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
823 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
150 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
161 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
362 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
250 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
129 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
722 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
164 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
111 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
484 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
109 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 ...
0
votes
0answers
277 views

How are multiplication and addition defined?

Many people argue that multiplication is not necessarily repeated addition, indeed in the field of order 4 given on Wikipedia, adding one to itself A times will never yield A*1. See: ...
2
votes
2answers
201 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
163 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
122 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
55 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
65 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
176 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
137 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
928 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
84 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
100 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
85 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
381 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 ...
10
votes
6answers
307 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
244 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 ...
0
votes
4answers
93 views

Replacing Axiom of Extensionality with a logical formalism

Is it possible to replace the Axiom of Extensionality with a formalism from logic, namely the following one: $\forall a \forall b (a=b\Leftrightarrow \forall P (P (a)\Leftrightarrow P (b)))$ ($ P $ is ...
1
vote
1answer
93 views

Apparent equivalent notations for the axiom of infinity

I'm just begining to build the systems of numbers based on the axioms of set theory ($\mathsf{ZF}$). Accordingly the axiom of infinity is no more than assuming the existence of $\mathbb{N}$ (of course ...
3
votes
3answers
145 views

If we abandon the axiom of regularity, can the cumulative hierarchy just become a definition?

In ZFC, the axiom of regularity is used to prove that every set is an element of some stage of the cumulative hierarchy. The index of the least such stage is, by definition, the rank of that set. Now ...
2
votes
1answer
64 views

Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
1
vote
3answers
110 views

Is there a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?

Is it possible to create a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?
3
votes
2answers
364 views

Why is matrix multiplication defined a certain way? [duplicate]

Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined ...
7
votes
0answers
172 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
11
votes
2answers
576 views

Which axioms of ZFC or PA are known to not be derivable from the others?

Which, if any, axioms of ZFC are known to not be derivable from the other axioms? Which, if any, axioms of PA are known to not be derivable from the other axioms?
6
votes
1answer
145 views

An Exercise in Kunen (A Model for Foundation, Pairing,…)

This is exercise I.4.18 in Kunen's Set Theory. Derive $\forall y (y \notin y)$ from the Axioms of Comprehension and Foundation. Don't use the Pairing or Extensionality Axioms. Then find a 2 element ...
2
votes
1answer
337 views

What's the differences between naive and axiomatic set theory?

I'm at a loss studying math.Recently I decided to begin with set theory as it seems the most fundamental for math.I found the book Naive Set Theory by Halmos,and began to read it because it's so thin ...
2
votes
1answer
242 views

Axioms and deduction rules of Propositional Calculus

I'm looking for a list of axioms and fundamental deduction rules of Propositional Calculus.
26
votes
6answers
1k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
7
votes
3answers
250 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y ...
2
votes
1answer
148 views

Pythagorean theorem proof by dissection

I have this proof of the Pythagorean theorem, but in the last two lines of the fourth paragraph I can't seem to find geometrically how the congruency between $1$, $2$, $3$, $4$ and $1'$, $2'$, $3'$, ...
4
votes
1answer
167 views

Understanding the Definition of the Axiom Schema of Specification

Consider the Axiom Schema of Separation: If $P$ is a property (with paramter $p$), then for any $X$ and $p$ there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those $u \in X$ that ...
2
votes
2answers
58 views

Question Regarding the Replacement Schema

For each formula $\phi(x,y,p)$, we have the following axiom: $\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z) \rightarrow \forall X \exists Y \forall y (y \in Y ...